Optimal. Leaf size=652 \[ \frac{3 i b^2 d (c d-e) (c d+e) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}-\frac{3 i b^3 e \left (6 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{4 c^4}+\frac{3 b^3 d (c d-e) (c d+e) \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^3}+\frac{i b^3 e^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{4 c^4}-\frac{3 b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}+\frac{3 a b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac{b^2 e^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}-\frac{\left (-6 c^2 d^2 e^2+c^4 d^4+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e x \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac{i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}+\frac{3 b d (c d-e) (c d+e) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}+\frac{i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac{3 b^3 d e^2 \log \left (c^2 x^2+1\right )}{2 c^3}+\frac{3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}-\frac{b^3 e^3 x}{4 c^3}+\frac{b^3 e^3 \tan ^{-1}(c x)}{4 c^4} \]
[Out]
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Rubi [A] time = 1.20167, antiderivative size = 652, normalized size of antiderivative = 1., number of steps used = 29, number of rules used = 15, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {4864, 4846, 4920, 4854, 2402, 2315, 4852, 4916, 260, 4884, 321, 203, 4984, 4994, 6610} \[ \frac{3 i b^2 d (c d-e) (c d+e) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}-\frac{3 i b^3 e \left (6 c^2 d^2-e^2\right ) \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{4 c^4}+\frac{3 b^3 d (c d-e) (c d+e) \text{PolyLog}\left (3,1-\frac{2}{1+i c x}\right )}{2 c^3}+\frac{i b^3 e^3 \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{4 c^4}-\frac{3 b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}+\frac{3 a b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac{b^2 e^3 \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c^4}-\frac{\left (-6 c^2 d^2 e^2+c^4 d^4+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e x \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac{i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}+\frac{3 b d (c d-e) (c d+e) \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}+\frac{i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac{3 b^3 d e^2 \log \left (c^2 x^2+1\right )}{2 c^3}+\frac{3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}-\frac{b^3 e^3 x}{4 c^3}+\frac{b^3 e^3 \tan ^{-1}(c x)}{4 c^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4864
Rule 4846
Rule 4920
Rule 4854
Rule 2402
Rule 2315
Rule 4852
Rule 4916
Rule 260
Rule 4884
Rule 321
Rule 203
Rule 4984
Rule 4994
Rule 6610
Rubi steps
\begin{align*} \int (d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac{(3 b c) \int \left (\frac{e^2 \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac{4 d e^3 x \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}+\frac{e^4 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}+\frac{\left (c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 e}\\ &=\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac{(3 b) \int \frac{\left (c^4 d^4-6 c^2 d^2 e^2+e^4+4 c^2 d (c d-e) e (c d+e) x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{4 c^3 e}-\frac{\left (3 b d e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c}-\frac{\left (3 b e^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{4 c}-\frac{\left (3 b e \left (6 c^2 d^2-e^2\right )\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{4 c^3}\\ &=-\frac{3 b e \left (6 c^2 d^2-e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac{(3 b) \int \left (\frac{c^4 d^4 \left (1+\frac{-6 c^2 d^2 e^2+e^4}{c^4 d^4}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}+\frac{4 c^2 d (c d-e) e (c d+e) x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{4 c^3 e}+\left (3 b^2 d e^2\right ) \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{1}{2} \left (b^2 e^3\right ) \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac{\left (3 b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c^2}\\ &=-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e \left (6 c^2 d^2-e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}+\frac{\left (3 b^2 d e^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2}-\frac{\left (3 b^2 d e^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^2}+\frac{\left (b^2 e^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac{\left (b^2 e^3\right ) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c^2}-\frac{(3 b d (c d-e) (c d+e)) \int \frac{x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{c}-\frac{\left (3 b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 c^3}-\frac{\left (3 b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right )\right ) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{4 c^3 e}\\ &=\frac{3 a b^2 d e^2 x}{c^2}+\frac{b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac{i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e \left (6 c^2 d^2-e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac{i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}-\frac{\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}-\frac{3 b^2 e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}+\frac{\left (3 b^3 d e^2\right ) \int \tan ^{-1}(c x) \, dx}{c^2}+\frac{\left (b^2 e^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 c^3}-\frac{\left (b^3 e^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx}{4 c}+\frac{(3 b d (c d-e) (c d+e)) \int \frac{\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx}{c^2}+\frac{\left (3 b^3 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}\\ &=\frac{3 a b^2 d e^2 x}{c^2}-\frac{b^3 e^3 x}{4 c^3}+\frac{3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}+\frac{b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac{i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e \left (6 c^2 d^2-e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac{i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}-\frac{\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}+\frac{b^2 e^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}-\frac{3 b^2 e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}+\frac{3 b d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3}-\frac{\left (3 b^3 d e^2\right ) \int \frac{x}{1+c^2 x^2} \, dx}{c}+\frac{\left (b^3 e^3\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 