Optimal. Leaf size=36 \[ \frac{1}{2} \coth (x) \sqrt{a \sinh ^4(x)}-\frac{1}{2} x \text{csch}^2(x) \sqrt{a \sinh ^4(x)} \]
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Rubi [A] time = 0.0149813, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 8} \[ \frac{1}{2} \coth (x) \sqrt{a \sinh ^4(x)}-\frac{1}{2} x \text{csch}^2(x) \sqrt{a \sinh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \sqrt{a \sinh ^4(x)} \, dx &=\left (\text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int \sinh ^2(x) \, dx\\ &=\frac{1}{2} \coth (x) \sqrt{a \sinh ^4(x)}-\frac{1}{2} \left (\text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int 1 \, dx\\ &=\frac{1}{2} \coth (x) \sqrt{a \sinh ^4(x)}-\frac{1}{2} x \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\\ \end{align*}
Mathematica [A] time = 0.0380094, size = 24, normalized size = 0.67 \[ \frac{1}{2} \sqrt{a \sinh ^4(x)} \left (\coth (x)-x \text{csch}^2(x)\right ) \]
Antiderivative was successfully verified.
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Maple [B] time = 0.095, size = 90, normalized size = 2.5 \begin{align*}{\frac{\sqrt{8} \left ( -1+\cosh \left ( 2\,x \right ) \right ) \sqrt{2}}{16\,\sinh \left ( 2\,x \right ) }\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) } \left ( \sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}-\ln \left ( \sqrt{a}\cosh \left ( 2\,x \right ) +\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}} \right ) a \right ){\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) ^{2}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.87253, size = 36, normalized size = 1. \begin{align*} -\frac{1}{8} \,{\left (\sqrt{a} e^{\left (-4 \, x\right )} - \sqrt{a}\right )} e^{\left (2 \, x\right )} - \frac{1}{2} \, \sqrt{a} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.7905, size = 551, normalized size = 15.31 \begin{align*} \frac{{\left (4 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} \sinh \left (x\right )^{3} + e^{\left (2 \, x\right )} \sinh \left (x\right )^{4} + 2 \,{\left (3 \, \cosh \left (x\right )^{2} - 2 \, x\right )} e^{\left (2 \, x\right )} \sinh \left (x\right )^{2} + 4 \,{\left (\cosh \left (x\right )^{3} - 2 \, x \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} \sinh \left (x\right ) +{\left (\cosh \left (x\right )^{4} - 4 \, x \cosh \left (x\right )^{2} - 1\right )} e^{\left (2 \, x\right )}\right )} \sqrt{a e^{\left (8 \, x\right )} - 4 \, a e^{\left (6 \, x\right )} + 6 \, a e^{\left (4 \, x\right )} - 4 \, a e^{\left (2 \, x\right )} + a} e^{\left (-2 \, x\right )}}{8 \,{\left (\cosh \left (x\right )^{2} e^{\left (4 \, x\right )} - 2 \, \cosh \left (x\right )^{2} e^{\left (2 \, x\right )} +{\left (e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} + \cosh \left (x\right )^{2} + 2 \,{\left (\cosh \left (x\right ) e^{\left (4 \, x\right )} - 2 \, \cosh \left (x\right ) e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right )\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 ^{4}{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.22562, size = 35, normalized size = 0.97 \begin{align*} \frac{1}{8} \,{\left ({\left (2 \, e^{\left (2 \, x\right )} - 1\right )} e^{\left (-2 \, x\right )} - 4 \, x + e^{\left (2 \, x\right )}\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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