Optimal. Leaf size=33 \[ -\frac{1}{9} \text{csch}^9(x)-\frac{3 \text{csch}^7(x)}{7}-\frac{3 \text{csch}^5(x)}{5}-\frac{\text{csch}^3(x)}{3} \]
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Rubi [A] time = 0.0336737, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2606, 270} \[ -\frac{1}{9} \text{csch}^9(x)-\frac{3 \text{csch}^7(x)}{7}-\frac{3 \text{csch}^5(x)}{5}-\frac{\text{csch}^3(x)}{3} \]
Antiderivative was successfully verified.
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Rule 2606
Rule 270
Rubi steps
\begin{align*} \int \coth ^7(x) \text{csch}^3(x) \, dx &=-\left (i \operatorname{Subst}\left (\int x^2 \left (-1+x^2\right )^3 \, dx,x,-i \text{csch}(x)\right )\right )\\ &=-\left (i \operatorname{Subst}\left (\int \left (-x^2+3 x^4-3 x^6+x^8\right ) \, dx,x,-i \text{csch}(x)\right )\right )\\ &=-\frac{1}{3} \text{csch}^3(x)-\frac{3 \text{csch}^5(x)}{5}-\frac{3 \text{csch}^7(x)}{7}-\frac{\text{csch}^9(x)}{9}\\ \end{align*}
Mathematica [A] time = 0.0117182, size = 33, normalized size = 1. \[ -\frac{1}{9} \text{csch}^9(x)-\frac{3 \text{csch}^7(x)}{7}-\frac{3 \text{csch}^5(x)}{5}-\frac{\text{csch}^3(x)}{3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.015, size = 76, normalized size = 2.3 \begin{align*} -{\frac{ \left ( \cosh \left ( x \right ) \right ) ^{6}}{3\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}+{\frac{2\, \left ( \cosh \left ( x \right ) \right ) ^{4}}{5\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}-{\frac{8\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{45\, \left ( \sinh \left ( x \right ) \right ) ^{9}}}-{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\, \left ( \sinh \left ( x \right ) \right ) ^{7}}}+{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\, \left ( \sinh \left ( x \right ) \right ) ^{5}}}-{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\, \left ( \sinh \left ( x \right ) \right ) ^{3}}}+{\frac{16\, \left ( \cosh \left ( x \right ) \right ) ^{2}}{315\,\sinh \left ( x \right ) }}-{\frac{16\,\sinh \left ( x \right ) }{315}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.06554, size = 587, normalized size = 17.79 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.77903, size = 1571, normalized size = 47.61 \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 [B] time = 1.17492, size = 73, normalized size = 2.21 \begin{align*} \frac{8 \,{\left (105 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{6} + 756 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{4} + 2160 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 2240\right )}}{315 \,{\left (e^{\left (-x\right )} - e^{x}\right )}^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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