Optimal. Leaf size=169 \[ \frac {1}{2} c \log \left (a^2 x^2+1\right )+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^2-\frac {1}{2} c \text {Li}_3\left (1-\frac {2}{i a x+1}\right )+\frac {1}{2} c \text {Li}_3\left (\frac {2}{i a x+1}-1\right )-i c \text {Li}_2\left (1-\frac {2}{i a x+1}\right ) \tan ^{-1}(a x)+i c \text {Li}_2\left (\frac {2}{i a x+1}-1\right ) \tan ^{-1}(a x)+\frac {1}{2} c \tan ^{-1}(a x)^2-a c x \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.31, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {4950, 4850, 4988, 4884, 4994, 6610, 4852, 4916, 4846, 260} \[ -\frac {1}{2} c \text {PolyLog}\left (3,1-\frac {2}{1+i a x}\right )+\frac {1}{2} c \text {PolyLog}\left (3,-1+\frac {2}{1+i a x}\right )-i c \tan ^{-1}(a x) \text {PolyLog}\left (2,1-\frac {2}{1+i a x}\right )+i c \tan ^{-1}(a x) \text {PolyLog}\left (2,-1+\frac {2}{1+i a x}\right )+\frac {1}{2} c \log \left (a^2 x^2+1\right )+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+\frac {1}{2} c \tan ^{-1}(a x)^2-a c x \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 260
Rule 4846
Rule 4850
Rule 4852
Rule 4884
Rule 4916
Rule 4950
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^2}{x} \, dx &=c \int \frac {\tan ^{-1}(a x)^2}{x} \, dx+\left (a^2 c\right ) \int x \tan ^{-1}(a x)^2 \, dx\\ &=\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-(4 a c) \int \frac {\tan ^{-1}(a x) \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^3 c\right ) \int \frac {x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-(a c) \int \tan ^{-1}(a x) \, dx+(a c) \int \frac {\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+(2 a c) \int \frac {\tan ^{-1}(a x) \log \left (\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(2 a c) \int \frac {\tan ^{-1}(a x) \log \left (2-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-a c x \tan ^{-1}(a x)+\frac {1}{2} c \tan ^{-1}(a x)^2+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )-i c \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )+(i a c) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx-(i a c) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^2 c\right ) \int \frac {x}{1+a^2 x^2} \, dx\\ &=-a c x \tan ^{-1}(a x)+\frac {1}{2} c \tan ^{-1}(a x)^2+\frac {1}{2} a^2 c x^2 \tan ^{-1}(a x)^2+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c \log \left (1+a^2 x^2\right )-i c \tan ^{-1}(a x) \text {Li}_2\left (1-\frac {2}{1+i a x}\right )+i c \tan ^{-1}(a x) \text {Li}_2\left (-1+\frac {2}{1+i a x}\right )-\frac {1}{2} c \text {Li}_3\left (1-\frac {2}{1+i a x}\right )+\frac {1}{2} c \text {Li}_3\left (-1+\frac {2}{1+i a x}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.05, size = 177, normalized size = 1.05 \[ \frac {1}{2} c \log \left (a^2 x^2+1\right )+\frac {1}{2} c \left (a^2 x^2+1\right ) \tan ^{-1}(a x)^2+\frac {1}{2} c \text {Li}_3\left (\frac {-a x-i}{a x-i}\right )-\frac {1}{2} c \text {Li}_3\left (\frac {a x+i}{a x-i}\right )+i c \text {Li}_2\left (\frac {-a x-i}{a x-i}\right ) \tan ^{-1}(a x)-i c \text {Li}_2\left (\frac {a x+i}{a x-i}\right ) \tan ^{-1}(a x)-a c x \tan ^{-1}(a x)+2 c \tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac {2 i}{-a x+i}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 3.75, size = 1078, normalized size = 6.38 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {1}{8} \, a^{2} c x^{2} \arctan \left (a x\right )^{2} - \frac {1}{32} \, a^{2} c x^{2} \log \left (a^{2} x^{2} + 1\right )^{2} + 12 \, a^{4} c \int \frac {x^{4} \arctan \left (a x\right )^{2}}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + a^{4} c \int \frac {x^{4} \log \left (a^{2} x^{2} + 1\right )^{2}}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 2 \, a^{4} c \int \frac {x^{4} \log \left (a^{2} x^{2} + 1\right )}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} - 4 \, a^{3} c \int \frac {x^{3} \arctan \left (a x\right )}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + 24 \, a^{2} c \int \frac {x^{2} \arctan \left (a x\right )^{2}}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + \frac {1}{48} \, c \log \left (a^{2} x^{2} + 1\right )^{3} + 12 \, c \int \frac {\arctan \left (a x\right )^{2}}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} + c \int \frac {\log \left (a^{2} x^{2} + 1\right )^{2}}{16 \, {\left (a^{2} x^{3} + x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {atan}\left (a\,x\right )}^2\,\left (c\,a^2\,x^2+c\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ c \left (\int \frac {\operatorname {atan}^{2}{\left (a x \right )}}{x}\, dx + \int a^{2} x \operatorname {atan}^{2}{\left (a x \right )}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________