Optimal. Leaf size=194 \[ -\frac {b^3 x}{4 c^3}+\frac {b^3 \text {ArcTan}(c x)}{4 c^4}+\frac {b^2 x^2 (a+b \text {ArcTan}(c x))}{4 c^2}+\frac {i b (a+b \text {ArcTan}(c x))^2}{c^4}+\frac {3 b x (a+b \text {ArcTan}(c x))^2}{4 c^3}-\frac {b x^3 (a+b \text {ArcTan}(c x))^2}{4 c}-\frac {(a+b \text {ArcTan}(c x))^3}{4 c^4}+\frac {1}{4} x^4 (a+b \text {ArcTan}(c x))^3+\frac {2 b^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {i b^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4} \]
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Rubi [A]
time = 0.38, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 18, number of rules used = 10, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {4946, 5036,
327, 209, 5040, 4964, 2449, 2352, 4930, 5004} \begin {gather*} \frac {2 b^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{c^4}+\frac {b^2 x^2 (a+b \text {ArcTan}(c x))}{4 c^2}-\frac {(a+b \text {ArcTan}(c x))^3}{4 c^4}+\frac {i b (a+b \text {ArcTan}(c x))^2}{c^4}+\frac {3 b x (a+b \text {ArcTan}(c x))^2}{4 c^3}+\frac {1}{4} x^4 (a+b \text {ArcTan}(c x))^3-\frac {b x^3 (a+b \text {ArcTan}(c x))^2}{4 c}+\frac {b^3 \text {ArcTan}(c x)}{4 c^4}+\frac {i b^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c^4}-\frac {b^3 x}{4 c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 209
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 5004
Rule 5036
Rule 5040
Rubi steps
\begin {align*} \int x^3 \left (a+b \tan ^{-1}(c x)\right )^3 \, dx &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {1}{4} (3 b c) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx\\ &=\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3-\frac {(3 b) \int x^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{4 c}+\frac {(3 b) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {1}{2} b^2 \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {(3 b) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{4 c^3}-\frac {(3 b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{1+c^2 x^2} \, dx}{4 c^3}\\ &=\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {b^2 \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^2}-\frac {b^2 \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c^2}-\frac {\left (3 b^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c^2}\\ &=\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {b^2 \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 c^3}+\frac {\left (3 b^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{2 c^3}-\frac {b^3 \int \frac {x^2}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac {b^3 x}{4 c^3}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {2 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {b^3 \int \frac {1}{1+c^2 x^2} \, dx}{4 c^3}-\frac {b^3 \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}-\frac {\left (3 b^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 c^3}\\ &=-\frac {b^3 x}{4 c^3}+\frac {b^3 \tan ^{-1}(c x)}{4 c^4}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {2 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {\left (i b^3\right ) \text {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\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{2 c^4}\\ &=-\frac {b^3 x}{4 c^3}+\frac {b^3 \tan ^{-1}(c x)}{4 c^4}+\frac {b^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )^2}{c^4}+\frac {3 b x \left (a+b \tan ^{-1}(c x)\right )^2}{4 c^3}-\frac {b x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{4 c}-\frac {\left (a+b \tan ^{-1}(c x)\right )^3}{4 c^4}+\frac {1}{4} x^4 \left (a+b \tan ^{-1}(c x)\right )^3+\frac {2 b^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4}+\frac {i b^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4}\\ \end {align*}
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Mathematica [A]
time = 0.41, size = 225, normalized size = 1.16 \begin {gather*} \frac {a b^2+3 a^2 b c x-b^3 c x+a b^2 c^2 x^2-a^2 b c^3 x^3+a^3 c^4 x^4-b^2 \left (b \left (4 i-3 c x+c^3 x^3\right )+a \left (3-3 c^4 x^4\right )\right ) \text {ArcTan}(c x)^2+b^3 \left (-1+c^4 x^4\right ) \text {ArcTan}(c x)^3+b \text {ArcTan}(c x) \left (-2 a b c x \left (-3+c^2 x^2\right )+b^2 \left (1+c^2 x^2\right )+3 a^2 \left (-1+c^4 x^4\right )+8 b^2 \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-4 a b^2 \log \left (1+c^2 x^2\right )-4 i b^3 \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )}{4 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 410 vs. \(2 (176 ) = 352\).
time = 0.54, size = 411, normalized size = 2.12 Too large to display
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 x^{3} \left (a + b \operatorname {atan}{\left (c x \right )}\right )^{3}\, 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 = -0.01 \begin {gather*} \int x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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