Optimal. Leaf size=35 \[ \frac {i}{2 x^2}-i e^{-2 i a} \log \left (1+\frac {e^{2 i a}}{x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\tan (a+i \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\tan (a+i \log (x))}{x^3} \, dx &=\int \frac {\tan (a+i \log (x))}{x^3} \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 0.04, size = 132, normalized size = 3.77 \[ \cos (2 a) \left (-\tan ^{-1}\left (\frac {\left (x^2+1\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right )\right )+i \sin (2 a) \tan ^{-1}\left (\frac {\left (x^2+1\right ) \cos (a)}{\sin (a)-x^2 \sin (a)}\right )-\frac {1}{2} i \cos (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )-\frac {1}{2} \sin (2 a) \log \left (2 x^2 \cos (2 a)+x^4+1\right )+2 \sin (2 a) \log (x)+2 i \cos (2 a) \log (x)+\frac {i}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.46, size = 37, normalized size = 1.06 \[ \frac {{\left (-2 i \, x^{2} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) + 4 i \, x^{2} \log \relax (x) + i \, e^{\left (2 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.53, size = 33, normalized size = 0.94 \[ -i \, e^{\left (-2 i \, a\right )} \log \left (-i \, x^{2} - i \, e^{\left (2 i \, a\right )}\right ) + 2 i \, e^{\left (-2 i \, a\right )} \log \relax (x) + \frac {i}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 36, normalized size = 1.03 \[ \frac {i}{2 x^{2}}+2 i {\mathrm e}^{-2 i a} \ln \relax (x )-i {\mathrm e}^{-2 i a} \ln \left ({\mathrm e}^{2 i a}+x^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.36, size = 96, normalized size = 2.74 \[ -\frac {x^{2} {\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{4} + 2 \, x^{2} \cos \left (2 \, a\right ) + \cos \left (2 \, a\right )^{2} + \sin \left (2 \, a\right )^{2}\right ) - {\left ({\left (2 \, \cos \left (2 \, a\right ) - 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (2 \, a\right ), x^{2} + \cos \left (2 \, a\right )\right ) + 4 \, {\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \relax (x)\right )} x^{2} - i}{2 \, x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.29, size = 35, normalized size = 1.00 \[ -{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\ln \left (x^2+{\mathrm {e}}^{a\,2{}\mathrm {i}}\right )\,1{}\mathrm {i}+{\mathrm {e}}^{-a\,2{}\mathrm {i}}\,\ln \relax (x)\,2{}\mathrm {i}+\frac {1{}\mathrm {i}}{2\,x^2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.35, size = 39, normalized size = 1.11 \[ 2 i e^{- 2 i a} \log {\relax (x )} - i e^{- 2 i a} \log {\left (x^{2} + e^{2 i a} \right )} + \frac {i}{2 x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________