Optimal. Leaf size=116 \[ -\frac {1}{2} \text {Li}_3\left (1-\frac {2 i}{a x+i}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2 a x}{a x+i}\right )-i \text {Li}_2\left (1-\frac {2 i}{a x+i}\right ) \cot ^{-1}(a x)+i \text {Li}_2\left (1-\frac {2 a x}{a x+i}\right ) \cot ^{-1}(a x)+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.21, antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4851, 4989, 4885, 4993, 6610} \[ -\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2 i}{a x+i}\right )+\frac {1}{2} \text {PolyLog}\left (3,1-\frac {2 a x}{a x+i}\right )-i \cot ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2 i}{a x+i}\right )+i \cot ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2 a x}{a x+i}\right )+2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4851
Rule 4885
Rule 4989
Rule 4993
Rule 6610
Rubi steps
\begin {align*} \int \frac {\cot ^{-1}(a x)^2}{x} \, dx &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )+(4 a) \int \frac {\cot ^{-1}(a x) \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-(2 a) \int \frac {\cot ^{-1}(a x) \log \left (\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(2 a) \int \frac {\cot ^{-1}(a x) \log \left (\frac {2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-i \cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2 i}{i+a x}\right )+i \cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )-(i a) \int \frac {\text {Li}_2\left (1-\frac {2 i}{i+a x}\right )}{1+a^2 x^2} \, dx+(i a) \int \frac {\text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )}{1+a^2 x^2} \, dx\\ &=2 \cot ^{-1}(a x)^2 \coth ^{-1}\left (1-\frac {2}{1+i a x}\right )-i \cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2 i}{i+a x}\right )+i \cot ^{-1}(a x) \text {Li}_2\left (1-\frac {2 a x}{i+a x}\right )-\frac {1}{2} \text {Li}_3\left (1-\frac {2 i}{i+a x}\right )+\frac {1}{2} \text {Li}_3\left (1-\frac {2 a x}{i+a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 132, normalized size = 1.14 \[ -i \cot ^{-1}(a x) \text {Li}_2\left (e^{-2 i \cot ^{-1}(a x)}\right )-i \cot ^{-1}(a x) \text {Li}_2\left (-e^{2 i \cot ^{-1}(a x)}\right )-\frac {1}{2} \text {Li}_3\left (e^{-2 i \cot ^{-1}(a x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{2 i \cot ^{-1}(a x)}\right )-\frac {2}{3} i \cot ^{-1}(a x)^3-\cot ^{-1}(a x)^2 \log \left (1-e^{-2 i \cot ^{-1}(a x)}\right )+\cot ^{-1}(a x)^2 \log \left (1+e^{2 i \cot ^{-1}(a x)}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arccot}\left (a x\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.88, size = 959, normalized size = 8.27 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arccot}\left (a x\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acot}\left (a\,x\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acot}^{2}{\left (a x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________