Integrand size = 7, antiderivative size = 49 \[ \int \log \left (a \tan ^2(x)\right ) \, dx=4 x \text {arctanh}\left (e^{2 i x}\right )+x \log \left (a \tan ^2(x)\right )-i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
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Time = 0.03 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.857, Rules used = {2628, 12, 4504, 4268, 2317, 2438} \[ \int \log \left (a \tan ^2(x)\right ) \, dx=x \log \left (a \tan ^2(x)\right )+4 x \text {arctanh}\left (e^{2 i x}\right )-i \operatorname {PolyLog}\left (2,-e^{2 i x}\right )+i \operatorname {PolyLog}\left (2,e^{2 i x}\right ) \]
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Rule 12
Rule 2317
Rule 2438
Rule 2628
Rule 4268
Rule 4504
Rubi steps \begin{align*} \text {integral}& = x \log \left (a \tan ^2(x)\right )-\int 2 x \csc (x) \sec (x) \, dx \\ & = x \log \left (a \tan ^2(x)\right )-2 \int x \csc (x) \sec (x) \, dx \\ & = x \log \left (a \tan ^2(x)\right )-4 \int x \csc (2 x) \, dx \\ & = 4 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \tan ^2(x)\right )+2 \int \log \left (1-e^{2 i x}\right ) \, dx-2 \int \log \left (1+e^{2 i x}\right ) \, dx \\ & = 4 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \tan ^2(x)\right )-i \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 i x}\right )+i \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i x}\right ) \\ & = 4 x \tanh ^{-1}\left (e^{2 i x}\right )+x \log \left (a \tan ^2(x)\right )-i \text {Li}_2\left (-e^{2 i x}\right )+i \text {Li}_2\left (e^{2 i x}\right ) \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.53 \[ \int \log \left (a \tan ^2(x)\right ) \, dx=-\frac {1}{2} i \log (-i (i-\tan (x))) \log \left (a \tan ^2(x)\right )+\frac {1}{2} i \log \left (a \tan ^2(x)\right ) \log (-i (i+\tan (x)))-i \operatorname {PolyLog}(2,-i \tan (x))+i \operatorname {PolyLog}(2,i \tan (x)) \]
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Time = 1.14 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.67
method | result | size |
derivativedivides | \(-\frac {i \left (\ln \left (\tan \left (x \right )-i\right ) \ln \left (a \left (\tan ^{2}\left (x \right )\right )\right )-2 \operatorname {dilog}\left (-i \tan \left (x \right )\right )-2 \ln \left (\tan \left (x \right )-i\right ) \ln \left (-i \tan \left (x \right )\right )\right )}{2}+\frac {i \left (\ln \left (\tan \left (x \right )+i\right ) \ln \left (a \left (\tan ^{2}\left (x \right )\right )\right )-2 \operatorname {dilog}\left (i \tan \left (x \right )\right )-2 \ln \left (\tan \left (x \right )+i\right ) \ln \left (i \tan \left (x \right )\right )\right )}{2}\) | \(82\) |
default | \(-\frac {i \left (\ln \left (\tan \left (x \right )-i\right ) \ln \left (a \left (\tan ^{2}\left (x \right )\right )\right )-2 \operatorname {dilog}\left (-i \tan \left (x \right )\right )-2 \ln \left (\tan \left (x \right )-i\right ) \ln \left (-i \tan \left (x \right )\right )\right )}{2}+\frac {i \left (\ln \left (\tan \left (x \right )+i\right ) \ln \left (a \left (\tan ^{2}\left (x \right )\right )\right )-2 \operatorname {dilog}\left (i \tan \left (x \right )\right )-2 \ln \left (\tan \left (x \right )+i\right ) \ln \left (i \tan \left (x \right )\right )\right )}{2}\) | \(82\) |
risch | \(\text {Expression too large to display}\) | \(664\) |
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Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 184 vs. \(2 (34) = 68\).
Time = 0.32 (sec) , antiderivative size = 184, normalized size of antiderivative = 3.76 \[ \int \log \left (a \tan ^2(x)\right ) \, dx=x \log \left (a \tan \left (x\right )^{2}\right ) - x \log \left (\frac {2 \, {\left (\tan \left (x\right )^{2} + i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1}\right ) - x \log \left (\frac {2 \, {\left (\tan \left (x\right )^{2} - i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1}\right ) + x \log \left (-\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) + x \log \left (-\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1}\right ) - \frac {1}{2} i \, {\rm Li}_2\left (-\frac {2 \, {\left (\tan \left (x\right )^{2} + i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (-\frac {2 \, {\left (\tan \left (x\right )^{2} - i \, \tan \left (x\right )\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) + \frac {1}{2} i \, {\rm Li}_2\left (\frac {2 \, {\left (i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) - \frac {1}{2} i \, {\rm Li}_2\left (\frac {2 \, {\left (-i \, \tan \left (x\right ) - 1\right )}}{\tan \left (x\right )^{2} + 1} + 1\right ) \]
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\[ \int \log \left (a \tan ^2(x)\right ) \, dx=\int \log {\left (a \tan ^{2}{\left (x \right )} \right )}\, dx \]
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none
Time = 0.30 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.90 \[ \int \log \left (a \tan ^2(x)\right ) \, dx=x \log \left (a \tan \left (x\right )^{2}\right ) + \frac {1}{2} \, \pi \log \left (\tan \left (x\right )^{2} + 1\right ) - 2 \, x \log \left (\tan \left (x\right )\right ) + i \, {\rm Li}_2\left (i \, \tan \left (x\right ) + 1\right ) - i \, {\rm Li}_2\left (-i \, \tan \left (x\right ) + 1\right ) \]
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\[ \int \log \left (a \tan ^2(x)\right ) \, dx=\int { \log \left (a \tan \left (x\right )^{2}\right ) \,d x } \]
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Time = 0.07 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.84 \[ \int \log \left (a \tan ^2(x)\right ) \, dx=x\,\ln \left (a\,{\mathrm {tan}\left (x\right )}^2\right )-\mathrm {polylog}\left (2,-{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}+4\,x\,\mathrm {atanh}\left ({\mathrm {e}}^{x\,2{}\mathrm {i}}\right )+\mathrm {polylog}\left (2,{\mathrm {e}}^{x\,2{}\mathrm {i}}\right )\,1{}\mathrm {i} \]
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