c^3}-\frac{\left (b^3 e^3\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}-\frac{\left (6 b^2 d (c d-e) (c d+e)\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}-\frac{\left (3 i b^3 e \left (6 c^2 d^2-e^2\right )\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{2 c^4}\\ &=\frac{3 a b^2 d e^2 x}{c^2}-\frac{b^3 e^3 x}{4 c^3}+\frac{b^3 e^3 \tan ^{-1}(c x)}{4 c^4}+\frac{3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}+\frac{b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac{i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e \left (6 c^2 d^2-e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac{i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}-\frac{\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}+\frac{b^2 e^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}-\frac{3 b^2 e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}+\frac{3 b d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3}-\frac{3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}-\frac{3 i b^3 e \left (6 c^2 d^2-e^2\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{4 c^4}+\frac{3 i b^2 d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3}+\frac{\left (i b^3 e^3\right ) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{2 c^4}-\frac{\left (3 i b^3 d (c d-e) (c d+e)\right ) \int \frac{\text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2}\\ &=\frac{3 a b^2 d e^2 x}{c^2}-\frac{b^3 e^3 x}{4 c^3}+\frac{b^3 e^3 \tan ^{-1}(c x)}{4 c^4}+\frac{3 b^3 d e^2 x \tan ^{-1}(c x)}{c^2}+\frac{b^2 e^3 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{3 b d e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3}+\frac{i b e^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 i b e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4}-\frac{3 b e \left (6 c^2 d^2-e^2\right ) x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac{3 b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}-\frac{b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac{i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^3}{c^3}-\frac{\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4 e}+\frac{(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^3}{4 e}+\frac{b^2 e^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}-\frac{3 b^2 e \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{2 c^4}+\frac{3 b d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac{2}{1+i c x}\right )}{c^3}-\frac{3 b^3 d e^2 \log \left (1+c^2 x^2\right )}{2 c^3}+\frac{i b^3 e^3 \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{4 c^4}-\frac{3 i b^3 e \left (6 c^2 d^2-e^2\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{4 c^4}+\frac{3 i b^2 d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right ) \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{c^3}+\frac{3 b^3 d (c d-e) (c d+e) \text{Li}_3\left (1-\frac{2}{1+i c x}\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 1.84167, size = 855, normalized size = 1.31 \[ \frac{a^3 e^3 x^4 c^4+3 a^2 b x \left (4 d^3+6 e x d^2+4 e^2 x^2 d+e^3 x^3\right ) \tan ^{-1}(c x) c^4+a^2 e^2 (4 a c d-b e) x^3 c^3+6 a^2 d e (a c d-b e) x^2 c^3+12 a b^2 d^3 \left (\tan ^{-1}(c x) \left ((c x-i) \tan ^{-1}(c x)+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right ) c^3+2 b^3 d^3 \left (2 \left ((c x-i) \tan ^{-1}(c x)+3 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right ) \tan ^{-1}(c x)^2-6 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right ) \tan ^{-1}(c x)+3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )\right ) c^3+18 a b^2 d^2 e \left (\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2-2 c x \tan ^{-1}(c x)+\log \left (c^2 x^2+1\right )\right ) c^2+6 b^3 d^2 e \left (\tan ^{-1}(c x) \left (\left (c^2 x^2+1\right ) \tan ^{-1}(c x)^2+(3 i-3 c x) \tan ^{-1}(c x)-6 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+3 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right ) c^2+a^2 \left (4 a c^3 d^3+3 b e \left (e^2-6 c^2 d^2\right )\right ) x c-6 a^2 b d \left (c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right ) c+12 a b^2 d e^2 \left (\left (c^3 x^3+i\right ) \tan ^{-1}(c x)^2-\left (c^2 x^2+2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right ) \tan ^{-1}(c x)+c x+i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right ) c+2 b^3 d e^2 \left (2 c^3 x^3 \tan ^{-1}(c x)^3+2 i \tan ^{-1}(c x)^3-3 c^2 x^2 \tan ^{-1}(c x)^2-6 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right ) \tan ^{-1}(c x)^2-3 \tan ^{-1}(c x)^2+6 c x \tan ^{-1}(c x)+6 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right ) \tan ^{-1}(c x)-3 \log \left (c^2 x^2+1\right )-3 \text{PolyLog}\left (3,-e^{2 i \tan ^{-1}(c x)}\right )\right ) c+3 a^2 b \left (6 c^2 d^2 e-e^3\right ) \tan ^{-1}(c x)+a b^2 e^3 \left (c^2 x^2+3 \left (c^4 x^4-1\right ) \tan ^{-1}(c x)^2+\left (6 c x-2 c^3 x^3\right ) \tan ^{-1}(c x)-4 \log \left (c^2 x^2+1\right )+1\right )+b^3 e^3 \left (\left (c^4 x^4-1\right ) \tan ^{-1}(c x)^3-\left (c^3 x^3-3 c x+4 i\right ) \tan ^{-1}(c x)^2+\left (c^2 x^2+8 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+1\right ) \tan ^{-1}(c x)-c x-4 i \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )\right )}{4 c^4} \]
Warning: Unable to verify antiderivative.
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Maple [C] time = 3.813, size = 3577, normalized size = 5.5 \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*} \text{result too large to display} \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 (a^{3} e^{3} x^{3} + 3 \, a^{3} d e^{2} x^{2} + 3 \, a^{3} d^{2} e x + a^{3} d^{3} +{\left (b^{3} e^{3} x^{3} + 3 \, b^{3} d e^{2} x^{2} + 3 \, b^{3} d^{2} e x + b^{3} d^{3}\right )} \arctan \left (c x\right )^{3} + 3 \,{\left (a b^{2} e^{3} x^{3} + 3 \, a b^{2} d e^{2} x^{2} + 3 \, a b^{2} d^{2} e x + a b^{2} d^{3}\right )} \arctan \left (c x\right )^{2} + 3 \,{\left (a^{2} b e^{3} x^{3} + 3 \, a^{2} b d e^{2} x^{2} + 3 \, a^{2} b d^{2} e x + a^{2} b d^{3}\right )} \arctan \left (c x\right ), 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 \left (a + b \operatorname{atan}{\left (c x \right )}\right )^{3} \left (d + e x\right )^{3}\, 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{\left (e x + d\right )}^{3}{\left (b \arctan \left (c x\right ) + a\right )}^{3}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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