Optimal. Leaf size=87 \[ \frac{10 i \sqrt{i \sinh (x)} \sinh (x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right )}{21 a \sqrt{a \sinh ^3(x)}}+\frac{10 \cosh (x)}{21 a \sqrt{a \sinh ^3(x)}}-\frac{2 \coth (x) \text{csch}(x)}{7 a \sqrt{a \sinh ^3(x)}} \]
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Rubi [A] time = 0.0419059, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3207, 2636, 2642, 2641} \[ \frac{10 \cosh (x)}{21 a \sqrt{a \sinh ^3(x)}}+\frac{10 i \sqrt{i \sinh (x)} \sinh (x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right )}{21 a \sqrt{a \sinh ^3(x)}}-\frac{2 \coth (x) \text{csch}(x)}{7 a \sqrt{a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2636
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\left (a \sinh ^3(x)\right )^{3/2}} \, dx &=\frac{\sinh ^{\frac{3}{2}}(x) \int \frac{1}{\sinh ^{\frac{9}{2}}(x)} \, dx}{a \sqrt{a \sinh ^3(x)}}\\ &=-\frac{2 \coth (x) \text{csch}(x)}{7 a \sqrt{a \sinh ^3(x)}}-\frac{\left (5 \sinh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sinh ^{\frac{5}{2}}(x)} \, dx}{7 a \sqrt{a \sinh ^3(x)}}\\ &=\frac{10 \cosh (x)}{21 a \sqrt{a \sinh ^3(x)}}-\frac{2 \coth (x) \text{csch}(x)}{7 a \sqrt{a \sinh ^3(x)}}+\frac{\left (5 \sinh ^{\frac{3}{2}}(x)\right ) \int \frac{1}{\sqrt{\sinh (x)}} \, dx}{21 a \sqrt{a \sinh ^3(x)}}\\ &=\frac{10 \cosh (x)}{21 a \sqrt{a \sinh ^3(x)}}-\frac{2 \coth (x) \text{csch}(x)}{7 a \sqrt{a \sinh ^3(x)}}+\frac{\left (5 \sqrt{i \sinh (x)} \sinh (x)\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx}{21 a \sqrt{a \sinh ^3(x)}}\\ &=\frac{10 \cosh (x)}{21 a \sqrt{a \sinh ^3(x)}}-\frac{2 \coth (x) \text{csch}(x)}{7 a \sqrt{a \sinh ^3(x)}}+\frac{10 i F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)} \sinh (x)}{21 a \sqrt{a \sinh ^3(x)}}\\ \end{align*}
Mathematica [A] time = 0.0820777, size = 53, normalized size = 0.61 \[ \frac{2 \left (5 (i \sinh (x))^{3/2} \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),2\right )+5 \cosh (x)-3 \coth (x) \text{csch}(x)\right )}{21 a \sqrt{a \sinh ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.062, size = 0, normalized size = 0. \begin{align*} \int \left ( a \left ( \sinh \left ( x \right ) \right ) ^{3} \right ) ^{-{\frac{3}{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{3}{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^{2} \sinh \left (x\right )^{6}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a \sinh ^{3}{\left (x \right )}\right )^{\frac{3}{2}}}\, dx \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{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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