Optimal. Leaf size=36 \[ -i e^{-2 i a} \log \left (1-\frac{e^{2 i a}}{x^2}\right )-\frac{i}{2 x^2} \]
[Out]
________________________________________________________________________________________
Rubi [F] time = 0.0240748, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\cot (a+i \log (x))}{x^3} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
Rubi steps
\begin{align*} \int \frac{\cot (a+i \log (x))}{x^3} \, dx &=\int \frac{\cot (a+i \log (x))}{x^3} \, dx\\ \end{align*}
Mathematica [B] time = 0.0297716, size = 136, normalized size = 3.78 \[ -\frac{1}{2} i \cos (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )-\frac{1}{2} \sin (2 a) \log \left (-2 x^2 \cos (2 a)+x^4+1\right )+\cos (2 a) \left (-\tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )\right )+i \sin (2 a) \tan ^{-1}\left (\frac{\left (x^2-1\right ) \cos (a)}{x^2 (-\sin (a))-\sin (a)}\right )+2 \sin (2 a) \log (x)+2 i \cos (2 a) \log (x)-\frac{i}{2 x^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.066, size = 59, normalized size = 1.6 \begin{align*}{\frac{-{\frac{i}{2}}}{{x}^{2}}}+i \left ( 2\,{\frac{\ln \left ( x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}-{\frac{\ln \left ({{\rm e}^{ia}}-x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}}-{\frac{\ln \left ({{\rm e}^{ia}}+x \right ) }{ \left ({{\rm e}^{ia}} \right ) ^{2}}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.38878, size = 188, normalized size = 5.22 \begin{align*} -\frac{x^{2}{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} + 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) + x^{2}{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x^{2} - 2 \, x \cos \left (a\right ) + \cos \left (a\right )^{2} + \sin \left (a\right )^{2}\right ) -{\left ({\left (2 \, \cos \left (2 \, a\right ) - 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x + \cos \left (a\right )\right ) -{\left (2 \, \cos \left (2 \, a\right ) - 2 i \, \sin \left (2 \, a\right )\right )} \arctan \left (\sin \left (a\right ), x - \cos \left (a\right )\right ) + 4 \,{\left (i \, \cos \left (2 \, a\right ) + \sin \left (2 \, a\right )\right )} \log \left (x\right )\right )} x^{2} + i}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.697061, size = 39, normalized size = 1.08 \begin{align*} 2 i e^{- 2 i a} \log{\left (x \right )} - i e^{- 2 i a} \log{\left (x^{2} - e^{2 i a} \right )} - \frac{i}{2 x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.3175, size = 66, normalized size = 1.83 \begin{align*} \frac{1}{2} \, \pi e^{\left (-2 i \, a\right )} - i \, e^{\left (-2 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) + 2 i \, e^{\left (-2 i \, a\right )} \log \left (x\right ) - i \, e^{\left (-2 i \, a\right )} \log \left (-x + e^{\left (i \, a\right )}\right ) - \frac{i}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]