Optimal. Leaf size=37 \[ -\frac{2}{7} (\text{csch}(x)+1)^{7/2}+\frac{4}{5} (\text{csch}(x)+1)^{5/2}-\frac{4}{3} (\text{csch}(x)+1)^{3/2} \]
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Rubi [A] time = 0.105672, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {4372, 1570, 1469, 697} \[ -\frac{2}{7} (\text{csch}(x)+1)^{7/2}+\frac{4}{5} (\text{csch}(x)+1)^{5/2}-\frac{4}{3} (\text{csch}(x)+1)^{3/2} \]
Antiderivative was successfully verified.
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Rule 4372
Rule 1570
Rule 1469
Rule 697
Rubi steps
\begin{align*} \int \coth ^3(x) \text{csch}(x) \sqrt{1+\text{csch}(x)} \, dx &=\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{1}{x}} \left (1+x^2\right )}{x^4} \, dx,x,\sinh (x)\right )\\ &=\operatorname{Subst}\left (\int \frac{\left (1+\frac{1}{x^2}\right ) \sqrt{1+\frac{1}{x}}}{x^2} \, dx,x,\sinh (x)\right )\\ &=-\operatorname{Subst}\left (\int \sqrt{1+x} \left (1+x^2\right ) \, dx,x,\text{csch}(x)\right )\\ &=-\operatorname{Subst}\left (\int \left (2 \sqrt{1+x}-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\text{csch}(x)\right )\\ &=-\frac{4}{3} (1+\text{csch}(x))^{3/2}+\frac{4}{5} (1+\text{csch}(x))^{5/2}-\frac{2}{7} (1+\text{csch}(x))^{7/2}\\ \end{align*}
Mathematica [A] time = 0.0585229, size = 34, normalized size = 0.92 \[ -\frac{1}{210} \text{csch}^3(x) \sqrt{\text{csch}(x)+1} (-117 \sinh (x)+43 \sinh (3 x)+62 \cosh (2 x)-2) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.113, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm coth} \left (x\right ) \right ) ^{3}{\rm csch} \left (x\right )\sqrt{1+{\rm csch} \left (x\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.44684, size = 525, normalized size = 14.19 \begin{align*} \frac{124 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac{78 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-2 \, x\right )}}{35 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac{8 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-3 \, x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{78 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-4 \, x\right )}}{35 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{124 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-5 \, x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} - \frac{86 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1} e^{\left (-6 \, x\right )}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} + \frac{86 \, \sqrt{-2 \, e^{\left (-x\right )} + e^{\left (-2 \, x\right )} - 1}}{105 \, \sqrt{e^{\left (-x\right )} + 1} \sqrt{e^{\left (-x\right )} - 1}{\left (3 \, e^{\left (-2 \, x\right )} - 3 \, e^{\left (-4 \, x\right )} + e^{\left (-6 \, x\right )} - 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09575, size = 953, normalized size = 25.76 \begin{align*} -\frac{2 \,{\left (43 \, \cosh \left (x\right )^{6} + 2 \,{\left (129 \, \cosh \left (x\right ) + 31\right )} \sinh \left (x\right )^{5} + 43 \, \sinh \left (x\right )^{6} + 62 \, \cosh \left (x\right )^{5} +{\left (645 \, \cosh \left (x\right )^{2} + 310 \, \cosh \left (x\right ) - 117\right )} \sinh \left (x\right )^{4} - 117 \, \cosh \left (x\right )^{4} + 4 \,{\left (215 \, \cosh \left (x\right )^{3} + 155 \, \cosh \left (x\right )^{2} - 117 \, \cosh \left (x\right ) - 1\right )} \sinh \left (x\right )^{3} - 4 \, \cosh \left (x\right )^{3} +{\left (645 \, \cosh \left (x\right )^{4} + 620 \, \cosh \left (x\right )^{3} - 702 \, \cosh \left (x\right )^{2} - 12 \, \cosh \left (x\right ) + 117\right )} \sinh \left (x\right )^{2} + 117 \, \cosh \left (x\right )^{2} + 2 \,{\left (129 \, \cosh \left (x\right )^{5} + 155 \, \cosh \left (x\right )^{4} - 234 \, \cosh \left (x\right )^{3} - 6 \, \cosh \left (x\right )^{2} + 117 \, \cosh \left (x\right ) + 31\right )} \sinh \left (x\right ) + 62 \, \cosh \left (x\right ) - 43\right )} \sqrt{\frac{\sinh \left (x\right ) + 1}{\sinh \left (x\right )}}}{105 \,{\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \,{\left (5 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{4} - 3 \, \cosh \left (x\right )^{4} + 4 \,{\left (5 \, \cosh \left (x\right )^{3} - 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \,{\left (5 \, \cosh \left (x\right )^{4} - 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \,{\left (\cosh \left (x\right )^{5} - 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right )}} \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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\operatorname{csch}\left (x\right ) + 1} \coth \left (x\right )^{3} \operatorname{csch}\left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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