Integrand size = 13, antiderivative size = 14 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\log (x)-\frac {1}{2} \tanh (a+2 \log (x)) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {3554, 8} \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\log (x)-\frac {1}{2} \tanh (a+2 \log (x)) \]
[In]
[Out]
Rule 8
Rule 3554
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \tanh ^2(a+2 x) \, dx,x,\log (x)\right ) \\ & = -\frac {1}{2} \tanh (a+2 \log (x))+\text {Subst}(\int 1 \, dx,x,\log (x)) \\ & = \log (x)-\frac {1}{2} \tanh (a+2 \log (x)) \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.71 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\frac {1}{2} \text {arctanh}(\tanh (a+2 \log (x)))-\frac {1}{2} \tanh (a+2 \log (x)) \]
[In]
[Out]
Time = 0.04 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93
method | result | size |
parallelrisch | \(\ln \left (x \right )-\frac {\tanh \left (a +2 \ln \left (x \right )\right )}{2}\) | \(13\) |
risch | \(\frac {1}{1+{\mathrm e}^{2 a} x^{4}}+\ln \left (x \right )\) | \(16\) |
derivativedivides | \(-\frac {\tanh \left (a +2 \ln \left (x \right )\right )}{2}-\frac {\ln \left (\tanh \left (a +2 \ln \left (x \right )\right )-1\right )}{4}+\frac {\ln \left (\tanh \left (a +2 \ln \left (x \right )\right )+1\right )}{4}\) | \(35\) |
default | \(-\frac {\tanh \left (a +2 \ln \left (x \right )\right )}{2}-\frac {\ln \left (\tanh \left (a +2 \ln \left (x \right )\right )-1\right )}{4}+\frac {\ln \left (\tanh \left (a +2 \ln \left (x \right )\right )+1\right )}{4}\) | \(35\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).
Time = 0.24 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\frac {{\left (x^{4} e^{\left (2 \, a\right )} + 1\right )} \log \left (x\right ) + 1}{x^{4} e^{\left (2 \, a\right )} + 1} \]
[In]
[Out]
Time = 0.12 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.86 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\log {\left (x \right )} - \frac {\tanh {\left (a + 2 \log {\left (x \right )} \right )}}{2} \]
[In]
[Out]
none
Time = 0.18 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.50 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\frac {1}{2} \, a - \frac {1}{e^{\left (-2 \, a - 4 \, \log \left (x\right )\right )} + 1} + \log \left (x\right ) \]
[In]
[Out]
none
Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.36 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\frac {1}{x^{4} e^{\left (2 \, a\right )} + 1} + \frac {1}{4} \, \log \left (x^{4}\right ) \]
[In]
[Out]
Time = 1.69 (sec) , antiderivative size = 28, normalized size of antiderivative = 2.00 \[ \int \frac {\tanh ^2(a+2 \log (x))}{x} \, dx=\ln \left (x\right )-\frac {x^4\,{\mathrm {e}}^{2\,a}-1}{2\,\left ({\mathrm {e}}^{2\,a}\,x^4+1\right )} \]
[In]
[Out]