Optimal. Leaf size=376 \[ \frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}+\frac {2 b d (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}-\frac {\left (c^4 d^4-6 c^2 d^2 e^2+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4 e}-\frac {a b e x \left (6 c^2 d^2-e^2\right )}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac {b^2 d e^2 \tan ^{-1}(c x)}{c^3}+\frac {i b^2 d (c d-e) (c d+e) \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c^3}+\frac {b^2 d e^2 x}{c^2}+\frac {b^2 e^3 x^2}{12 c^2}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{4 c^4}-\frac {b^2 e^3 \log \left (c^2 x^2+1\right )}{12 c^4}-\frac {b^2 e x \left (6 c^2 d^2-e^2\right ) \tan ^{-1}(c x)}{2 c^3} \]
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Rubi [A] time = 0.57, antiderivative size = 376, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 14, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.778, Rules used = {4864, 4846, 260, 4852, 321, 203, 266, 43, 4984, 4884, 4920, 4854, 2402, 2315} \[ \frac {i b^2 d (c d-e) (c d+e) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^3}-\frac {a b e x \left (6 c^2 d^2-e^2\right )}{2 c^3}-\frac {\left (-6 c^2 d^2 e^2+c^4 d^4+e^4\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^4 e}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3}+\frac {2 b d (c d-e) (c d+e) \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (c^2 x^2+1\right )}{4 c^4}-\frac {b^2 e x \left (6 c^2 d^2-e^2\right ) \tan ^{-1}(c x)}{2 c^3}+\frac {b^2 d e^2 x}{c^2}-\frac {b^2 d e^2 \tan ^{-1}(c x)}{c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 e^3 \log \left (c^2 x^2+1\right )}{12 c^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 203
Rule 260
Rule 266
Rule 321
Rule 2315
Rule 2402
Rule 4846
Rule 4852
Rule 4854
Rule 4864
Rule 4884
Rule 4920
Rule 4984
Rubi steps
\begin {align*} \int (d+e x)^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}-\frac {(b c) \int \left (\frac {e^2 \left (6 c^2 d^2-e^2\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4}+\frac {4 d e^3 x \left (a+b \tan ^{-1}(c x)\right )}{c^2}+\frac {e^4 x^2 \left (a+b \tan ^{-1}(c x)\right )}{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 )}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}-\frac {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 )}{1+c^2 x^2} \, dx}{2 c^3 e}-\frac {\left (2 b d e^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c}-\frac {\left (b e^3\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}-\frac {\left (b e \left (6 c^2 d^2-e^2\right )\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}\\ &=-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}-\frac {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 )}{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 )}{1+c^2 x^2}\right ) \, dx}{2 c^3 e}+\left (b^2 d e^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx+\frac {1}{6} \left (b^2 e^3\right ) \int \frac {x^3}{1+c^2 x^2} \, dx-\frac {\left (b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \tan ^{-1}(c x) \, dx}{2 c^3}\\ &=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \tan ^{-1}(c x)}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}-\frac {\left (b^2 d e^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{c^2}+\frac {1}{12} \left (b^2 e^3\right ) \operatorname {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )-\frac {(2 b d (c d-e) (c d+e)) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c}+\frac {\left (b^2 e \left (6 c^2 d^2-e^2\right )\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (b \left (c^4 d^4-6 c^2 d^2 e^2+e^4\right )\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3 e}\\ &=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}-\frac {b^2 d e^2 \tan ^{-1}(c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \tan ^{-1}(c x)}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{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 )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {1}{12} \left (b^2 e^3\right ) \operatorname {Subst}\left (\int \left (\frac {1}{c^2}-\frac {1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )+\frac {(2 b d (c d-e) (c d+e)) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^2}\\ &=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tan ^{-1}(c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \tan ^{-1}(c x)}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{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 )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}+\frac {2 b d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}-\frac {\left (2 b^2 d (c d-e) (c d+e)\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2}\\ &=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tan ^{-1}(c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \tan ^{-1}(c x)}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{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 )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}+\frac {2 b d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {\left (2 i b^2 d (c d-e) (c d+e)\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^3}\\ &=\frac {b^2 d e^2 x}{c^2}-\frac {a b e \left (6 c^2 d^2-e^2\right ) x}{2 c^3}+\frac {b^2 e^3 x^2}{12 c^2}-\frac {b^2 d e^2 \tan ^{-1}(c x)}{c^3}-\frac {b^2 e \left (6 c^2 d^2-e^2\right ) x \tan ^{-1}(c x)}{2 c^3}-\frac {b d e^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{c}-\frac {b e^3 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}+\frac {i d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right )^2}{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 )^2}{4 c^4 e}+\frac {(d+e x)^4 \left (a+b \tan ^{-1}(c x)\right )^2}{4 e}+\frac {2 b d (c d-e) (c d+e) \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^3}-\frac {b^2 e^3 \log \left (1+c^2 x^2\right )}{12 c^4}+\frac {b^2 e \left (6 c^2 d^2-e^2\right ) \log \left (1+c^2 x^2\right )}{4 c^4}+\frac {i b^2 d (c d-e) (c d+e) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 1.00, size = 472, normalized size = 1.26 \[ \frac {12 a^2 c^4 d^3 x+18 a^2 c^4 d^2 e x^2+12 a^2 c^4 d e^2 x^3+3 a^2 c^4 e^3 x^4-36 a b c^3 d^2 e x-12 a b c^3 d e^2 x^2-2 a b c^3 e^3 x^3+12 a b c d e^2 \log \left (c^2 x^2+1\right )+2 b \tan ^{-1}(c x) \left (3 a \left (c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )+6 c^2 d^2 e-e^3\right )-b c e \left (18 c^2 d^2 x+6 d \left (c^2 e x^2+e\right )+e^2 x \left (c^2 x^2-3\right )\right )+12 b c d \left (c^2 d^2-e^2\right ) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )-12 a b c^3 d^3 \log \left (c^2 x^2+1\right )+6 a b c e^3 x-12 i b^2 c d \left (c^2 d^2-e^2\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+18 b^2 c^2 d^2 e \log \left (c^2 x^2+1\right )+12 b^2 c^2 d e^2 x+b^2 c^2 e^3 x^2-4 b^2 e^3 \log \left (c^2 x^2+1\right )+3 b^2 \tan ^{-1}(c x)^2 \left (c^4 x \left (4 d^3+6 d^2 e x+4 d e^2 x^2+e^3 x^3\right )-4 i c^3 d^3+6 c^2 d^2 e+4 i c d e^2-e^3\right )+b^2 e^3}{12 c^4} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (a^{2} e^{3} x^{3} + 