Optimal. Leaf size=499 \[ \frac{3 i b^2 c \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac{3 b c \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac{3 b c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)} \]
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Rubi [A] time = 0.53011, antiderivative size = 499, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {4864, 4858, 4984, 4884, 4920, 4854, 4994, 6610} \[ \frac{3 i b^2 c \text{PolyLog}\left (2,1-\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{PolyLog}\left (2,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{PolyLog}\left (3,1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{PolyLog}\left (3,1-\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac{3 b c \log \left (\frac{2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac{3 b c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{c^2 d^2+e^2}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 4864
Rule 4858
Rule 4984
Rule 4884
Rule 4920
Rule 4854
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{(d+e x)^2} \, dx &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}+\frac{(3 b c) \int \left (\frac{e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right ) (d+e x)}+\frac{c^2 (d-e x) \left (a+b \tan ^{-1}(c x)\right )^2}{\left (c^2 d^2+e^2\right ) \left (1+c^2 x^2\right )}\right ) \, dx}{e}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}+\frac{\left (3 b c^3\right ) \int \frac{(d-e x) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )}+\frac{(3 b c e) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{d+e x} \, dx}{c^2 d^2+e^2}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{\left (3 b c^3\right ) \int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}-\frac{e x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{e \left (c^2 d^2+e^2\right )}\\ &=-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac{\left (3 b c^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c^2 d^2+e^2}+\frac{\left (3 b c^3 d\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{e \left (c^2 d^2+e^2\right )}\\ &=\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{\left (3 b c^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{c^2 d^2+e^2}\\ &=\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac{\left (6 b^2 c^2\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^2+e^2}\\ &=\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}-\frac{\left (3 i b^3 c^2\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^2+e^2}\\ &=\frac{i c \left (a+b \tan ^{-1}(c x)\right )^3}{c^2 d^2+e^2}+\frac{c^2 d \left (a+b \tan ^{-1}(c x)\right )^3}{e \left (c^2 d^2+e^2\right )}-\frac{\left (a+b \tan ^{-1}(c x)\right )^3}{e (d+e x)}-\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^2 d^2+e^2}+\frac{3 b c \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1-i c x}\right )}{c^2 d^2+e^2}+\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^2 d^2+e^2}-\frac{3 i b^2 c \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{c^2 d^2+e^2}-\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1-i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 \left (c^2 d^2+e^2\right )}+\frac{3 b^3 c \text{Li}_3\left (1-\frac{2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 \left (c^2 d^2+e^2\right )}\\ \end{align*}
Mathematica [F] time = 123.733, size = 0, normalized size = 0. \[ \int \frac{\left (a+b \tan ^{-1}(c x)\right )^3}{(d+e x)^2} \, dx \]
Verification is Not applicable to the result.
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Maple [C] time = 0.774, size = 2960, normalized size = 5.9 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{3}{2} \,{\left ({\left (\frac{2 \, c d \arctan \left (c x\right )}{c^{2} d^{2} e + e^{3}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{2} d^{2} + e^{2}} + \frac{2 \, \log \left (e x + d\right )}{c^{2} d^{2} + e^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{e^{2} x + d e}\right )} a^{2} b - \frac{a^{3}}{e^{2} x + d e} - \frac{\frac{15}{2} \, b^{3} \arctan \left (c x\right )^{3} - \frac{21}{8} \, b^{3} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )^{2} -{\left (e^{2} x + d e\right )} \int \frac{196 \,{\left (b^{3} c^{2} e x^{2} + b^{3} e\right )} \arctan \left (c x\right )^{3} + 12 \,{\left (64 \, a b^{2} c^{2} e x^{2} + 15 \, b^{3} c e x + 15 \, b^{3} c d + 64 \, a b^{2} e\right )} \arctan \left (c x\right )^{2} - 84 \,{\left (b^{3} c^{2} e x^{2} + b^{3} c^{2} d x\right )} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) - 21 \,{\left (b^{3} c e x + b^{3} c d -{\left (b^{3} c^{2} e x^{2} + b^{3} e\right )} \arctan \left (c x\right )\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{8 \,{\left (c^{2} e^{3} x^{4} + 2 \, c^{2} d e^{2} x^{3} + 2 \, d e^{2} x + d^{2} e +{\left (c^{2} d^{2} e + e^{3}\right )} x^{2}\right )}}\,{d x}}{32 \,{\left (e^{2} x + d e\right )}} \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{b^{3} \arctan \left (c x\right )^{3} + 3 \, a b^{2} \arctan \left (c x\right )^{2} + 3 \, a^{2} b \arctan \left (c x\right ) + a^{3}}{e^{2} x^{2} + 2 \, d e x + d^{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*} \int \frac{\left (a + b \operatorname{atan}{\left (c x \right )}\right )^{3}}{\left (d + e x\right )^{2}}\, dx \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 (b \arctan \left (c x\right ) + a\right )}^{3}}{{\left (e x + d\right )}^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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