Optimal. Leaf size=354 \[ \frac{3}{2} a c^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{20} a^3 c^3 x^2-a c^3 \log \left (a^2 x^2+1\right )+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{x}+\frac{33}{5} a c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
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Rubi [A] time = 1.28021, antiderivative size = 354, normalized size of antiderivative = 1., number of steps used = 45, number of rules used = 15, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.682, Rules used = {4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4924, 4868, 4992, 4916, 260, 266, 43} \[ \frac{3}{2} a c^3 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{20} a^3 c^3 x^2-a c^3 \log \left (a^2 x^2+1\right )+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2+3 a^2 c^3 x \tan ^{-1}(a x)^3+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{c^3 \tan ^{-1}(a x)^3}{x}+\frac{33}{5} a c^3 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^3 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
Antiderivative was successfully verified.
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Rule 4948
Rule 4846
Rule 4920
Rule 4854
Rule 4884
Rule 4994
Rule 6610
Rule 4852
Rule 4924
Rule 4868
Rule 4992
Rule 4916
Rule 260
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^3 \tan ^{-1}(a x)^3}{x^2} \, dx &=\int \left (3 a^2 c^3 \tan ^{-1}(a x)^3+\frac{c^3 \tan ^{-1}(a x)^3}{x^2}+3 a^4 c^3 x^2 \tan ^{-1}(a x)^3+a^6 c^3 x^4 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^3 \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx+\left (3 a^2 c^3\right ) \int \tan ^{-1}(a x)^3 \, dx+\left (3 a^4 c^3\right ) \int x^2 \tan ^{-1}(a x)^3 \, dx+\left (a^6 c^3\right ) \int x^4 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\left (3 a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (9 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\left (3 a^5 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^7 c^3\right ) \int \frac{x^5 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=2 i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\left (3 i a c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx+\left (9 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (3 a^3 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (3 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^5 c^3\right ) \int x^3 \tan ^{-1}(a x)^2 \, dx+\frac{1}{5} \left (3 a^5 c^3\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{3}{2} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+9 a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (18 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^3 c^3\right ) \int x \tan ^{-1}(a x)^2 \, dx-\frac{1}{5} \left (3 a^3 c^3\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx+\left (3 a^4 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^6 c^3\right ) \int \frac{x^4 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+6 a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+9 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (9 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx+\left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx-\left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^4 c^3\right ) \int x^2 \tan ^{-1}(a x) \, dx-\frac{1}{10} \left (3 a^4 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^4 c^3\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=3 a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{3}{2} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+6 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{9}{2} a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\left (3 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\frac{1}{10} \left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{10} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (3 a^2 c^3\right ) \int \tan ^{-1}(a x) \, dx+\frac{1}{5} \left (3 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx-\frac{1}{5} \left (6 a^2 c^3\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{10} \left (a^5 c^3\right ) \int \frac{x^3}{1+a^2 x^2} \, dx\\ &=\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-\frac{3}{2} a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+3 a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )-\frac{1}{5} \left (3 i a^2 c^3\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\frac{1}{10} \left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx+\frac{1}{5} \left (3 a^3 c^3\right ) \int \frac{x}{1+a^2 x^2} \, dx-\frac{1}{20} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \frac{x}{1+a^2 x} \, dx,x,x^2\right )\\ &=\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-\frac{21}{20} a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )-\frac{1}{20} \left (a^5 c^3\right ) \operatorname{Subst}\left (\int \left (\frac{1}{a^2}-\frac{1}{a^2 \left (1+a^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{20} a^3 c^3 x^2+\frac{21}{10} a^2 c^3 x \tan ^{-1}(a x)+\frac{1}{10} a^4 c^3 x^3 \tan ^{-1}(a x)-\frac{21}{20} a c^3 \tan ^{-1}(a x)^2-\frac{6}{5} a^3 c^3 x^2 \tan ^{-1}(a x)^2-\frac{3}{20} a^5 c^3 x^4 \tan ^{-1}(a x)^2+\frac{6}{5} i a c^3 \tan ^{-1}(a x)^3-\frac{c^3 \tan ^{-1}(a x)^3}{x}+3 a^2 c^3 x \tan ^{-1}(a x)^3+a^4 c^3 x^3 \tan ^{-1}(a x)^3+\frac{1}{5} a^6 c^3 x^5 \tan ^{-1}(a x)^3+\frac{33}{5} a c^3 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-a c^3 \log \left (1+a^2 x^2\right )+3 a c^3 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{5} i a c^3 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^3 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{33}{10} a c^3 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.72241, size = 298, normalized size = 0.84 \[ \frac{c^3 \left (120 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-264 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+60 a x \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+132 a x \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-2 a^3 x^3-40 a x \log \left (a^2 x^2+1\right )+8 a^6 x^6 \tan ^{-1}(a x)^3-6 a^5 x^5 \tan ^{-1}(a x)^2+40 a^4 x^4 \tan ^{-1}(a x)^3+4 a^4 x^4 \tan ^{-1}(a x)-48 a^3 x^3 \tan ^{-1}(a x)^2+120 a^2 x^2 \tan ^{-1}(a x)^3+84 a^2 x^2 \tan ^{-1}(a x)-2 a x-5 i \pi ^3 a x-48 i a x \tan ^{-1}(a x)^3-42 a x \tan ^{-1}(a x)^2-40 \tan ^{-1}(a x)^3+120 a x \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+264 a x \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{40 x} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 10.748, size = 10139, normalized size = 28.6 \begin{align*} \text{output too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}{x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} c^{3} \left (\int 3 a^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int 3 a^{4} x^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int a^{6} x^{4} \operatorname{atan}^{3}{\left (a x \right )}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a^{2} c x^{2} + c\right )}^{3} \arctan \left (a x\right )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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