Optimal. Leaf size=310 \[ -\frac{3}{2} i a^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{3}{4} i a^2 c \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i a^2 c \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2+3 a^2 c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c \tan ^{-1}(a x)^2}{2 x} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.556854, antiderivative size = 310, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {4950, 4852, 4918, 4924, 4868, 2447, 4884, 4850, 4988, 4994, 4998, 6610} \[ -\frac{3}{2} i a^2 c \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{3}{4} i a^2 c \text{PolyLog}\left (4,1-\frac{2}{1+i a x}\right )-\frac{3}{4} i a^2 c \text{PolyLog}\left (4,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left (3,-1+\frac{2}{1+i a x}\right )-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2+3 a^2 c \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{c \tan ^{-1}(a x)^3}{2 x^2}-\frac{3 a c \tan ^{-1}(a x)^2}{2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4950
Rule 4852
Rule 4918
Rule 4924
Rule 4868
Rule 2447
Rule 4884
Rule 4850
Rule 4988
Rule 4994
Rule 4998
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right ) \tan ^{-1}(a x)^3}{x^3} \, dx &=c \int \frac{\tan ^{-1}(a x)^3}{x^3} \, dx+\left (a^2 c\right ) \int \frac{\tan ^{-1}(a x)^3}{x} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} (3 a c) \int \frac{\tan ^{-1}(a x)^2}{x^2 \left (1+a^2 x^2\right )} \, dx-\left (6 a^3 c\right ) \int \frac{\tan ^{-1}(a x)^2 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+\frac{1}{2} (3 a c) \int \frac{\tan ^{-1}(a x)^2}{x^2} \, dx-\frac{1}{2} \left (3 a^3 c\right ) \int \frac{\tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (3 a^3 c\right ) \int \frac{\tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c\right ) \int \frac{\tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{3 a c \tan ^{-1}(a x)^2}{2 x}-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )+\left (3 a^2 c\right ) \int \frac{\tan ^{-1}(a x)}{x \left (1+a^2 x^2\right )} \, dx+\left (3 i a^3 c\right ) \int \frac{\tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 i a^3 c\right ) \int \frac{\tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2-\frac{3 a c \tan ^{-1}(a x)^2}{2 x}-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\left (3 i a^2 c\right ) \int \frac{\tan ^{-1}(a x)}{x (i+a x)} \, dx+\frac{1}{2} \left (3 a^3 c\right ) \int \frac{\text{Li}_3\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{2} \left (3 a^3 c\right ) \int \frac{\text{Li}_3\left (-1+\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2-\frac{3 a c \tan ^{-1}(a x)^2}{2 x}-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+3 a^2 c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{4} i a^2 c \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{4} i a^2 c \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )-\left (3 a^3 c\right ) \int \frac{\log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2-\frac{3 a c \tan ^{-1}(a x)^2}{2 x}-\frac{1}{2} a^2 c \tan ^{-1}(a x)^3-\frac{c \tan ^{-1}(a x)^3}{2 x^2}+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2}{1+i a x}\right )+3 a^2 c \tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{Li}_2\left (-1+\frac{2}{1+i a x}\right )-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{Li}_3\left (-1+\frac{2}{1+i a x}\right )+\frac{3}{4} i a^2 c \text{Li}_4\left (1-\frac{2}{1+i a x}\right )-\frac{3}{4} i a^2 c \text{Li}_4\left (-1+\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.240957, size = 337, normalized size = 1.09 \[ -\frac{3}{4} i a^2 c \text{PolyLog}\left (4,\frac{-a x-i}{a x-i}\right )+\frac{3}{4} i a^2 c \text{PolyLog}\left (4,\frac{a x+i}{a x-i}\right )+\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{-a x-i}{a x-i}\right )-\frac{3}{2} i a^2 c \tan ^{-1}(a x)^2 \text{PolyLog}\left (2,\frac{a x+i}{a x-i}\right )+\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left (3,\frac{-a x-i}{a x-i}\right )-\frac{3}{2} a^2 c \tan ^{-1}(a x) \text{PolyLog}\left (3,\frac{a x+i}{a x-i}\right )+\frac{3}{2} a^2 c \left (-i \text{PolyLog}\left (2,e^{2 i \tan ^{-1}(a x)}\right )-\frac{1}{3} \tan ^{-1}(a x) \left (\left (\tan ^{-1}(a x)+3 i\right ) \tan ^{-1}(a x)+\frac{3 \tan ^{-1}(a x)}{a x}-6 \log \left (1-e^{2 i \tan ^{-1}(a x)}\right )\right )\right )+\frac{c \left (-a^2 x^2-1\right ) \tan ^{-1}(a x)^3}{2 x^2}+\frac{1}{2} a^2 c \tan ^{-1}(a x)^3+2 a^2 c \tan ^{-1}(a x)^3 \tanh ^{-1}\left (1-\frac{2 i}{-a x+i}\right ) \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.037, size = 568, normalized size = 1.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{4 \, c \arctan \left (a x\right )^{3} - 3 \, c \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right )^{2} + x^{2} \int \frac{12 \, a^{2} c x^{2} \arctan \left (a x\right ) \log \left (a^{2} x^{2} + 1\right ) - 12 \, a c x \arctan \left (a x\right )^{2} - 56 \,{\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3} + 3 \,{\left (a c x - 2 \,{\left (a^{4} c x^{4} + 2 \, a^{2} c x^{2} + c\right )} \arctan \left (a x\right )\right )} \log \left (a^{2} x^{2} + 1\right )^{2}}{a^{2} x^{5} + x^{3}}\,{d x}}{64 \, 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{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{3}}, 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*} c \left (\int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{x^{3}}\, dx + \int \frac{a^{2} \operatorname{atan}^{3}{\left (a x \right )}}{x}\, dx\right ) \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{{\left (a^{2} c x^{2} + c\right )} \arctan \left (a x\right )^{3}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]