Optimal. Leaf size=118 \[ -\frac{1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt{a \text{csch}^4(x)}+\frac{4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt{a \text{csch}^4(x)}-\frac{6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt{a \text{csch}^4(x)}+\frac{4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt{a \text{csch}^4(x)}-a^2 \sinh (x) \cosh (x) \sqrt{a \text{csch}^4(x)} \]
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Rubi [A] time = 0.0322146, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4123, 3767} \[ -\frac{1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt{a \text{csch}^4(x)}+\frac{4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt{a \text{csch}^4(x)}-\frac{6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt{a \text{csch}^4(x)}+\frac{4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt{a \text{csch}^4(x)}-a^2 \sinh (x) \cosh (x) \sqrt{a \text{csch}^4(x)} \]
Antiderivative was successfully verified.
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Rule 4123
Rule 3767
Rubi steps
\begin{align*} \int \left (a \text{csch}^4(x)\right )^{5/2} \, dx &=\left (a^2 \sqrt{a \text{csch}^4(x)} \sinh ^2(x)\right ) \int \text{csch}^{10}(x) \, dx\\ &=-\left (\left (i a^2 \sqrt{a \text{csch}^4(x)} \sinh ^2(x)\right ) \operatorname{Subst}\left (\int \left (1+4 x^2+6 x^4+4 x^6+x^8\right ) \, dx,x,-i \coth (x)\right )\right )\\ &=\frac{4}{3} a^2 \cosh ^2(x) \coth (x) \sqrt{a \text{csch}^4(x)}-\frac{6}{5} a^2 \cosh ^2(x) \coth ^3(x) \sqrt{a \text{csch}^4(x)}+\frac{4}{7} a^2 \cosh ^2(x) \coth ^5(x) \sqrt{a \text{csch}^4(x)}-\frac{1}{9} a^2 \cosh ^2(x) \coth ^7(x) \sqrt{a \text{csch}^4(x)}-a^2 \cosh (x) \sqrt{a \text{csch}^4(x)} \sinh (x)\\ \end{align*}
Mathematica [A] time = 0.0324982, size = 47, normalized size = 0.4 \[ -\frac{1}{315} a^2 \sinh (x) \cosh (x) \left (35 \text{csch}^8(x)-40 \text{csch}^6(x)+48 \text{csch}^4(x)-64 \text{csch}^2(x)+128\right ) \sqrt{a \text{csch}^4(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.059, size = 60, normalized size = 0.5 \begin{align*} -{\frac{256\,{a}^{2}{{\rm e}^{-2\,x}} \left ( 126\,{{\rm e}^{8\,x}}-84\,{{\rm e}^{6\,x}}+36\,{{\rm e}^{4\,x}}-9\,{{\rm e}^{2\,x}}+1 \right ) }{315\, \left ({{\rm e}^{2\,x}}-1 \right ) ^{7}}\sqrt{{\frac{{{\rm e}^{4\,x}}a}{ \left ({{\rm e}^{2\,x}}-1 \right ) ^{4}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.78231, size = 435, normalized size = 3.69 \begin{align*} -\frac{256 \, a^{\frac{5}{2}} e^{\left (-2 \, x\right )}}{35 \,{\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac{1024 \, a^{\frac{5}{2}} e^{\left (-4 \, x\right )}}{35 \,{\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} - \frac{1024 \, a^{\frac{5}{2}} e^{\left (-6 \, x\right )}}{15 \,{\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac{512 \, a^{\frac{5}{2}} e^{\left (-8 \, x\right )}}{5 \,{\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} + \frac{256 \, a^{\frac{5}{2}}}{315 \,{\left (9 \, e^{\left (-2 \, x\right )} - 36 \, e^{\left (-4 \, x\right )} + 84 \, e^{\left (-6 \, x\right )} - 126 \, e^{\left (-8 \, x\right )} + 126 \, e^{\left (-10 \, x\right )} - 84 \, e^{\left (-12 \, x\right )} + 36 \, e^{\left (-14 \, x\right )} - 9 \, e^{\left (-16 \, x\right )} + e^{\left (-18 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.91199, size = 4632, normalized size = 39.25 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \operatorname{csch}^{4}{\left (x \right )}\right )^{\frac{5}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1685, size = 53, normalized size = 0.45 \begin{align*} -\frac{256 \, a^{\frac{5}{2}}{\left (126 \, e^{\left (8 \, x\right )} - 84 \, e^{\left (6 \, x\right )} + 36 \, e^{\left (4 \, x\right )} - 9 \, e^{\left (2 \, x\right )} + 1\right )}}{315 \,{\left (e^{\left (2 \, x\right )} - 1\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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