Optimal. Leaf size=66 \[ -i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.105241, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {4853, 4925, 4869, 2447} \[ -i a \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \log \left (2-\frac{2}{1-i a x}\right ) \cot ^{-1}(a x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4853
Rule 4925
Rule 4869
Rule 2447
Rubi steps
\begin{align*} \int \frac{\cot ^{-1}(a x)^2}{x^2} \, dx &=-\frac{\cot ^{-1}(a x)^2}{x}-(2 a) \int \frac{\cot ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-(2 i a) \int \frac{\cot ^{-1}(a x)}{x (i+a x)} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\left (2 a^2\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-i a \cot ^{-1}(a x)^2-\frac{\cot ^{-1}(a x)^2}{x}-2 a \cot ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-i a \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.0439344, size = 64, normalized size = 0.97 \[ a \left (i \text{PolyLog}\left (2,-e^{2 i \cot ^{-1}(a x)}\right )-\frac{\cot ^{-1}(a x)^2}{a x}+i \cot ^{-1}(a x)^2-2 \cot ^{-1}(a x) \log \left (1+e^{2 i \cot ^{-1}(a x)}\right )\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.138, size = 234, normalized size = 3.6 \begin{align*} -{\frac{ \left ({\rm arccot} \left (ax\right ) \right ) ^{2}}{x}}+a{\rm arccot} \left (ax\right )\ln \left ({a}^{2}{x}^{2}+1 \right ) -2\,a\ln \left ( ax \right ){\rm arccot} \left (ax\right )+ia\ln \left ( ax \right ) \ln \left ( 1+iax \right ) -ia\ln \left ( ax \right ) \ln \left ( 1-iax \right ) +ia{\it dilog} \left ( 1+iax \right ) -ia{\it dilog} \left ( 1-iax \right ) +{\frac{i}{4}}a \left ( \ln \left ( ax-i \right ) \right ) ^{2}+{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{i}{2}}a\ln \left ( ax-i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) +{\frac{i}{2}}a{\it dilog} \left ( -{\frac{i}{2}} \left ( ax+i \right ) \right ) -{\frac{i}{4}}a \left ( \ln \left ( ax+i \right ) \right ) ^{2}-{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) +{\frac{i}{2}}a\ln \left ( ax+i \right ) \ln \left ({a}^{2}{x}^{2}+1 \right ) -{\frac{i}{2}}a{\it dilog} \left ({\frac{i}{2}} \left ( ax-i \right ) \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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{\operatorname{arccot}\left (a x\right )^{2}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{acot}^{2}{\left (a x \right )}}{x^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arccot}\left (a x\right )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]