Integrand size = 11, antiderivative size = 45 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-2 \sqrt {1+\tanh (x)}-\frac {2}{3} (1+\tanh (x))^{3/2} \]
[Out]
Time = 0.04 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {3608, 3559, 3561, 212} \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {\tanh (x)+1}}{\sqrt {2}}\right )-\frac {2}{3} (\tanh (x)+1)^{3/2}-2 \sqrt {\tanh (x)+1} \]
[In]
[Out]
Rule 212
Rule 3559
Rule 3561
Rule 3608
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3} (1+\tanh (x))^{3/2}+\int (1+\tanh (x))^{3/2} \, dx \\ & = -2 \sqrt {1+\tanh (x)}-\frac {2}{3} (1+\tanh (x))^{3/2}+2 \int \sqrt {1+\tanh (x)} \, dx \\ & = -2 \sqrt {1+\tanh (x)}-\frac {2}{3} (1+\tanh (x))^{3/2}+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}{3} (1+\tanh (x))^{3/2} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.87 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {1+\tanh (x)}}{\sqrt {2}}\right )-\frac {2}{3} \sqrt {1+\tanh (x)} (4+\tanh (x)) \]
[In]
[Out]
Time = 0.06 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.78
method | result | size |
derivativedivides | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \left (x \right )}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(35\) |
default | \(2 \,\operatorname {arctanh}\left (\frac {\sqrt {1+\tanh \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {1+\tanh \left (x \right )}-\frac {2 \left (1+\tanh \left (x \right )\right )^{\frac {3}{2}}}{3}\) | \(35\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 252 vs. \(2 (34) = 68\).
Time = 0.27 (sec) , antiderivative size = 252, normalized size of antiderivative = 5.60 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=-\frac {2 \, \sqrt {2} {\left (5 \, \sqrt {2} \cosh \left (x\right )^{3} + 15 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{2} + 5 \, \sqrt {2} \sinh \left (x\right )^{3} + 3 \, {\left (5 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right ) + 3 \, \sqrt {2} \cosh \left (x\right )\right )} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} - 3 \, {\left (\sqrt {2} \cosh \left (x\right )^{4} + 4 \, \sqrt {2} \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sqrt {2} \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \sqrt {2} \cosh \left (x\right )^{2} + \sqrt {2}\right )} \sinh \left (x\right )^{2} + 2 \, \sqrt {2} \cosh \left (x\right )^{2} + 4 \, {\left (\sqrt {2} \cosh \left (x\right )^{3} + \sqrt {2} \cosh \left (x\right )\right )} \sinh \left (x\right ) + \sqrt {2}\right )} \log \left (-2 \, \sqrt {2} \sqrt {\frac {\cosh \left (x\right )}{\cosh \left (x\right ) - \sinh \left (x\right )}} {\left (\cosh \left (x\right ) + \sinh \left (x\right )\right )} - 2 \, \cosh \left (x\right )^{2} - 4 \, \cosh \left (x\right ) \sinh \left (x\right ) - 2 \, \sinh \left (x\right )^{2} - 1\right )}{3 \, {\left (\cosh \left (x\right )^{4} + 4 \, \cosh \left (x\right ) \sinh \left (x\right )^{3} + \sinh \left (x\right )^{4} + 2 \, {\left (3 \, \cosh \left (x\right )^{2} + 1\right )} \sinh \left (x\right )^{2} + 2 \, \cosh \left (x\right )^{2} + 4 \, {\left (\cosh \left (x\right )^{3} + \cosh \left (x\right )\right )} \sinh \left (x\right ) + 1\right )}} \]
[In]
[Out]
Time = 3.90 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.29 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=- \sqrt {2} \left (\log {\left (\sqrt {\tanh {\left (x \right )} + 1} - \sqrt {2} \right )} - \log {\left (\sqrt {\tanh {\left (x \right )} + 1} + \sqrt {2} \right )}\right ) - \frac {2 \left (\tanh {\left (x \right )} + 1\right )^{\frac {3}{2}}}{3} - 2 \sqrt {\tanh {\left (x \right )} + 1} \]
[In]
[Out]
\[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=\int { {\left (\tanh \left (x\right ) + 1\right )}^{\frac {3}{2}} \tanh \left (x\right ) \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (34) = 68\).
Time = 0.29 (sec) , antiderivative size = 96, normalized size of antiderivative = 2.13 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=\frac {1}{3} \, \sqrt {2} {\left (\frac {2 \, {\left (9 \, {\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 12 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 12 \, e^{\left (2 \, x\right )} + 5\right )}}{{\left (\sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{3}} - 3 \, \log \left (-2 \, \sqrt {e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \]
[In]
[Out]
Time = 1.74 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.76 \[ \int \tanh (x) (1+\tanh (x))^{3/2} \, dx=2\,\sqrt {2}\,\mathrm {atanh}\left (\frac {\sqrt {2}\,\sqrt {\mathrm {tanh}\left (x\right )+1}}{2}\right )-2\,\sqrt {\mathrm {tanh}\left (x\right )+1}-\frac {2\,{\left (\mathrm {tanh}\left (x\right )+1\right )}^{3/2}}{3} \]
[In]
[Out]