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http://functions.wolfram.com/01.03.21.0521.01
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Integrate[Sqrt[a + b E^(e z)]/Sqrt[c + d E^(e z)], z] ==
(1/e)
(-((Sqrt[a] Log[(e (2 a c + b c E^(e z) + a d E^(e z) +
2 Sqrt[a] Sqrt[c] Sqrt[a + b E^(e z)] Sqrt[c + d E^(e z)]))/
(E^(e z) (a^(3/2) Sqrt[c]))])/Sqrt[c]) +
(Sqrt[b] Log[2 Sqrt[a + b E^(e z)] Sqrt[c + d E^(e z)] +
(a d + b (c + 2 d E^(e z)))/(Sqrt[b] Sqrt[d])])/Sqrt[d])
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Cell[BoxData[RowBox[List[RowBox[List["\[Integral]", RowBox[List[FractionBox[SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]], RowBox[List["\[DifferentialD]", "z"]]]]]], "\[Equal]", RowBox[List[FractionBox["1", "e"], RowBox[List["(", RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["e", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "c"]], "+", RowBox[List["b", " ", "c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["a", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox["c"], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SqrtBox["c"]]]], "]"]]]], SqrtBox["c"]]]], "+", FractionBox[RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["a", " ", "d"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], ")"]]]]]], RowBox[List[SqrtBox["b"], " ", SqrtBox["d"]]]]]], "]"]]]], SqrtBox["d"]]]], ")"]]]]]]]]
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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> <mfrac> <msqrt> <mrow> <mi> a </mi> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </msqrt> <msqrt> <mrow> <mi> c </mi> <mo> + </mo> <mrow> <mi> d </mi> <mo> ⁢ </mo> <msup> <mi> ⅇ </mi> <mrow> <mi> e </mi> <mo> ⁢ </mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> </msqrt> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ⅆ </mo> <mi> z </mi> </mrow> </mrow> </mrow> <mo> ⩵ </mo> <mrow> <mfrac> <mn> 1 </mn> <mi> e </mi> </mfrac> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mfrac> <mrow> <msqrt> <mi> b </mi> </msqrt> <mo> ⁢ </mo> <mrow> <mi> log </mi> <mo> ⁡ </mo> <mo> ( </mo> <mrow> <mfrac> <mrow> <mrow> <mi> a </mi> <mo> ⁢ </mo> <mi> d </mi> </mrow> <mo> + </mo> <mrow> <mi> b </mi> <mo> ⁢ </mo> <mrow> <mo> ( </mo> <mrow> <mi> c </mi> 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</mo> <mi> z </mi> </mrow> </msup> </mrow> </mrow> <mo> ) </mo> </mrow> </mrow> <mrow> <msup> <mi> a </mi> <mrow> <mn> 3 </mn> <mo> / </mo> <mn> 2 </mn> </mrow> </msup> <mo> ⁢ </mo> <msqrt> <mi> c </mi> </msqrt> </mrow> </mfrac> <mo> ) </mo> </mrow> </mrow> <msqrt> <mi> c </mi> </msqrt> </mfrac> </mrow> <mo> ) </mo> </mrow> </mrow> </mrow> <annotation-xml encoding='MathML-Content'> <apply> <eq /> <apply> <int /> <bvar> <ci> z </ci> </bvar> <apply> <times /> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> 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<apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> d </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </apply> </apply> <apply> <times /> <cn type='integer'> -1 </cn> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <ln /> <apply> <times /> <ci> e </ci> <apply> <power /> <exponentiale /> <apply> <times /> <apply> <times /> <cn type='integer'> -1 </cn> <ci> e </ci> </apply> <ci> z </ci> </apply> </apply> <apply> <plus /> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <ci> a </ci> <ci> c </ci> </apply> <apply> <times /> <cn type='integer'> 2 </cn> <apply> <power /> <ci> a </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> a </ci> <apply> <times /> <ci> b </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <apply> <plus /> <ci> c </ci> <apply> <times /> <ci> d </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <apply> <times /> <ci> a </ci> <ci> d </ci> <apply> <power /> <exponentiale /> <apply> <times /> <ci> e </ci> <ci> z </ci> </apply> </apply> </apply> </apply> <apply> <power /> <apply> <times /> <apply> <power /> <ci> a </ci> <cn type='rational'> 3 <sep /> 2 </cn> </apply> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> </apply> <cn type='integer'> -1 </cn> </apply> </apply> </apply> <apply> <power /> <apply> <power /> <ci> c </ci> <cn type='rational'> 1 <sep /> 2 </cn> </apply> <cn type='integer'> -1 </cn> </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[FractionBox[SqrtBox[RowBox[List["a_", "+", RowBox[List["b_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e_", " ", "z_"]]]]]]]], SqrtBox[RowBox[List["c_", "+", RowBox[List["d_", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e_", " ", "z_"]]]]]]]]], RowBox[List["\[DifferentialD]", "z_"]]]]]], "]"]], "\[RuleDelayed]", FractionBox[RowBox[List[RowBox[List["-", FractionBox[RowBox[List[SqrtBox["a"], " ", RowBox[List["Log", "[", FractionBox[RowBox[List["e", " ", SuperscriptBox["\[ExponentialE]", RowBox[List[RowBox[List["-", "e"]], " ", "z"]]], " ", RowBox[List["(", RowBox[List[RowBox[List["2", " ", "a", " ", "c"]], "+", RowBox[List["b", " ", "c", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["a", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]], "+", RowBox[List["2", " ", SqrtBox["a"], " ", SqrtBox["c"], " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]]]], ")"]]]], RowBox[List[SuperscriptBox["a", RowBox[List["3", "/", "2"]]], " ", SqrtBox["c"]]]], "]"]]]], SqrtBox["c"]]]], "+", FractionBox[RowBox[List[SqrtBox["b"], " ", RowBox[List["Log", "[", RowBox[List[RowBox[List["2", " ", SqrtBox[RowBox[List["a", "+", RowBox[List["b", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]], " ", SqrtBox[RowBox[List["c", "+", RowBox[List["d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]]]]], "+", FractionBox[RowBox[List[RowBox[List["a", " ", "d"]], "+", RowBox[List["b", " ", RowBox[List["(", RowBox[List["c", "+", RowBox[List["2", " ", "d", " ", SuperscriptBox["\[ExponentialE]", RowBox[List["e", " ", "z"]]]]]]], ")"]]]]]], RowBox[List[SqrtBox["b"], " ", SqrtBox["d"]]]]]], "]"]]]], SqrtBox["d"]]]], "e"]]]]] |
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Date Added to functions.wolfram.com (modification date)
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){kind=small}
