Optimal. Leaf size=46 \[ 2 \text {ArcSin}\left (\frac {\coth (x)}{\sqrt {2}}\right )+\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right )+\frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)} \]
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Rubi [A]
time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4213, 427, 537,
222, 385, 212} \begin {gather*} 2 \text {ArcSin}\left (\frac {\coth (x)}{\sqrt {2}}\right )+\frac {1}{2} \coth (x) \sqrt {2-\coth ^2(x)}+\tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {2-\coth ^2(x)}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 222
Rule 385
Rule 427
Rule 537
Rule 4213
Rubi steps
\begin {align*} \int \left (1-\text {csch}^2(x)\right )^{3/2} \, dx &=\text {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} \text {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 \text {Subst}\left (\int \frac {1}{\sqrt {2-x^2}} \, dx,x,\coth (x)\right )+\text {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)}+\text {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*}
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Mathematica [A]
time = 0.15, size = 92, normalized size = 2.00 \begin {gather*} \frac {\left (1-\text {csch}^2(x)\right )^{3/2} \left (\sqrt {-3+\cosh (2 x)} \coth (x) \text {csch}(x)+2 \sqrt {2} \left (2 \text {ArcTan}\left (\frac {\sqrt {2} \cosh (x)}{\sqrt {-3+\cosh (2 x)}}\right )+\log \left (\sqrt {2} \cosh (x)+\sqrt {-3+\cosh (2 x)}\right )\right )\right ) \sinh ^3(x)}{(-3+\cosh (2 x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 1.57, size = 0, normalized size = 0.00 \[\int \left (1-\mathrm {csch}\left (x \right )^{2}\right )^{\frac {3}{2}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 528 vs.
\(2 (39) = 78\).
time = 0.37, size = 528, normalized size = 11.48 \begin {gather*} \frac {\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} + \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} + 1\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 8 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \arctan \left (-\frac {1}{2} \, \cosh \left (x\right )^{2} - \cosh \left (x\right ) \sinh \left (x\right ) - \frac {1}{2} \, \sinh \left (x\right )^{2} + \frac {1}{2} \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + \frac {1}{2}\right ) - 2 \, \cosh \left (x\right )^{2} - {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 2\right )} \sinh \left (x\right )^{2} - \sqrt {2} {\left (\cosh \left (x\right )^{2} + 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2} - 1\right )} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} - 4 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - 2 \, \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1\right ) + {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )} \log \left (-\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) - \sinh \left (x\right )^{2} + \sqrt {2} \sqrt {\frac {\cosh \left (x\right )^{2} + \sinh \left (x\right )^{2} - 3}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}} + 1\right ) + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1}{2 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} - 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} - \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \end {gather*}
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 \begin {gather*} \int \left (1 - \operatorname {csch}^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 253 vs.
\(2 (39) = 78\).
time = 0.42, size = 253, normalized size = 5.50 \begin {gather*} -4 \, \arctan \left (\frac {1}{2} \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - \frac {1}{2} \, e^{\left (2 \, x\right )} + \frac {1}{2}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {1}{2} \, \log \left (-\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) - \frac {1}{2} \, \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 3 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {1}{2} \, \log \left ({\left | \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1 \right |}\right ) \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \frac {16 \, {\left ({\left (\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + 2 \, {\left (\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )} \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right ) + \mathrm {sgn}\left (e^{\left (2 \, x\right )} - 1\right )\right )}}{{\left ({\left (\sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} + 2 \, \sqrt {e^{\left (4 \, x\right )} - 6 \, e^{\left (2 \, x\right )} + 1} - 2 \, e^{\left (2 \, x\right )} + 5\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (1-\frac {1}{{\mathrm {sinh}\left (x\right )}^2}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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