Optimal. Leaf size=47 \[ -\frac {1}{2} \coth (x) \sqrt {\coth ^2(x)-2}+\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )+2 \tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]
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Rubi [A] time = 0.04, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {4128, 416, 523, 217, 206, 377, 203} \[ -\frac {1}{2} \coth (x) \sqrt {\coth ^2(x)-2}+\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right )+2 \tanh ^{-1}\left (\frac {\coth (x)}{\sqrt {\coth ^2(x)-2}}\right ) \]
Antiderivative was successfully verified.
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Rule 203
Rule 206
Rule 217
Rule 377
Rule 416
Rule 523
Rule 4128
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 )\\ &=-\frac {1}{2} \coth (x) \sqrt {-2+\coth ^2(x)}+2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )+\operatorname {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )\\ &=\tan ^{-1}\left (\frac {\coth (x)}{\sqrt {-2+\coth ^2(x)}}\right )+2 \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 = 0.10, size = 90, normalized size = 1.91 \[ \frac {\sinh ^3(x) \left (\text {csch}^2(x)-1\right )^{3/2} \left (2 \sqrt {2} \left (\log \left (\sqrt {2} \cosh (x)+\sqrt {\cosh (2 x)-3}\right )+2 \tan ^{-1}\left (\frac {\sqrt {2} \cosh (x)}{\sqrt {\cosh (2 x)-3}}\right )\right )+\sqrt {\cosh (2 x)-3} \coth (x) \text {csch}(x)\right )}{(\cosh (2 x)-3)^{3/2}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 666, normalized size = 14.17 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 412, normalized size = 8.77 \[ -\frac {1}{2} \, \arcsin \left (\frac {1}{4} \, \sqrt {2} {\left (e^{\left (2 \, x\right )} - 3\right )}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) - \arctan \left (-2 \, \sqrt {2} - \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} + 1 \right |}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) - 2 \, \log \left ({\left | -\sqrt {2} - \frac {2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}}{e^{\left (2 \, x\right )} - 3} - 1 \right |}\right ) \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac {2 \, {\left (\sqrt {2} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) + \frac {5 \, \sqrt {2} {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{2} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{2}} + \frac {3 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{3} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{3}} + \frac {5 \, {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right )}{e^{\left (2 \, x\right )} - 3}\right )}}{{\left (\frac {2 \, \sqrt {2} {\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}}{e^{\left (2 \, x\right )} - 3} + \frac {{\left (2 \, \sqrt {2} - \sqrt {-e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\right )}^{2}}{{\left (e^{\left (2 \, x\right )} - 3\right )}^{2}} + 1\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.35, size = 0, normalized size = 0.00 \[ \int \left (-1+\mathrm {csch}\relax (x )^{2}\right )^{\frac {3}{2}}\, dx \]
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 \[ \int {\left (\operatorname {csch}\relax (x)^{2} - 1\right )}^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\left (\frac {1}{{\mathrm {sinh}\relax (x)}^2}-1\right )}^{3/2} \,d x \]
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 \[ \int \left (\operatorname {csch}^{2}{\relax (x )} - 1\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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