Optimal. Leaf size=132 \[ -\frac {21}{128} a^2 \sinh (x) \cosh (x) \sqrt {a \sinh ^4(x)}+\frac {1}{10} a^2 \sinh ^7(x) \cosh (x) \sqrt {a \sinh ^4(x)}-\frac {9}{80} a^2 \sinh ^5(x) \cosh (x) \sqrt {a \sinh ^4(x)}+\frac {21}{160} a^2 \sinh ^3(x) \cosh (x) \sqrt {a \sinh ^4(x)}+\frac {63}{256} a^2 \coth (x) \sqrt {a \sinh ^4(x)}-\frac {63}{256} a^2 x \text {csch}^2(x) \sqrt {a \sinh ^4(x)} \]
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Rubi [A] time = 0.05, antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 8
Rule 2635
Rule 3207
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*}
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Mathematica [A] time = 0.16, size = 53, normalized size = 0.40 \[ \frac {a (-2520 x+2100 \sinh (2 x)-600 \sinh (4 x)+150 \sinh (6 x)-25 \sinh (8 x)+2 \sinh (10 x)) \text {csch}^6(x) \left (a \sinh ^4(x)\right )^{3/2}}{10240} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.58, size = 1597, normalized size = 12.10 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.30, size = 114, normalized size = 0.86 \[ -\frac {1}{20480} \, {\left (5040 \, a^{2} x - 2 \, a^{2} e^{\left (10 \, x\right )} + 25 \, a^{2} e^{\left (8 \, x\right )} - 150 \, a^{2} e^{\left (6 \, x\right )} + 600 \, a^{2} e^{\left (4 \, x\right )} - 2100 \, a^{2} e^{\left (2 \, x\right )} - {\left (5754 \, a^{2} e^{\left (10 \, x\right )} - 2100 \, a^{2} e^{\left (8 \, x\right )} + 600 \, a^{2} e^{\left (6 \, x\right )} - 150 \, a^{2} e^{\left (4 \, x\right )} + 25 \, a^{2} e^{\left (2 \, x\right )} - 2 \, a^{2}\right )} e^{\left (-10 \, x\right )}\right )} \sqrt {a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.20, size = 171, normalized size = 1.30 \[ \frac {\left (-1+\cosh \left (2 x \right )\right ) \sqrt {a \left (-1+\cosh \left (2 x \right )\right ) \left (\cosh \left (2 x \right )+1\right )}\, a^{\frac {3}{2}} \left (8 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \left (\sinh ^{4}\left (2 x \right )\right )-50 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \cosh \left (2 x \right ) \left (\sinh ^{2}\left (2 x \right )\right )+160 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}\, \left (\sinh ^{2}\left (2 x \right )\right )-325 \cosh \left (2 x \right ) \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}+640 \sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\, \sqrt {a}-315 \ln \left (\sqrt {a}\, \cosh \left (2 x \right )+\sqrt {a \left (\sinh ^{2}\left (2 x \right )\right )}\right ) a \right )}{2560 \sinh \left (2 x \right ) \sqrt {\left (-1+\cosh \left (2 x \right )\right )^{2} a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 100, normalized size = 0.76 \[ -\frac {63}{256} \, a^{\frac {5}{2}} x - \frac {1}{20480} \, {\left (25 \, a^{\frac {5}{2}} e^{\left (-2 \, x\right )} - 150 \, a^{\frac {5}{2}} e^{\left (-4 \, x\right )} + 600 \, a^{\frac {5}{2}} e^{\left (-6 \, x\right )} - 2100 \, a^{\frac {5}{2}} e^{\left (-8 \, x\right )} + 2100 \, a^{\frac {5}{2}} e^{\left (-12 \, x\right )} - 600 \, a^{\frac {5}{2}} e^{\left (-14 \, x\right )} + 150 \, a^{\frac {5}{2}} e^{\left (-16 \, x\right )} - 25 \, a^{\frac {5}{2}} e^{\left (-18 \, x\right )} + 2 \, a^{\frac {5}{2}} e^{\left (-20 \, x\right )} - 2 \, a^{\frac {5}{2}}\right )} e^{\left (10 \, x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (a\,{\mathrm {sinh}\relax (x)}^4\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \sinh ^{4}{\relax (x )}\right )^{\frac {5}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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