Optimal. Leaf size=132 \[ \frac{1}{10} a^2 \sinh ^7(x) \cosh (x) \sqrt{a \sinh ^4(x)}-\frac{9}{80} a^2 \sinh ^5(x) \cosh (x) \sqrt{a \sinh ^4(x)}+\frac{21}{160} a^2 \sinh ^3(x) \cosh (x) \sqrt{a \sinh ^4(x)}-\frac{21}{128} a^2 \sinh (x) \cosh (x) \sqrt{a \sinh ^4(x)}+\frac{63}{256} a^2 \coth (x) \sqrt{a \sinh ^4(x)}-\frac{63}{256} a^2 x \text{csch}^2(x) \sqrt{a \sinh ^4(x)} \]
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Rubi [A] time = 0.0488886, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3207, 2635, 8} \[ \frac{1}{10} a^2 \sinh ^7(x) \cosh (x) \sqrt{a \sinh ^4(x)}-\frac{9}{80} a^2 \sinh ^5(x) \cosh (x) \sqrt{a \sinh ^4(x)}+\frac{21}{160} a^2 \sinh ^3(x) \cosh (x) \sqrt{a \sinh ^4(x)}-\frac{21}{128} a^2 \sinh (x) \cosh (x) \sqrt{a \sinh ^4(x)}+\frac{63}{256} a^2 \coth (x) \sqrt{a \sinh ^4(x)}-\frac{63}{256} a^2 x \text{csch}^2(x) \sqrt{a \sinh ^4(x)} \]
Antiderivative was successfully verified.
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Rule 3207
Rule 2635
Rule 8
Rubi steps
\begin{align*} \int \left (a \sinh ^4(x)\right )^{5/2} \, dx &=\left (a^2 \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int \sinh ^{10}(x) \, dx\\ &=\frac{1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt{a \sinh ^4(x)}-\frac{1}{10} \left (9 a^2 \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int \sinh ^8(x) \, dx\\ &=-\frac{9}{80} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^4(x)}+\frac{1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt{a \sinh ^4(x)}+\frac{1}{80} \left (63 a^2 \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int \sinh ^6(x) \, dx\\ &=\frac{21}{160} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^4(x)}-\frac{9}{80} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^4(x)}+\frac{1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt{a \sinh ^4(x)}-\frac{1}{32} \left (21 a^2 \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int \sinh ^4(x) \, dx\\ &=-\frac{21}{128} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^4(x)}+\frac{21}{160} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^4(x)}-\frac{9}{80} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^4(x)}+\frac{1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt{a \sinh ^4(x)}+\frac{1}{128} \left (63 a^2 \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int \sinh ^2(x) \, dx\\ &=\frac{63}{256} a^2 \coth (x) \sqrt{a \sinh ^4(x)}-\frac{21}{128} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^4(x)}+\frac{21}{160} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^4(x)}-\frac{9}{80} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^4(x)}+\frac{1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt{a \sinh ^4(x)}-\frac{1}{256} \left (63 a^2 \text{csch}^2(x) \sqrt{a \sinh ^4(x)}\right ) \int 1 \, dx\\ &=\frac{63}{256} a^2 \coth (x) \sqrt{a \sinh ^4(x)}-\frac{63}{256} a^2 x \text{csch}^2(x) \sqrt{a \sinh ^4(x)}-\frac{21}{128} a^2 \cosh (x) \sinh (x) \sqrt{a \sinh ^4(x)}+\frac{21}{160} a^2 \cosh (x) \sinh ^3(x) \sqrt{a \sinh ^4(x)}-\frac{9}{80} a^2 \cosh (x) \sinh ^5(x) \sqrt{a \sinh ^4(x)}+\frac{1}{10} a^2 \cosh (x) \sinh ^7(x) \sqrt{a \sinh ^4(x)}\\ \end{align*}
Mathematica [A] time = 0.158968, size = 53, normalized size = 0.4 \[ \frac{a (-2520 x+2100 \sinh (2 x)-600 \sinh (4 x)+150 \sinh (6 x)-25 \sinh (8 x)+2 \sinh (10 x)) \text{csch}^6(x) \left (a \sinh ^4(x)\right )^{3/2}}{10240} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.115, size = 177, normalized size = 1.3 \begin{align*}{\frac{\sqrt{8} \left ( -1+\cosh \left ( 2\,x \right ) \right ) \sqrt{2}}{10240\,\sinh \left ( 2\,x \right ) }\sqrt{a \left ( -1+\cosh \left ( 2\,x \right ) \right ) \left ( \cosh \left ( 2\,x \right ) +1 \right ) }{a}^{{\frac{3}{2}}} \left ( 8\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a} \left ( \sinh \left ( 2\,x \right ) \right ) ^{4}-50\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}\cosh \left ( 2\,x \right ) \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}+160\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a} \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}-325\,\cosh \left ( 2\,x \right ) \sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}+640\,\sqrt{a \left ( \sinh \left ( 2\,x \right ) \right ) ^{2}}\sqrt{a}-315\,\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 \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 = 1.73714, size = 135, normalized size = 1.02 \begin{align*} -\frac{63}{256} \, a^{\frac{5}{2}} x - \frac{1}{20480} \,{\left (25 \, a^{\frac{5}{2}} e^{\left (-2 \, x\right )} - 150 \, a^{\frac{5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac{5}{2}} e^{\left (-6 \, x\right )} - 2100 \, a^{\frac{5}{2}} e^{\left (-8 \, x\right )} + 2100 \, a^{\frac{5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac{5}{2}} e^{\left (-14 \, x\right )} + 150 \, a^{\frac{5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac{5}{2}} e^{\left (-18 \, x\right )} + 2 \, a^{\frac{5}{2}} e^{\left (-20 \, x\right )} - 2 \, a^{\frac{5}{2}}\right )} e^{\left (10 \, x\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.0534, size = 4963, normalized size = 37.6 \begin{align*} \text{result too large to display} \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 [A] time = 1.27297, size = 154, normalized size = 1.17 \begin{align*} -\frac{1}{20480} \,{\left (5040 \, a^{2} x - 2 \, a^{2} e^{\left (10 \, x\right )} + 25 \, a^{2} e^{\left (8 \, x\right )} - 150 \, a^{2} e^{\left (6 \, x\right )} + 600 \, a^{2} e^{\left (4 \, x\right )} - 2100 \, a^{2} e^{\left (2 \, x\right )} -{\left (5754 \, a^{2} e^{\left (10 \, x\right )} - 2100 \, a^{2} e^{\left (8 \, x\right )} + 600 \, a^{2} e^{\left (6 \, x\right )} - 150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} - 2 \, a^{2}\right )} e^{\left (-10 \, x\right )}\right )} \sqrt{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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