Optimal. Leaf size=24 \[ \frac {1}{2} \sin ^{-1}(\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {3657, 4122, 195, 216} \[ \frac {1}{2} \sin ^{-1}(\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)} \]
Antiderivative was successfully verified.
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Rule 195
Rule 216
Rule 3657
Rule 4122
Rubi steps
\begin {align*} \int \left (1-\coth ^2(x)\right )^{3/2} \, dx &=\int \left (-\text {csch}^2(x)\right )^{3/2} \, dx\\ &=\operatorname {Subst}\left (\int \sqrt {1-x^2} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)}+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,\coth (x)\right )\\ &=\frac {1}{2} \sin ^{-1}(\coth (x))+\frac {1}{2} \coth (x) \sqrt {-\text {csch}^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 41, normalized size = 1.71 \[ \frac {1}{4} \text {csch}\left (\frac {x}{2}\right ) \sqrt {-\text {csch}^2(x)} \text {sech}\left (\frac {x}{2}\right ) \left (\cosh (x)+\sinh ^2(x) \log \left (\tanh \left (\frac {x}{2}\right )\right )\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 1, normalized size = 0.04 \[ 0 \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [C] time = 0.13, size = 60, normalized size = 2.50 \[ -\frac {1}{4} \, {\left (\frac {4 \, {\left (i \, e^{\left (-x\right )} + i \, e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - i \, \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + i \, \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (-e^{\left (2 \, x\right )} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 21, normalized size = 0.88 \[ \frac {\coth \relax (x ) \sqrt {1-\left (\coth ^{2}\relax (x )\right )}}{2}+\frac {\arcsin \left (\coth \relax (x )\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [C] time = 0.42, size = 49, normalized size = 2.04 \[ \frac {i \, e^{\left (-x\right )} + i \, e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} + \frac {1}{2} i \, \log \left (e^{\left (-x\right )} + 1\right ) - \frac {1}{2} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.21, size = 20, normalized size = 0.83 \[ \frac {\mathrm {asin}\left (\mathrm {coth}\relax (x)\right )}{2}+\frac {\mathrm {coth}\relax (x)\,\sqrt {1-{\mathrm {coth}\relax (x)}^2}}{2} \]
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 \[ \int \left (1 - \coth ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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