Optimal. Leaf size=284 \[ \frac{3}{2} a c^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{5}{2} a c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{2} a c^2 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{x}+5 a c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
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Rubi [A] time = 0.751286, antiderivative size = 284, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 13, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.591, Rules used = {4948, 4846, 4920, 4854, 4884, 4994, 6610, 4852, 4924, 4868, 4992, 4916, 260} \[ \frac{3}{2} a c^2 \text{PolyLog}\left (3,-1+\frac{2}{1-i a x}\right )+\frac{5}{2} a c^2 \text{PolyLog}\left (3,1-\frac{2}{1+i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,-1+\frac{2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text{PolyLog}\left (2,1-\frac{2}{1+i a x}\right )-\frac{1}{2} a c^2 \log \left (a^2 x^2+1\right )+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+2 a^2 c^2 x \tan ^{-1}(a x)^3+a^2 c^2 x \tan ^{-1}(a x)+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{1}{2} a c^2 \tan ^{-1}(a x)^2-\frac{c^2 \tan ^{-1}(a x)^3}{x}+5 a c^2 \log \left (\frac{2}{1+i a x}\right ) \tan ^{-1}(a x)^2+3 a c^2 \log \left (2-\frac{2}{1-i a x}\right ) \tan ^{-1}(a x)^2 \]
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
Rubi steps
\begin{align*} \int \frac{\left (c+a^2 c x^2\right )^2 \tan ^{-1}(a x)^3}{x^2} \, dx &=\int \left (2 a^2 c^2 \tan ^{-1}(a x)^3+\frac{c^2 \tan ^{-1}(a x)^3}{x^2}+a^4 c^2 x^2 \tan ^{-1}(a x)^3\right ) \, dx\\ &=c^2 \int \frac{\tan ^{-1}(a x)^3}{x^2} \, dx+\left (2 a^2 c^2\right ) \int \tan ^{-1}(a x)^3 \, dx+\left (a^4 c^2\right ) \int x^2 \tan ^{-1}(a x)^3 \, dx\\ &=-\frac{c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+\left (3 a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x \left (1+a^2 x^2\right )} \, dx-\left (6 a^3 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx-\left (a^5 c^2\right ) \int \frac{x^3 \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=i a c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+\left (3 i a c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{x (i+a x)} \, dx+\left (6 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (a^3 c^2\right ) \int x \tan ^{-1}(a x)^2 \, dx+\left (a^3 c^2\right ) \int \frac{x \tan ^{-1}(a x)^2}{1+a^2 x^2} \, dx\\ &=-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+6 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)^2}{i-a x} \, dx-\left (6 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (2-\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (12 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^4 c^2\right ) \int \frac{x^2 \tan ^{-1}(a x)}{1+a^2 x^2} \, dx\\ &=-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+5 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+6 i a c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\left (3 i a^2 c^2\right ) \int \frac{\text{Li}_2\left (-1+\frac{2}{1-i a x}\right )}{1+a^2 x^2} \, dx-\left (6 i a^2 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx+\left (a^2 c^2\right ) \int \tan ^{-1}(a x) \, dx-\left (a^2 c^2\right ) \int \frac{\tan ^{-1}(a x)}{1+a^2 x^2} \, dx+\left (2 a^2 c^2\right ) \int \frac{\tan ^{-1}(a x) \log \left (\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx\\ &=a^2 c^2 x \tan ^{-1}(a x)-\frac{1}{2} a c^2 \tan ^{-1}(a x)^2-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+5 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+3 a c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )+\left (i a^2 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i a x}\right )}{1+a^2 x^2} \, dx-\left (a^3 c^2\right ) \int \frac{x}{1+a^2 x^2} \, dx\\ &=a^2 c^2 x \tan ^{-1}(a x)-\frac{1}{2} a c^2 \tan ^{-1}(a x)^2-\frac{1}{2} a^3 c^2 x^2 \tan ^{-1}(a x)^2+\frac{2}{3} i a c^2 \tan ^{-1}(a x)^3-\frac{c^2 \tan ^{-1}(a x)^3}{x}+2 a^2 c^2 x \tan ^{-1}(a x)^3+\frac{1}{3} a^4 c^2 x^3 \tan ^{-1}(a x)^3+5 a c^2 \tan ^{-1}(a x)^2 \log \left (\frac{2}{1+i a x}\right )-\frac{1}{2} a c^2 \log \left (1+a^2 x^2\right )+3 a c^2 \tan ^{-1}(a x)^2 \log \left (2-\frac{2}{1-i a x}\right )-3 i a c^2 \tan ^{-1}(a x) \text{Li}_2\left (-1+\frac{2}{1-i a x}\right )+5 i a c^2 \tan ^{-1}(a x) \text{Li}_2\left (1-\frac{2}{1+i a x}\right )+\frac{3}{2} a c^2 \text{Li}_3\left (-1+\frac{2}{1-i a x}\right )+\frac{5}{2} a c^2 \text{Li}_3\left (1-\frac{2}{1+i a x}\right )\\ \end{align*}
Mathematica [A] time = 0.388447, size = 246, normalized size = 0.87 \[ \frac{c^2 \left (72 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,e^{-2 i \tan ^{-1}(a x)}\right )-120 i a x \tan ^{-1}(a x) \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(a x)}\right )+36 a x \text{PolyLog}\left (3,e^{-2 i \tan ^{-1}(a x)}\right )+60 a x \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(a x)}\right )-12 a x \log \left (a^2 x^2+1\right )+8 a^4 x^4 \tan ^{-1}(a x)^3-12 a^3 x^3 \tan ^{-1}(a x)^2+48 a^2 x^2 \tan ^{-1}(a x)^3+24 a^2 x^2 \tan ^{-1}(a x)-3 i \pi ^3 a x-16 i a x \tan ^{-1}(a x)^3-12 a x \tan ^{-1}(a x)^2-24 \tan ^{-1}(a x)^3+72 a x \tan ^{-1}(a x)^2 \log \left (1-e^{-2 i \tan ^{-1}(a x)}\right )+120 a x \tan ^{-1}(a x)^2 \log \left (1+e^{2 i \tan ^{-1}(a x)}\right )\right )}{24 x} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 2.39, size = 5486, normalized size = 19.3 \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^{4} c^{2} x^{4} + 2 \, a^{2} c^{2} x^{2} + c^{2}\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^{2} \left (\int 2 a^{2} \operatorname{atan}^{3}{\left (a x \right )}\, dx + \int \frac{\operatorname{atan}^{3}{\left (a x \right )}}{x^{2}}\, dx + \int a^{4} x^{2} \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 )}^{2} \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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