Optimal. Leaf size=135 \[ \frac{26}{77} i a^2 \sqrt{i \sinh (x)} \text{csch}^2(x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right ) \sqrt{a \sinh ^3(x)}+\frac{2}{15} a^2 \sinh ^5(x) \cosh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{165} a^2 \sinh ^3(x) \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{78}{385} a^2 \sinh (x) \cosh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{77} a^2 \coth (x) \sqrt{a \sinh ^3(x)} \]
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Rubi [A] time = 0.0605775, 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, 2635, 2642, 2641} \[ \frac{2}{15} a^2 \sinh ^5(x) \cosh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{165} a^2 \sinh ^3(x) \cosh (x) \sqrt{a \sinh ^3(x)}+\frac{78}{385} a^2 \sinh (x) \cosh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{77} a^2 \coth (x) \sqrt{a \sinh ^3(x)}+\frac{26}{77} i a^2 \sqrt{i \sinh (x)} \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \sinh ^3(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \left (a \sinh ^3(x)\right )^{5/2} \, dx &=\frac{\left (a^2 \sqrt{a \sinh ^3(x)}\right ) \int \sinh ^{\frac{15}{2}}(x) \, dx}{\sinh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^3(x)}-\frac{\left (13 a^2 \sqrt{a \sinh ^3(x)}\right ) \int \sinh ^{\frac{11}{2}}(x) \, dx}{15 \sinh ^{\frac{3}{2}}(x)}\\ &=-\frac{26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^3(x)}+\frac{2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^3(x)}+\frac{\left (39 a^2 \sqrt{a \sinh ^3(x)}\right ) \int \sinh ^{\frac{7}{2}}(x) \, dx}{55 \sinh ^{\frac{3}{2}}(x)}\\ &=\frac{78}{385} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^3(x)}+\frac{2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^3(x)}-\frac{\left (39 a^2 \sqrt{a \sinh ^3(x)}\right ) \int \sinh ^{\frac{3}{2}}(x) \, dx}{77 \sinh ^{\frac{3}{2}}(x)}\\ &=-\frac{26}{77} a^2 \coth (x) \sqrt{a \sinh ^3(x)}+\frac{78}{385} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^3(x)}+\frac{2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^3(x)}+\frac{\left (13 a^2 \sqrt{a \sinh ^3(x)}\right ) \int \frac{1}{\sqrt{\sinh (x)}} \, dx}{77 \sinh ^{\frac{3}{2}}(x)}\\ &=-\frac{26}{77} a^2 \coth (x) \sqrt{a \sinh ^3(x)}+\frac{78}{385} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^3(x)}+\frac{2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^3(x)}+\frac{1}{77} \left (13 a^2 \text{csch}^2(x) \sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx\\ &=-\frac{26}{77} a^2 \coth (x) \sqrt{a \sinh ^3(x)}+\frac{26}{77} i a^2 \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}+\frac{78}{385} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^3(x)}-\frac{26}{165} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^3(x)}+\frac{2}{15} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^3(x)}\\ \end{align*}
Mathematica [A] time = 0.182696, size = 67, normalized size = 0.5 \[ \frac{a^2 \text{csch}(x) \sqrt{a \sinh ^3(x)} \left (-\frac{12480 \text{EllipticF}\left (\frac{1}{4} (\pi -2 i x),2\right )}{\sqrt{i \sinh (x)}}-15465 \cosh (x)+3657 \cosh (3 x)-749 \cosh (5 x)+77 \cosh (7 x)\right )}{36960} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.076, 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 \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 (\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(-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 \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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