Optimal. Leaf size=11 \[ \left (\log (x)+\log \left (-x^2\right )\right )^2 \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 19, normalized size of antiderivative = 1.73, number of steps
used = 4, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {14, 2338}
\begin {gather*} \frac {3}{2} \log ^2\left (-x^2\right )+3 \log ^2(x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2338
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {6 \log (x)}{x}+\frac {6 \log \left (-x^2\right )}{x}\right ) \, dx\\ &=6 \int \frac {\log (x)}{x} \, dx+6 \int \frac {\log \left (-x^2\right )}{x} \, dx\\ &=3 \log ^2(x)+\frac {3}{2} \log ^2\left (-x^2\right )\\ \end {aligned} \end {gather*}
________________________________________________________________________________________
Mathematica [A]
time = 0.01, size = 19, normalized size = 1.73 \begin {gather*} 3 \log ^2(x)+\frac {3}{2} \log ^2\left (-x^2\right ) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.07, size = 18, normalized size = 1.64
method | result | size |
default | \(\frac {3 \ln \left (-x^{2}\right )^{2}}{2}+3 \ln \left (x \right )^{2}\) | \(18\) |
norman | \(-3 \ln \left (x \right )^{2}+6 \ln \left (x \right ) \ln \left (-x^{2}\right )\) | \(18\) |
risch | \(9 \ln \left (x \right )^{2}-3 i \pi \left (\mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-\mathrm {csgn}\left (i x^{2}\right )^{3}+2 \mathrm {csgn}\left (i x^{2}\right )^{2}-2\right ) \ln \left (x \right )\) | \(69\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 11, normalized size = 1.00 \begin {gather*} {\left (\log \left (-x^{2}\right ) + \log \left (x\right )\right )}^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [C] Result contains complex when optimal does not.
time = 0.39, size = 20, normalized size = 1.82 \begin {gather*} \frac {3}{2} i \, \pi \log \left (-x^{2}\right ) + \frac {9}{4} \, \log \left (-x^{2}\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.04, size = 14, normalized size = 1.27 \begin {gather*} 9 \log {\left (x \right )}^{2} + 6 i \pi \log {\left (x \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [C] Result contains complex when optimal does not.
time = 0.39, size = 12, normalized size = 1.09 \begin {gather*} 6 i \, \pi \log \left (x\right ) + 9 \, \log \left (x\right )^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.03, size = 16, normalized size = 1.45 \begin {gather*} \ln \left (x\right )\,\left (\ln \left (x^{12}\right )-3\,\ln \left (x\right )+\pi \,6{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________