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 = 0.0607026, antiderivative size = 135, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, 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 3207
Rule 2636
Rule 2640
Rule 2639
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*}
Mathematica [A] time = 0.196194, 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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Maple [F] time = 0.06, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( \sinh \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sinh \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a \sinh \left (x\right )^{3}}}{a^{3} \sinh \left (x\right )^{9}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sinh \left (x\right )^{3}\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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