3 \, a^{2} d e^{2} x^{2} + 3 \, a^{2} d^{2} e x + a^{2} d^{3} + {\left (b^{2} e^{3} x^{3} + 3 \, b^{2} d e^{2} x^{2} + 3 \, b^{2} d^{2} e x + b^{2} d^{3}\right )} \arctan \left (c x\right )^{2} + 2 \, {\left (a b e^{3} x^{3} + 3 \, a b d e^{2} x^{2} + 3 \, a b d^{2} e x + a b d^{3}\right )} \arctan \left (c x\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 948, normalized size = 2.52 \[ \frac {a^{2} e^{3} x^{4}}{4}+a^{2} x \,d^{3}-\frac {i b^{2} e^{2} d \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 c^{3}}+\frac {i b^{2} e^{2} d \ln \left (\frac {i \left (c x -i\right )}{2}\right ) \ln \left (c x +i\right )}{2 c^{3}}+\frac {i b^{2} e^{2} d \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x -i\right )}{2 c^{3}}-\frac {i b^{2} e^{2} d \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x +i\right )}{2 c^{3}}+2 a b \,e^{2} \arctan \left (c x \right ) x^{3} d +3 a b e \arctan \left (c x \right ) x^{2} d^{2}+\frac {b^{2} e^{3} x^{2}}{12 c^{2}}-\frac {i b^{2} d^{3} \ln \left (c x +i\right )^{2}}{4 c}+\frac {i b^{2} d^{3} \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c}-\frac {i b^{2} d^{3} \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2 c}+\frac {i b^{2} d^{3} \ln \left (c x -i\right )^{2}}{4 c}+\frac {b^{2} d \,e^{2} x}{c^{2}}-\frac {b^{2} d \,e^{2} \arctan \left (c x \right )}{c^{3}}-\frac {a b \,e^{2} d \,x^{2}}{c}-\frac {3 b^{2} e \arctan \left (c x \right ) d^{2} x}{c}-\frac {b^{2} e^{2} \arctan \left (c x \right ) d \,x^{2}}{c}+\frac {b^{2} e^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) d}{c^{3}}+\frac {a b \,e^{2} \ln \left (c^{2} x^{2}+1\right ) d}{c^{3}}+\frac {3 a b e \arctan \left (c x \right ) d^{2}}{c^{2}}-\frac {3 a b e \,d^{2} x}{c}+\frac {3 b^{2} e \arctan \left (c x \right )^{2} d^{2}}{2 c^{2}}-\frac {a b \,e^{3} \arctan \left (c x \right )}{2 c^{4}}-\frac {b^{2} e^{3} \arctan \left (c x \right ) x^{3}}{6 c}-\frac {b^{2} \arctan \left (c x \right ) \ln \left (c^{2} x^{2}+1\right ) d^{3}}{c}-\frac {a b \ln \left (c^{2} x^{2}+1\right ) d^{3}}{c}-\frac {a b \,e^{3} x^{3}}{6 c}+\frac {a b \,e^{3} x}{2 c^{3}}+\frac {a b \,e^{3} \arctan \left (c x \right ) x^{4}}{2}+b^{2} e^{2} \arctan \left (c x \right )^{2} x^{3} d +\frac {3 b^{2} e \arctan \left (c x \right )^{2} x^{2} d^{2}}{2}+2 a b \arctan \left (c x \right ) x \,d^{3}+\frac {b^{2} e^{3} \arctan \left (c x \right ) x}{2 c^{3}}+\frac {a^{2} d^{4}}{4 e}-\frac {b^{2} e^{3} \arctan \left (c x \right )^{2}}{4 c^{4}}+\frac {b^{2} e^{3} \arctan \left (c x \right )^{2} x^{4}}{4}+b^{2} \arctan \left (c x \right )^{2} x \,d^{3}+a^{2} e^{2} x^{3} d +\frac {3 a^{2} e \,x^{2} d^{2}}{2}+\frac {3 b^{2} e \ln \left (c^{2} x^{2}+1\right ) d^{2}}{2 c^{2}}-\frac {i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x -i\right )}{2 c}-\frac {i b^{2} d^{3} \ln \left (\frac {i \left (c x -i\right )}{2}\right ) \ln \left (c x +i\right )}{2 c}-\frac {i b^{2} e^{2} d \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 c^{3}}+\frac {i b^{2} e^{2} d \dilog \left (\frac {i \left (c x -i\right )}{2}\right )}{2 c^{3}}+\frac {i b^{2} d^{3} \ln \left (c^{2} x^{2}+1\right ) \ln \left (c x +i\right )}{2 c}+\frac {i b^{2} d^{3} \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 c}+\frac {i b^{2} e^{2} d \ln \left (c x +i\right )^{2}}{4 c^{3}}-\frac {i b^{2} e^{2} d \ln \left (c x -i\right )^{2}}{4 c^{3}}-\frac {b^{2} e^{3} \ln \left (c^{2} x^{2}+1\right )}{3 c^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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