Optimal. Leaf size=60 \[ -\frac {2 \sinh (x) \cosh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3207, 2636, 2640, 2639} \[ -\frac {2 \sinh (x) \cosh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {2 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {a \sinh ^3(x)}} \, dx &=\frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {3}{2}}(x)} \, dx}{\sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {\sinh ^{\frac {3}{2}}(x) \int \sqrt {\sinh (x)} \, dx}{\sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {\sinh ^2(x) \int \sqrt {i \sinh (x)} \, dx}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \cosh (x) \sinh (x)}{\sqrt {a \sinh ^3(x)}}+\frac {2 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{\sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 42, normalized size = 0.70 \[ -\frac {2 \sinh (x) \left (\cosh (x)-\sqrt {i \sinh (x)} E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )\right )}{\sqrt {a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sinh \relax (x)^{3}}}{a \sinh \relax (x)^{3}}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {a \sinh \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.10, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {a \left (\sinh ^{3}\relax (x )\right )}}\, dx \]
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 \[ \int \frac {1}{\sqrt {a \sinh \relax (x)^{3}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {a\,{\mathrm {sinh}\relax (x)}^3}} \,d x \]
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 \[ \int \frac {1}{\sqrt {a \sinh ^{3}{\relax (x )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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