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   http://functions.wolfram.com/01.09.21.0110.01
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    Integrate[Cos[c z] Sqrt[a + b Cos[2 c z]] Cot[c z], z] == 
 (1/(4 Sqrt[b] Sqrt[a + b] c)) 
  (-4 Sqrt[b] (a + b) ArcTanh[(Sqrt[a + b] Cos[c z])/
      Sqrt[a + b Cos[2 c z]]] + Sqrt[a + b] 
    (2 Sqrt[b] Cos[c z] Sqrt[a + b Cos[2 c z]] + Sqrt[2] (a + 3 b) 
      Log[Sqrt[2] Sqrt[b] Cos[c z] + Sqrt[a + b Cos[2 c z]]])) 
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   Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]], " ", RowBox[List["Cot", "[", RowBox[List["c", " ", "z"]], "]"]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", RowBox[List["4", " ", SqrtBox["b"], " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", "c"]]], RowBox[List["(", RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["b"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["b"], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["3", " ", "b"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], ")"]]]]]]]] 
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   <math xmlns='http://www.w3.org/1998/Math/MathML' mathematica:form='TraditionalForm' xmlns:mathematica='http://www.wolfram.com/XML/'>  <semantics>  <mrow>  <mrow>  <mo> ∫ </mo>  <mrow>  <mrow>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mi> b </mi>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mi> cot </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mo> ⅆ </mo>  <mi> z </mi>  </mrow>  </mrow>  </mrow>  <mo> ⩵ </mo>  <mrow>  <mfrac>  <mn> 1 </mn>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <msqrt>  <mi> b </mi>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> b </mi>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mi> c </mi>  </mrow>  </mfrac>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> b </mi>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <msqrt>  <mi> b </mi>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mi> b </mi>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <mrow>  <msqrt>  <mn> 2 </mn>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mn> 3 </mn>  <mo> ⁢ </mo>  <mi> b </mi>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <mi> log </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mrow>  <msqrt>  <mn> 2 </mn>  </msqrt>  <mo> ⁢ </mo>  <msqrt>  <mi> b </mi>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> + </mo>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mi> b </mi>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </msqrt>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <mo> - </mo>  <mrow>  <mn> 4 </mn>  <mo> ⁢ </mo>  <msqrt>  <mi> b </mi>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mo> ( </mo>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> b </mi>  </mrow>  <mo> ) </mo>  </mrow>  <mo> ⁢ </mo>  <mrow>  <msup>  <mi> tanh </mi>  <mrow>  <mo> - </mo>  <mn> 1 </mn>  </mrow>  </msup>  <mo> ( </mo>  <mfrac>  <mrow>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mi> b </mi>  </mrow>  </msqrt>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  <msqrt>  <mrow>  <mi> a </mi>  <mo> + </mo>  <mrow>  <mi> b </mi>  <mo> ⁢ </mo>  <mrow>  <mi> cos </mi>  <mo> ⁡ </mo>  <mo> ( </mo>  <mrow>  <mn> 2 </mn>  <mo> ⁢ </mo>  <mi> c </mi>  <mo> ⁢ </mo>  <mi> z </mi>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  </msqrt>  </mfrac>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <mo> ) </mo>  </mrow>  </mrow>  </mrow>  <annotation-xml encoding='MathML-Content'>  <apply>  <eq />  <apply>  <int />  <bvar>  <ci> z </ci>  </bvar>  <apply>  <times />  <apply>  <cos />  <apply>  <times />  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <ci> b </ci>  <apply>  <cos />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <cot />  <apply>  <times />  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <times />  <cn type='integer'> 1 </cn>  <apply>  <power />  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <power />  <ci> b </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <ci> b </ci>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <ci> c </ci>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  <apply>  <plus />  <apply>  <times />  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <ci> b </ci>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <apply>  <times />  <cn type='integer'> 2 </cn>  <apply>  <power />  <ci> b </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <ci> b </ci>  <apply>  <cos />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <cos />  <apply>  <times />  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <times />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <cn type='integer'> 3 </cn>  <ci> b </ci>  </apply>  </apply>  <apply>  <ln />  <apply>  <plus />  <apply>  <times />  <apply>  <power />  <cn type='integer'> 2 </cn>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <power />  <ci> b </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <cos />  <apply>  <times />  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <ci> b </ci>  <apply>  <cos />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  <apply>  <times />  <cn type='integer'> -1 </cn>  <apply>  <times />  <cn type='integer'> 4 </cn>  <apply>  <power />  <ci> b </ci>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <plus />  <ci> a </ci>  <ci> b </ci>  </apply>  <apply>  <arctanh />  <apply>  <times />  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <ci> b </ci>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <apply>  <cos />  <apply>  <times />  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  <apply>  <power />  <apply>  <power />  <apply>  <plus />  <ci> a </ci>  <apply>  <times />  <ci> b </ci>  <apply>  <cos />  <apply>  <times />  <cn type='integer'> 2 </cn>  <ci> c </ci>  <ci> z </ci>  </apply>  </apply>  </apply>  </apply>  <cn type='rational'> 1 <sep /> 2 </cn>  </apply>  <cn type='integer'> -1 </cn>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </apply>  </annotation-xml>  </semantics>  </math> 
   
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  | Cell[BoxData[RowBox[List[RowBox[List["HoldPattern", "[", RowBox[List["\[Integral]", RowBox[List[RowBox[List[RowBox[List["Cos", "[", RowBox[List["c_", " ", "z_"]], "]"]], " ", SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c_", " ", "z_"]], "]"]]]]]]], " ", RowBox[List["Cot", "[", RowBox[List["c_", " ", "z_"]], "]"]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List[RowBox[List["-", "4"]], " ", SqrtBox["b"], " ", RowBox[List["(", RowBox[List["a", "+", "b"]], ")"]], " ", RowBox[List["ArcTanh", "[", FractionBox[RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]], "]"]]]], "+", RowBox[List[SqrtBox[RowBox[List["a", "+", "b"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", SqrtBox["b"], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "+", RowBox[List[SqrtBox["2"], " ", RowBox[List["(", RowBox[List["a", "+", RowBox[List["3", " ", "b"]]]], ")"]], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List[SqrtBox["2"], " ", SqrtBox["b"], " ", RowBox[List["Cos", "[", RowBox[List["c", " ", "z"]], "]"]]]], "+", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", RowBox[List["Cos", "[", RowBox[List["2", " ", "c", " ", "z"]], "]"]]]]]]]]], "]"]]]]]], ")"]]]]]], RowBox[List["4", " ", SqrtBox["b"], " ", SqrtBox[RowBox[List["a", "+", "b"]]], " ", "c"]]]]]]]  |  
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   Date Added to functions.wolfram.com (modification date)
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