Optimal. Leaf size=112 \[ \frac {2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
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Rubi [A] time = 0.05, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3076, 3078, 2639} \[ \frac {2 (a \sinh (x)+b \cosh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {2 i \sqrt {a \cosh (x)+b \sinh (x)} E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right )}{\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 3076
Rule 3078
Rubi steps
\begin {align*} \int \frac {1}{(a \cosh (x)+b \sinh (x))^{3/2}} \, dx &=\frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {\int \sqrt {a \cosh (x)+b \sinh (x)} \, dx}{-a^2+b^2}\\ &=\frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {\sqrt {a \cosh (x)+b \sinh (x)} \int \sqrt {\cosh \left (x+i \tan ^{-1}(a,-i b)\right )} \, dx}{\left (-a^2+b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}\\ &=\frac {2 (b \cosh (x)+a \sinh (x))}{\left (a^2-b^2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}+\frac {2 i E\left (\left .\frac {1}{2} \left (i x-\tan ^{-1}(a,-i b)\right )\right |2\right ) \sqrt {a \cosh (x)+b \sinh (x)}}{\left (a^2-b^2\right ) \sqrt {\frac {a \cosh (x)+b \sinh (x)}{\sqrt {a^2-b^2}}}}\\ \end {align*}
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Mathematica [C] time = 0.46, size = 148, normalized size = 1.32 \[ \frac {b \sinh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right ) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cosh ^2\left (x+\tanh ^{-1}\left (\frac {b}{a}\right )\right )\right )-\sqrt {-\sinh ^2\left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )} \left (2 a \sqrt {1-\frac {b^2}{a^2}} \cosh (x)+b \sinh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )-2 a \cosh \left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )\right )}{a b \sqrt {1-\frac {b^2}{a^2}} \sqrt {-\sinh ^2\left (\tanh ^{-1}\left (\frac {b}{a}\right )+x\right )} \sqrt {a \cosh (x)+b \sinh (x)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \cosh \relax (x) + b \sinh \relax (x)}}{a^{2} \cosh \relax (x)^{2} + 2 \, a b \cosh \relax (x) \sinh \relax (x) + b^{2} \sinh \relax (x)^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.34, size = 33, normalized size = 0.29 \[ \frac {\arctanh \left (\cosh \relax (x )\right )}{\sqrt {a^{2}-b^{2}}\, \sqrt {-\sinh \relax (x ) \sqrt {a^{2}-b^{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a \cosh \relax (x) + b \sinh \relax (x)\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a\,\mathrm {cosh}\relax (x)+b\,\mathrm {sinh}\relax (x)\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \cosh {\relax (x )} + b \sinh {\relax (x )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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