Optimal. Leaf size=135 \[ \frac {154 \sinh (x) \cosh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}} \]
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Rubi [A] time = 0.06, antiderivative size = 135, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {3207, 2636, 2640, 2639} \[ \frac {154 \sinh (x) \cosh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i \sinh ^2(x) E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right )}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rule 2640
Rule 3207
Rubi steps
\begin {align*} \int \frac {1}{\left (a \sinh ^3(x)\right )^{5/2}} \, dx &=\frac {\sinh ^{\frac {3}{2}}(x) \int \frac {1}{\sinh ^{\frac {15}{2}}(x)} \, dx}{a^2 \sqrt {a \sinh ^3(x)}}\\ &=-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (11 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {11}{2}}(x)} \, dx}{13 a^2 \sqrt {a \sinh ^3(x)}}\\ &=\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {\left (77 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {7}{2}}(x)} \, dx}{117 a^2 \sqrt {a \sinh ^3(x)}}\\ &=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (77 \sinh ^{\frac {3}{2}}(x)\right ) \int \frac {1}{\sinh ^{\frac {3}{2}}(x)} \, dx}{195 a^2 \sqrt {a \sinh ^3(x)}}\\ &=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (77 \sinh ^{\frac {3}{2}}(x)\right ) \int \sqrt {\sinh (x)} \, dx}{195 a^2 \sqrt {a \sinh ^3(x)}}\\ &=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {\left (77 \sinh ^2(x)\right ) \int \sqrt {i \sinh (x)} \, dx}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}\\ &=-\frac {154 \coth (x)}{585 a^2 \sqrt {a \sinh ^3(x)}}+\frac {22 \coth (x) \text {csch}^2(x)}{117 a^2 \sqrt {a \sinh ^3(x)}}-\frac {2 \coth (x) \text {csch}^4(x)}{13 a^2 \sqrt {a \sinh ^3(x)}}+\frac {154 \cosh (x) \sinh (x)}{195 a^2 \sqrt {a \sinh ^3(x)}}-\frac {154 i E\left (\left .\frac {\pi }{4}-\frac {i x}{2}\right |2\right ) \sinh ^2(x)}{195 a^2 \sqrt {i \sinh (x)} \sqrt {a \sinh ^3(x)}}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 69, normalized size = 0.51 \[ \frac {462 \sinh (x) \cosh (x)-2 \coth (x) \left (45 \text {csch}^4(x)-55 \text {csch}^2(x)+77\right )+462 i (i \sinh (x))^{3/2} E\left (\left .\frac {1}{4} (\pi -2 i x)\right |2\right )}{585 a^2 \sqrt {a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.42, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a \sinh \relax (x)^{3}}}{a^{3} \sinh \relax (x)^{9}}, x\right ) \]
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 \[ \int \frac {1}{\left (a \sinh \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.09, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \left (\sinh ^{3}\relax (x )\right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {1}{\left (a \sinh \relax (x)^{3}\right )^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{{\left (a\,{\mathrm {sinh}\relax (x)}^3\right )}^{5/2}} \,d x \]
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 \[ \int \frac {1}{\left (a \sinh ^{3}{\relax (x )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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