Optimal. Leaf size=35 \[ a \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x))-\frac {1}{2} a \tanh (x) \sqrt {a \tanh ^2(x)} \]
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Rubi [A] time = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3658, 3473, 3475} \[ a \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x))-\frac {1}{2} a \tanh (x) \sqrt {a \tanh ^2(x)} \]
Antiderivative was successfully verified.
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Rule 3473
Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \left (a \tanh ^2(x)\right )^{3/2} \, dx &=\left (a \coth (x) \sqrt {a \tanh ^2(x)}\right ) \int \tanh ^3(x) \, dx\\ &=-\frac {1}{2} a \tanh (x) \sqrt {a \tanh ^2(x)}+\left (a \coth (x) \sqrt {a \tanh ^2(x)}\right ) \int \tanh (x) \, dx\\ &=a \coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)}-\frac {1}{2} a \tanh (x) \sqrt {a \tanh ^2(x)}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.80 \[ \frac {1}{2} a \sqrt {a \tanh ^2(x)} (\text {csch}(x) \text {sech}(x)+2 \coth (x) \log (\cosh (x))) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 467, normalized size = 13.34 \[ -\frac {{\left (a x \cosh \relax (x)^{4} + {\left (a x e^{\left (2 \, x\right )} + a x\right )} \sinh \relax (x)^{4} + 4 \, {\left (a x \cosh \relax (x) e^{\left (2 \, x\right )} + a x \cosh \relax (x)\right )} \sinh \relax (x)^{3} + 2 \, {\left (a x - a\right )} \cosh \relax (x)^{2} + 2 \, {\left (3 \, a x \cosh \relax (x)^{2} + a x + {\left (3 \, a x \cosh \relax (x)^{2} + a x - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \relax (x)^{2} + a x + {\left (a x \cosh \relax (x)^{4} + 2 \, {\left (a x - a\right )} \cosh \relax (x)^{2} + a 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 {2 \, \cosh \relax (x)}{\cosh \relax (x) - \sinh \relax (x)}\right ) + 4 \, {\left (a x \cosh \relax (x)^{3} + {\left (a x - a\right )} \cosh \relax (x) + {\left (a x \cosh \relax (x)^{3} + {\left (a x - a\right )} \cosh \relax (x)\right )} e^{\left (2 \, x\right )}\right )} \sinh \relax (x)\right )} \sqrt {\frac {a e^{\left (4 \, x\right )} - 2 \, a e^{\left (2 \, x\right )} + a}{e^{\left (4 \, x\right )} + 2 \, e^{\left (2 \, x\right )} + 1}}}{{\left (e^{\left (2 \, x\right )} - 1\right )} \sinh \relax (x)^{4} - \cosh \relax (x)^{4} + 4 \, {\left (\cosh \relax (x) e^{\left (2 \, x\right )} - \cosh \relax (x)\right )} \sinh \relax (x)^{3} - 2 \, {\left (3 \, \cosh \relax (x)^{2} - {\left (3 \, \cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} + 1\right )} \sinh \relax (x)^{2} - 2 \, \cosh \relax (x)^{2} + {\left (\cosh \relax (x)^{4} + 2 \, \cosh \relax (x)^{2} + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (\cosh \relax (x)^{3} - {\left (\cosh \relax (x)^{3} + \cosh \relax (x)\right )} e^{\left (2 \, x\right )} + \cosh \relax (x)\right )} \sinh \relax (x) - 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 5.01, size = 52, normalized size = 1.49 \[ -{\left (x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}}\right )} a^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 30, normalized size = 0.86 \[ -\frac {\left (a \left (\tanh ^{2}\relax (x )\right )\right )^{\frac {3}{2}} \left (\tanh ^{2}\relax (x )+\ln \left (\tanh \relax (x )-1\right )+\ln \left (1+\tanh \relax (x )\right )\right )}{2 \tanh \relax (x )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 42, normalized size = 1.20 \[ -a^{\frac {3}{2}} x - a^{\frac {3}{2}} \log \left (e^{\left (-2 \, x\right )} + 1\right ) - \frac {2 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int {\left (a\,{\mathrm {tanh}\relax (x)}^2\right )}^{3/2} \,d x \]
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 \[ \int \left (a \tanh ^{2}{\relax (x )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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