Integrand size = 13, antiderivative size = 34 \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-\frac {2}{3} (1+\tanh (x))^{3/2} \]
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Time = 0.03 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {3624, 3561, 212} \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-\frac {2}{3} (\tanh (x)+1)^{3/2} \]
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Rule 212
Rule 3561
Rule 3624
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3} (1+\tanh (x))^{3/2}+\int \sqrt {1+\tanh (x)} \, dx \\ & = -\frac {2}{3} (1+\tanh (x))^{3/2}+2 \text {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\sqrt {1+\tanh (x)}\right ) \\ & = \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-\frac {2}{3} (1+\tanh (x))^{3/2} \\ \end{align*}
Time = 0.42 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=\sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-\frac {2}{3} (1+\tanh (x))^{3/2} \]
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Time = 0.07 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76
method | result | size |
derivativedivides | \(\operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(26\) |
default | \(\operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(26\) |
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Leaf count of result is larger than twice the leaf count of optimal. 237 vs. \(2 (25) = 50\).
Time = 0.26 (sec) , antiderivative size = 237, normalized size of antiderivative = 6.97 \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=-\frac {8 \, \sqrt {2} {\left (\sqrt {2} \cosh \left (x\right )^{3} + 3 \, \sqrt {2} \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + \sqrt {2} \sinh \left (x\right )^{3}\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )}{6 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
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Time = 1.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.44 \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=- \frac {\sqrt {2} \left (\log {\left (\sqrt {\tanh {\left (x \right )} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {\tanh {\left (x \right )} + 1} + \sqrt {2} \right )}\right )}{2} - \frac {2 \left (\tanh {\left (x \right )} + 1\right )^{\frac {3}{2}}}{3} \]
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\[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=\int { \sqrt {\tanh \left (x\right ) + 1} \tanh \left (x\right )^{2} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (25) = 50\).
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.82 \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=\frac {1}{6} \, \sqrt {2} {\left (\frac {8 \, {\left (3 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 3 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 3 \, e^{\left (2 \, x\right )} + 1\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{3}} - 3 \, \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \]
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Time = 0.10 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int \tanh ^2(x) \sqrt {1+\tanh (x)} \, dx=\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )-\frac {2\,{\left (\mathrm {tanh}\left (x\right )+1\right )}^{3/2}}{3} \]
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