Integrand size = 9, antiderivative size = 23 \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {x^2}{2}-e^{-a} \arctan \left (e^a x^2\right ) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5656, 470, 281, 209} \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {x^2}{2}-e^{-a} \arctan \left (e^a x^2\right ) \]
[In]
[Out]
Rule 209
Rule 281
Rule 470
Rule 5656
Rubi steps \begin{align*} \text {integral}& = \int \frac {x \left (-1+e^{2 a} x^4\right )}{1+e^{2 a} x^4} \, dx \\ & = \frac {x^2}{2}-2 \int \frac {x}{1+e^{2 a} x^4} \, dx \\ & = \frac {x^2}{2}-\text {Subst}\left (\int \frac {1}{1+e^{2 a} x^2} \, dx,x,x^2\right ) \\ & = \frac {x^2}{2}-e^{-a} \arctan \left (e^a x^2\right ) \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.52 \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {x^2}{2}-\arctan \left (x^2 (\cosh (a)+\sinh (a))\right ) \cosh (a)+\arctan \left (x^2 (\cosh (a)+\sinh (a))\right ) \sinh (a) \]
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.09 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.78
method | result | size |
risch | \(\frac {x^{2}}{2}+\frac {i {\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}-i\right )}{2}-\frac {i {\mathrm e}^{-a} \ln \left ({\mathrm e}^{a} x^{2}+i\right )}{2}\) | \(41\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {1}{2} \, {\left (x^{2} e^{a} - 2 \, \arctan \left (x^{2} e^{a}\right )\right )} e^{\left (-a\right )} \]
[In]
[Out]
\[ \int x \tanh (a+2 \log (x)) \, dx=\int x \tanh {\left (a + 2 \log {\left (x \right )} \right )}\, dx \]
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {1}{2} \, x^{2} - \arctan \left (x^{2} e^{a}\right ) e^{\left (-a\right )} \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83 \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {1}{2} \, x^{2} - \arctan \left (x^{2} e^{a}\right ) e^{\left (-a\right )} \]
[In]
[Out]
Time = 1.69 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int x \tanh (a+2 \log (x)) \, dx=\frac {x^2}{2}-\frac {\mathrm {atan}\left (x^2\,\sqrt {{\mathrm {e}}^{2\,a}}\right )}{\sqrt {{\mathrm {e}}^{2\,a}}} \]
[In]
[Out]