Integrand size = 10, antiderivative size = 35 \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=a \coth (x) \log (\cosh (x)) \sqrt {a \tanh ^2(x)}-\frac {1}{2} a \tanh (x) \sqrt {a \tanh ^2(x)} \]
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Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3739, 3554, 3556} \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=a \coth (x) \sqrt {a \tanh ^2(x)} \log (\cosh (x))-\frac {1}{2} a \tanh (x) \sqrt {a \tanh ^2(x)} \]
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Rule 3554
Rule 3556
Rule 3739
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.03 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.77 \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=\frac {1}{2} \coth (x) \left (-1+2 \coth ^2(x) \log (\cosh (x))\right ) \left (a \tanh ^2(x)\right )^{3/2} \]
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Time = 0.09 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.86
method | result | size |
derivativedivides | \(-\frac {\left (a \tanh \left (x \right )^{2}\right )^{\frac {3}{2}} \left (\tanh \left (x \right )^{2}+\ln \left (\tanh \left (x \right )-1\right )+\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )^{3}}\) | \(30\) |
default | \(-\frac {\left (a \tanh \left (x \right )^{2}\right )^{\frac {3}{2}} \left (\tanh \left (x \right )^{2}+\ln \left (\tanh \left (x \right )-1\right )+\ln \left (1+\tanh \left (x \right )\right )\right )}{2 \tanh \left (x \right )^{3}}\) | \(30\) |
risch | \(\frac {a \sqrt {\frac {a \left ({\mathrm e}^{2 x}-1\right )^{2}}{\left (1+{\mathrm e}^{2 x}\right )^{2}}}\, \left ({\mathrm e}^{4 x} \ln \left (1+{\mathrm e}^{2 x}\right )-{\mathrm e}^{4 x} x +2 \,{\mathrm e}^{2 x} \ln \left (1+{\mathrm e}^{2 x}\right )-2 \,{\mathrm e}^{2 x} x +2 \,{\mathrm e}^{2 x}+\ln \left (1+{\mathrm e}^{2 x}\right )-x \right )}{\left ({\mathrm e}^{2 x}-1\right ) \left (1+{\mathrm e}^{2 x}\right )}\) | \(95\) |
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Leaf count of result is larger than twice the leaf count of optimal. 467 vs. \(2 (29) = 58\).
Time = 0.27 (sec) , antiderivative size = 467, normalized size of antiderivative = 13.34 \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=-\frac {{\left (a x \cosh \left (x\right )^{4} + {\left (a x e^{\left (2 \, x\right )} + a x\right )} \sinh \left (x\right )^{4} + 4 \, {\left (a x \cosh \left (x\right ) e^{\left (2 \, x\right )} + a x \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, {\left (a x - a\right )} \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a x \cosh \left (x\right )^{2} + a x + {\left (3 \, a x \cosh \left (x\right )^{2} + a x - a\right )} e^{\left (2 \, x\right )} - a\right )} \sinh \left (x\right )^{2} + a x + {\left (a x \cosh \left (x\right )^{4} + 2 \, {\left (a x - a\right )} \cosh \left (x\right )^{2} + a x\right )} e^{\left (2 \, x\right )} - {\left (a \cosh \left (x\right )^{4} + {\left (a e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{4} + 4 \, {\left (a \cosh \left (x\right ) e^{\left (2 \, x\right )} + a \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 2 \, a \cosh \left (x\right )^{2} + 2 \, {\left (3 \, a \cosh \left (x\right )^{2} + {\left (3 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + a\right )} \sinh \left (x\right )^{2} + {\left (a \cosh \left (x\right )^{4} + 2 \, a \cosh \left (x\right )^{2} + a\right )} e^{\left (2 \, x\right )} + 4 \, {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right ) + {\left (a \cosh \left (x\right )^{3} + a \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right ) + a\right )} \log \left (\frac {2 \, \cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}\right ) + 4 \, {\left (a x \cosh \left (x\right )^{3} + {\left (a x - a\right )} \cosh \left (x\right ) + {\left (a x \cosh \left (x\right )^{3} + {\left (a x - a\right )} \cosh \left (x\right )\right )} e^{\left (2 \, x\right )}\right )} \sinh \left (x\right )\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 \left (x\right )^{4} - \cosh \left (x\right )^{4} + 4 \, {\left (\cosh \left (x\right ) e^{\left (2 \, x\right )} - \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} - 2 \, {\left (3 \, \cosh \left (x\right )^{2} - {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} + 1\right )} \sinh \left (x\right )^{2} - 2 \, \cosh \left (x\right )^{2} + {\left (\cosh \left (x\right )^{4} + 2 \, \cosh \left (x\right )^{2} + 1\right )} e^{\left (2 \, x\right )} - 4 \, {\left (\cosh \left (x\right )^{3} - {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} e^{\left (2 \, x\right )} + \cosh \left (x\right )\right )} \sinh \left (x\right ) - 1} \]
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\[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=\int \left (a \tanh ^{2}{\left (x \right )}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.20 \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=-a^{\frac {3}{2}} x - a^{\frac {3}{2}} \log \left (e^{\left (-2 \, x\right )} + 1\right ) - \frac {2 \, a^{\frac {3}{2}} e^{\left (-2 \, x\right )}}{2 \, e^{\left (-2 \, x\right )} + e^{\left (-4 \, x\right )} + 1} \]
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Time = 0.27 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.49 \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=-{\left (x \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \log \left (e^{\left (2 \, x\right )} + 1\right ) \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right ) - \frac {2 \, e^{\left (2 \, x\right )} \mathrm {sgn}\left (e^{\left (4 \, x\right )} - 1\right )}{{\left (e^{\left (2 \, x\right )} + 1\right )}^{2}}\right )} a^{\frac {3}{2}} \]
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Timed out. \[ \int \left (a \tanh ^2(x)\right )^{3/2} \, dx=\int {\left (a\,{\mathrm {tanh}\left (x\right )}^2\right )}^{3/2} \,d x \]
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