Optimal. Leaf size=264 \[ -\frac {3 i b e (a+b \text {ArcTan}(c x))^2}{2 c^2}-\frac {3 b e x (a+b \text {ArcTan}(c x))^2}{2 c}+\frac {i d (a+b \text {ArcTan}(c x))^3}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \text {ArcTan}(c x))^3}{2 e}+\frac {(d+e x)^2 (a+b \text {ArcTan}(c x))^3}{2 e}-\frac {3 b^2 e (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {3 b d (a+b \text {ArcTan}(c x))^2 \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^3 e \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 c^2}+\frac {3 i b^2 d (a+b \text {ArcTan}(c x)) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c} \]
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Rubi [A]
time = 0.38, antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 10, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {4974, 4930,
5040, 4964, 2449, 2352, 5104, 5004, 5114, 6745} \begin {gather*} -\frac {3 b^2 e \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{c^2}+\frac {3 i b^2 d \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) (a+b \text {ArcTan}(c x))}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) (a+b \text {ArcTan}(c x))^3}{2 e}-\frac {3 i b e (a+b \text {ArcTan}(c x))^2}{2 c^2}+\frac {(d+e x)^2 (a+b \text {ArcTan}(c x))^3}{2 e}+\frac {i d (a+b \text {ArcTan}(c x))^3}{c}+\frac {3 b d \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))^2}{c}-\frac {3 b e x (a+b \text {ArcTan}(c x))^2}{2 c}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 c^2}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 2352
Rule 2449
Rule 4930
Rule 4964
Rule 4974
Rule 5004
Rule 5040
Rule 5104
Rule 5114
Rule 6745
Rubi steps
\begin {align*} \int (d+e x) \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {(3 b c) \int \left (\frac {e^2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^2}+\frac {\left (c^2 d^2-e^2+2 c^2 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx}{2 e}\\ &=\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {(3 b) \int \frac {\left (c^2 d^2-e^2+2 c^2 d e x\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c e}-\frac {(3 b e) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{2 c}\\ &=-\frac {3 b e x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {(3 b) \int \left (\frac {c^2 d^2 \left (1-\frac {e^2}{c^2 d^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}+\frac {2 c^2 d e x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2}\right ) \, dx}{2 c e}+\left (3 b^2 e\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b e x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-(3 b c d) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx-\frac {\left (3 b^2 e\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c}-\frac {(3 b (c d-e) (c d+e)) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{2 c e}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b e x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2}+(3 b d) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{i-c x} \, dx+\frac {\left (3 b^3 e\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b e x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {3 b d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c}-\left (6 b^2 d\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-\frac {\left (3 i b^3 e\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^2}\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b e x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {3 b d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^2}+\frac {3 i b^2 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}-\left (3 i b^3 d\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-\frac {3 i b e \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^2}-\frac {3 b e x \left (a+b \tan ^{-1}(c x)\right )^2}{2 c}+\frac {i d \left (a+b \tan ^{-1}(c x)\right )^3}{c}-\frac {\left (d^2-\frac {e^2}{c^2}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}+\frac {(d+e x)^2 \left (a+b \tan ^{-1}(c x)\right )^3}{2 e}-\frac {3 b^2 e \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^2}+\frac {3 b d \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c}-\frac {3 i b^3 e \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c^2}+\frac {3 i b^2 d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c}+\frac {3 b^3 d \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 342, normalized size = 1.30 \begin {gather*} \frac {a^2 c (2 a c d-3 b e) x+a^3 c^2 e x^2+3 a^2 b e \text {ArcTan}(c x)+3 a^2 b c^2 x (2 d+e x) \text {ArcTan}(c x)-3 a^2 b c d \log \left (1+c^2 x^2\right )+3 a b^2 e \left (-2 c x \text {ArcTan}(c x)+\left (1+c^2 x^2\right ) \text {ArcTan}(c x)^2+\log \left (1+c^2 x^2\right )\right )+6 a b^2 c d \left (\text {ArcTan}(c x) \left ((-i+c x) \text {ArcTan}(c x)+2 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )\right )+b^3 e \left (\text {ArcTan}(c x) \left ((3 i-3 c x) \text {ArcTan}(c x)+\left (1+c^2 x^2\right ) \text {ArcTan}(c x)^2-6 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )+3 i \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )\right )+b^3 c d \left (2 \text {ArcTan}(c x)^2 \left ((-i+c x) \text {ArcTan}(c x)+3 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-6 i \text {ArcTan}(c x) \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )+3 \text {PolyLog}\left (3,-e^{2 i \text {ArcTan}(c x)}\right )\right )}{2 c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 1.45, size = 7195, normalized size = 27.25
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(7195\) |
default | \(\text {Expression too large to display}\) | \(7195\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3} \left (d + e x\right )\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int {\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3\,\left (d+e\,x\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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