Integrand size = 12, antiderivative size = 3 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\arctan (\log (x)) \]
[Out]
Time = 0.02 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {209} \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\arctan (\log (x)) \]
[In]
[Out]
Rule 209
Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\log (x)\right ) \\ & = \arctan (\log (x)) \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\arctan (\log (x)) \]
[In]
[Out]
Time = 0.10 (sec) , antiderivative size = 4, normalized size of antiderivative = 1.33
method | result | size |
derivativedivides | \(\arctan \left (\ln \left (x \right )\right )\) | \(4\) |
default | \(\arctan \left (\ln \left (x \right )\right )\) | \(4\) |
risch | \(\frac {i \ln \left (\ln \left (x \right )+i\right )}{2}-\frac {i \ln \left (\ln \left (x \right )-i\right )}{2}\) | \(20\) |
parallelrisch | \(\frac {i \ln \left (\ln \left (x \right )+i\right )}{2}-\frac {i \ln \left (\ln \left (x \right )-i\right )}{2}\) | \(20\) |
[In]
[Out]
none
Time = 0.27 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\arctan \left (\log \left (x\right )\right ) \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 15 vs. \(2 (3) = 6\).
Time = 0.07 (sec) , antiderivative size = 15, normalized size of antiderivative = 5.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\operatorname {RootSum} {\left (4 z^{2} + 1, \left ( i \mapsto i \log {\left (2 i + \log {\left (x \right )} \right )} \right )\right )} \]
[In]
[Out]
none
Time = 0.28 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\arctan \left (\log \left (x\right )\right ) \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\arctan \left (\log \left (x\right )\right ) \]
[In]
[Out]
Time = 0.41 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \left (1+\log ^2(x)\right )} \, dx=\mathrm {atan}\left (\ln \left (x\right )\right ) \]
[In]
[Out]