Integrand size = 13, antiderivative size = 45 \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)}-\frac {2}{5} (1+\tanh (x))^{5/2} \]
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Time = 0.05 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {3624, 3559, 3561, 212} \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-\frac {2}{5} (\tanh (x)+1)^{5/2}-2 \sqrt {\tanh (x)+1} \]
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Rule 212
Rule 3559
Rule 3561
Rule 3624
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{5} (1+\tanh (x))^{5/2}+\int (1+\tanh (x))^{3/2} \, dx \\ & = -2 \sqrt {1+\tanh (x)}-\frac {2}{5} (1+\tanh (x))^{5/2}+2 \int \sqrt {1+\tanh (x)} \, dx \\ & = -2 \sqrt {1+\tanh (x)}-\frac {2}{5} (1+\tanh (x))^{5/2}+4 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right ) \\ & = 2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)}-\frac {2}{5} (1+\tanh (x))^{5/2} \\ \end{align*}
Time = 0.67 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00 \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)}-\frac {2}{5} (1+\tanh (x))^{5/2} \]
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Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \left (x \right )}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {5}{2}}}{5}\) | \(35\) |
default | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \left (x \right )}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {5}{2}}}{5}\) | \(35\) |
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Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 429, normalized size of antiderivative = 9.53 \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=-\frac {2 \, \sqrt {2} {\left (9 \, \sqrt {2} \cosh \left (x\right )^{5} + 45 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{4} + 9 \, \sqrt {2} \sinh \left (x\right )^{5} + 10 \, {\left (9 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{3} + 10 \, \sqrt {2} \cosh \left (x\right )^{3} + 30 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{2} + 5 \, {\left (9 \, \sqrt {2} \cosh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right ) + 5 \, \sqrt {2} \cosh \left (x\right )\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 5 \, {\left (\sqrt {2} \cosh \left (x\right )^{6} + 6 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sqrt {2} \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{4} + 3 \, \sqrt {2} \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{4} + 6 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} + 6 \, {\left (\sqrt {2} \cosh \left (x\right )^{5} + 2 \, \sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )}{5 \, {\left (\cosh \left (x\right )^{6} + 6 \, \cosh \left (x\right ) \sinh \left (x\right )^{5} + \sinh \left (x\right )^{6} + 3 \, {\left (5 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{4} + 3 \, \cosh \left (x\right )^{4} + 4 \, {\left (5 \, \cosh \left (x\right )^{3} + 3 \, \cosh \left (x\right )\right )} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \cosh \left (x\right )^{4} + 6 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 3 \, \cosh \left (x\right )^{2} + 6 \, {\left (\cosh \left (x\right )^{5} + 2 \, \cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
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Time = 5.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=- \sqrt {2} \left (\log {\left (\sqrt {\tanh {\left (x \right )} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {\tanh {\left (x \right )} + 1} + \sqrt {2} \right )}\right ) - \frac {2 \left (\tanh {\left (x \right )} + 1\right )^{\frac {5}{2}}}{5} - 2 \sqrt {\tanh {\left (x \right )} + 1} \]
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\[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=\int { {\left (\tanh \left (x\right ) + 1\right )}^{\frac {3}{2}} \tanh \left (x\right )^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 140 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 3.11 \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=\frac {1}{5} \, \sqrt {2} {\left (\frac {2 \, {\left (25 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} - 60 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 70 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 40 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 40 \, e^{\left (2 \, x\right )} + 9\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{5}} - 5 \, \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \]
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Time = 1.76 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx=2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )-2\,\sqrt {\mathrm {tanh}\left (x\right )+1}-\frac {2\,{\left (\mathrm {tanh}\left (x\right )+1\right )}^{5/2}}{5} \]
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