Optimal. Leaf size=62 \[ \frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{2}{3} i \sqrt{i \sinh (x)} \text{csch}^2(x) \text{EllipticF}\left (\frac{\pi }{4}-\frac{i x}{2},2\right ) \sqrt{a \sinh ^3(x)} \]
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Rubi [A] time = 0.0326089, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {3207, 2635, 2642, 2641} \[ \frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{2}{3} i \sqrt{i \sinh (x)} \text{csch}^2(x) F\left (\left .\frac{\pi }{4}-\frac{i x}{2}\right |2\right ) \sqrt{a \sinh ^3(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \sqrt{a \sinh ^3(x)} \, dx &=\frac{\sqrt{a \sinh ^3(x)} \int \sinh ^{\frac{3}{2}}(x) \, dx}{\sinh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{\sqrt{a \sinh ^3(x)} \int \frac{1}{\sqrt{\sinh (x)}} \, dx}{3 \sinh ^{\frac{3}{2}}(x)}\\ &=\frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{1}{3} \left (\text{csch}^2(x) \sqrt{i \sinh (x)} \sqrt{a \sinh ^3(x)}\right ) \int \frac{1}{\sqrt{i \sinh (x)}} \, dx\\ &=\frac{2}{3} \coth (x) \sqrt{a \sinh ^3(x)}-\frac{2}{3} i \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)}\\ \end{align*}
Mathematica [C] time = 0.0834606, size = 60, normalized size = 0.97 \[ \frac{2}{3} \sqrt{a \sinh ^3(x)} \left (\coth (x)-\sqrt{2} \text{csch}^2(x) \sqrt{-\sinh (x) (\sinh (x)+\cosh (x))} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\cosh (2 x)+\sinh (2 x)\right )\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.065, size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left ( \sinh \left ( x \right ) \right ) ^{3}}\, 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 \sqrt{a \sinh \left (x\right )^{3}}\,{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}}, 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 \sqrt{a \sinh ^{3}{\left (x \right )}}\, 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 \sqrt{a \sinh \left (x\right )^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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