Optimal. Leaf size=46 \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)} \]
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Rubi [A] time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4122, 195, 217, 206} \[ \frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)} \]
Antiderivative was successfully verified.
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Rule 195
Rule 206
Rule 217
Rule 4122
Rubi steps
\begin {align*} \int \left (a \text {csch}^2(x)\right )^{3/2} \, dx &=-\left (a \operatorname {Subst}\left (\int \sqrt {-a+a x^2} \, dx,x,\coth (x)\right )\right )\\ &=-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {-a+a x^2}} \, dx,x,\coth (x)\right )\\ &=-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)}+\frac {1}{2} a^2 \operatorname {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )\\ &=\frac {1}{2} a^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a} \coth (x)}{\sqrt {a \text {csch}^2(x)}}\right )-\frac {1}{2} a \coth (x) \sqrt {a \text {csch}^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 30, normalized size = 0.65 \[ -\frac {1}{2} a \sinh (x) \sqrt {a \text {csch}^2(x)} \left (\log \left (\tanh \left (\frac {x}{2}\right )\right )+\coth (x) \text {csch}(x)\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.57, size = 340, normalized size = 7.39 \[ \frac {{\left (2 \, a \cosh \relax (x)^{3} - 2 \, {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)^{3} - 6 \, {\left (a \cosh \relax (x) e^{\left (2 \, x\right )} - a \cosh \relax (x)\right )} \sinh \relax (x)^{2} + 2 \, a \cosh \relax (x) - 2 \, {\left (a \cosh \relax (x)^{3} + a \cosh \relax (x)\right )} e^{\left (2 \, x\right )} - {\left (a \cosh \relax (x)^{4} - {\left (a e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)^{4} - 4 \, {\left (a \cosh \relax (x) e^{\left (2 \, x\right )} - a \cosh \relax (x)\right )} \sinh \relax (x)^{3} - 2 \, a \cosh \relax (x)^{2} + 2 \, {\left (3 \, a \cosh \relax (x)^{2} - {\left (3 \, a \cosh \relax (x)^{2} - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)^{2} - {\left (a \cosh \relax (x)^{4} - 2 \, a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a \cosh \relax (x)^{3} - a \cosh \relax (x) - {\left (a \cosh \relax (x)^{3} - a \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x) + a\right )} \log \left (\frac {\cosh \relax (x) + \sinh \relax (x) + 1}{\cosh \relax (x) + \sinh \relax (x) - 1}\right ) + 2 \, {\left (3 \, a \cosh \relax (x)^{2} - {\left (3 \, a \cosh \relax (x)^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \relax (x)\right )} \sqrt {\frac {a}{e^{\left (4 \, x\right )} - 2 \, e^{\left (2 \, x\right )} + 1}} e^{x}}{2 \, {\left (4 \, \cosh \relax (x) e^{x} \sinh \relax (x)^{3} + e^{x} \sinh \relax (x)^{4} + 2 \, {\left (3 \, \cosh \relax (x)^{2} - 1\right )} e^{x} \sinh \relax (x)^{2} + 4 \, {\left (\cosh \relax (x)^{3} - \cosh \relax (x)\right )} e^{x} \sinh \relax (x) + {\left (\cosh \relax (x)^{4} - 2 \, \cosh \relax (x)^{2} + 1\right )} e^{x}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 58, normalized size = 1.26 \[ -\frac {1}{4} \, a^{\frac {3}{2}} {\left (\frac {4 \, {\left (e^{\left (-x\right )} + e^{x}\right )}}{{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4} - \log \left (e^{\left (-x\right )} + e^{x} + 2\right ) + \log \left (e^{\left (-x\right )} + e^{x} - 2\right )\right )} \mathrm {sgn}\left (e^{\left (3 \, x\right )} - e^{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 103, normalized size = 2.24 \[ -\frac {a \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \left (1+{\mathrm e}^{2 x}\right )}{{\mathrm e}^{2 x}-1}-\frac {a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}-1\right )}{2}+\frac {a \,{\mathrm e}^{-x} \left ({\mathrm e}^{2 x}-1\right ) \sqrt {\frac {a \,{\mathrm e}^{2 x}}{\left ({\mathrm e}^{2 x}-1\right )^{2}}}\, \ln \left ({\mathrm e}^{x}+1\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 60, normalized size = 1.30 \[ -\frac {1}{2} \, a^{\frac {3}{2}} \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{2} \, a^{\frac {3}{2}} \log \left (e^{\left (-x\right )} - 1\right ) - \frac {a^{\frac {3}{2}} e^{\left (-x\right )} + a^{\frac {3}{2}} e^{\left (-3 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} - e^{\left (-4 \, x\right )} - 1} \]
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 \[ \int {\left (\frac {a}{{\mathrm {sinh}\relax (x)}^2}\right )}^{3/2} \,d x \]
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 \[ \int \left (a \operatorname {csch}^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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