Optimal. Leaf size=46 \[ \frac{1}{2} \coth (x) \sqrt{2-\coth ^2(x)}+\tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{2-\coth ^2(x)}}\right )+2 \sin ^{-1}\left (\frac{\coth (x)}{\sqrt{2}}\right ) \]
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Rubi [A] time = 0.0472462, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {4128, 416, 523, 216, 377, 206} \[ \frac{1}{2} \coth (x) \sqrt{2-\coth ^2(x)}+\tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{2-\coth ^2(x)}}\right )+2 \sin ^{-1}\left (\frac{\coth (x)}{\sqrt{2}}\right ) \]
Antiderivative was successfully verified.
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Rule 4128
Rule 416
Rule 523
Rule 216
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \left (1-\text{csch}^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (2-x^2\right )^{3/2}}{1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \coth (x) \sqrt{2-\coth ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-6+4 x^2}{\left (1-x^2\right ) \sqrt{2-x^2}} \, dx,x,\coth (x)\right )\\ &=\frac{1}{2} \coth (x) \sqrt{2-\coth ^2(x)}+2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2}} \, dx,x,\coth (x)\right )+\operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{2-x^2}} \, dx,x,\coth (x)\right )\\ &=2 \sin ^{-1}\left (\frac{\coth (x)}{\sqrt{2}}\right )+\frac{1}{2} \coth (x) \sqrt{2-\coth ^2(x)}+\operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\frac{\coth (x)}{\sqrt{2-\coth ^2(x)}}\right )\\ &=2 \sin ^{-1}\left (\frac{\coth (x)}{\sqrt{2}}\right )+\tanh ^{-1}\left (\frac{\coth (x)}{\sqrt{2-\coth ^2(x)}}\right )+\frac{1}{2} \coth (x) \sqrt{2-\coth ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.206543, size = 92, normalized size = 2. \[ \frac{\sinh ^3(x) \left (1-\text{csch}^2(x)\right )^{3/2} \left (2 \sqrt{2} \left (\log \left (\sqrt{2} \cosh (x)+\sqrt{\cosh (2 x)-3}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \cosh (x)}{\sqrt{\cosh (2 x)-3}}\right )\right )+\sqrt{\cosh (2 x)-3} \coth (x) \text{csch}(x)\right )}{(\cosh (2 x)-3)^{3/2}} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.094, size = 0, normalized size = 0. \begin{align*} \int \left ( 1- \left ({\rm csch} \left (x\right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-\operatorname{csch}\left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19751, size = 1843, normalized size = 40.07 \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 (1 - \operatorname{csch}^{2}{\left (x \right )}\right )^{\frac{3}{2}}\, dx \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{\left (-\operatorname{csch}\left (x\right )^{2} + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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