Optimal. Leaf size=373 \[ \frac {5 a b d^2 x}{6 c^3}-\frac {3 i b^2 d^2 x}{5 c^3}+\frac {31 b^2 d^2 x^2}{180 c^2}+\frac {i b^2 d^2 x^3}{15 c}-\frac {1}{60} b^2 d^2 x^4+\frac {3 i b^2 d^2 \text {ArcTan}(c x)}{5 c^4}+\frac {5 b^2 d^2 x \text {ArcTan}(c x)}{6 c^3}+\frac {2 i b d^2 x^2 (a+b \text {ArcTan}(c x))}{5 c^2}-\frac {5 b d^2 x^3 (a+b \text {ArcTan}(c x))}{18 c}-\frac {1}{5} i b d^2 x^4 (a+b \text {ArcTan}(c x))+\frac {1}{15} b c d^2 x^5 (a+b \text {ArcTan}(c x))-\frac {49 d^2 (a+b \text {ArcTan}(c x))^2}{60 c^4}+\frac {1}{4} d^2 x^4 (a+b \text {ArcTan}(c x))^2+\frac {2}{5} i c d^2 x^5 (a+b \text {ArcTan}(c x))^2-\frac {1}{6} c^2 d^2 x^6 (a+b \text {ArcTan}(c x))^2+\frac {4 i b d^2 (a+b \text {ArcTan}(c x)) \log \left (\frac {2}{1+i c x}\right )}{5 c^4}-\frac {53 b^2 d^2 \log \left (1+c^2 x^2\right )}{90 c^4}-\frac {2 b^2 d^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{5 c^4} \]
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Rubi [A]
time = 0.70, antiderivative size = 373, normalized size of antiderivative = 1.00, number of steps
used = 43, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used =
{4996, 4946, 5036, 272, 45, 4930, 266, 5004, 308, 209, 327, 5040, 4964, 2449, 2352}
\begin {gather*} -\frac {49 d^2 (a+b \text {ArcTan}(c x))^2}{60 c^4}+\frac {4 i b d^2 \log \left (\frac {2}{1+i c x}\right ) (a+b \text {ArcTan}(c x))}{5 c^4}-\frac {1}{6} c^2 d^2 x^6 (a+b \text {ArcTan}(c x))^2+\frac {2 i b d^2 x^2 (a+b \text {ArcTan}(c x))}{5 c^2}+\frac {2}{5} i c d^2 x^5 (a+b \text {ArcTan}(c x))^2+\frac {1}{15} b c d^2 x^5 (a+b \text {ArcTan}(c x))+\frac {1}{4} d^2 x^4 (a+b \text {ArcTan}(c x))^2-\frac {1}{5} i b d^2 x^4 (a+b \text {ArcTan}(c x))-\frac {5 b d^2 x^3 (a+b \text {ArcTan}(c x))}{18 c}+\frac {5 a b d^2 x}{6 c^3}+\frac {3 i b^2 d^2 \text {ArcTan}(c x)}{5 c^4}+\frac {5 b^2 d^2 x \text {ArcTan}(c x)}{6 c^3}-\frac {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{5 c^4}-\frac {3 i b^2 d^2 x}{5 c^3}+\frac {31 b^2 d^2 x^2}{180 c^2}-\frac {53 b^2 d^2 \log \left (c^2 x^2+1\right )}{90 c^4}+\frac {i b^2 d^2 x^3}{15 c}-\frac {1}{60} b^2 d^2 x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 209
Rule 266
Rule 272
Rule 308
Rule 327
Rule 2352
Rule 2449
Rule 4930
Rule 4946
Rule 4964
Rule 4996
Rule 5004
Rule 5036
Rule 5040
Rubi steps
\begin {align*} \int x^3 (d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx &=\int \left (d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )^2+2 i c d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2-c^2 d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^2 \int x^3 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx+\left (2 i c d^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (c^2 d^2\right ) \int x^5 \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} \left (b c d^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {1}{5} \left (4 i b c^2 d^2\right ) \int \frac {x^5 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\frac {1}{3} \left (b c^3 d^2\right ) \int \frac {x^6 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{5} \left (4 i b d^2\right ) \int x^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx+\frac {1}{5} \left (4 i b d^2\right ) \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx-\frac {\left (b d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c}+\frac {\left (b d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{2 c}+\frac {1}{3} \left (b c d^2\right ) \int x^4 \left (a+b \tan ^{-1}(c x)\right ) \, dx-\frac {1}{3} \left (b c d^2\right ) \int \frac {x^4 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-\frac {b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{6 c}-\frac {1}{5} i b d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} b c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{6} \left (b^2 d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{2 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{2 c^3}+\frac {\left (4 i b d^2\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{5 c^2}-\frac {\left (4 i b d^2\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{5 c^2}-\frac {\left (b d^2\right ) \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c}+\frac {\left (b d^2\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{3 c}+\frac {1}{5} \left (i b^2 c d^2\right ) \int \frac {x^4}{1+c^2 x^2} \, dx-\frac {1}{15} \left (b^2 c^2 d^2\right ) \int \frac {x^5}{1+c^2 x^2} \, dx\\ &=\frac {a b d^2 x}{2 c^3}+\frac {2 i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac {5 b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{18 c}-\frac {1}{5} i b d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} b c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {13 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{20 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {1}{12} \left (b^2 d^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {1}{9} \left (b^2 d^2\right ) \int \frac {x^3}{1+c^2 x^2} \, dx+\frac {\left (4 i b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{5 c^3}+\frac {\left (b d^2\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{3 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{3 c^3}+\frac {\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{2 c^3}-\frac {\left (2 i b^2 d^2\right ) \int \frac {x^2}{1+c^2 x^2} \, dx}{5 c}+\frac {1}{5} \left (i b^2 c d^2\right ) \int \left (-\frac {1}{c^4}+\frac {x^2}{c^2}+\frac {1}{c^4 \left (1+c^2 x^2\right )}\right ) \, dx-\frac {1}{30} \left (b^2 c^2 d^2\right ) \text {Subst}\left (\int \frac {x^2}{1+c^2 x} \, dx,x,x^2\right )\\ &=\frac {5 a b d^2 x}{6 c^3}-\frac {3 i b^2 d^2 x}{5 c^3}+\frac {i b^2 d^2 x^3}{15 c}+\frac {b^2 d^2 x \tan ^{-1}(c x)}{2 c^3}+\frac {2 i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac {5 b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{18 c}-\frac {1}{5} i b d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} b c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {49 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{60 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^4}+\frac {1}{18} \left (b^2 d^2\right ) \text {Subst}\left (\int \frac {x}{1+c^2 x} \, dx,x,x^2\right )+\frac {1}{12} \left (b^2 d^2\right ) \text {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 {\left (i b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^3}+\frac {\left (2 i b^2 d^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{5 c^3}-\frac {\left (4 i b^2 d^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{5 c^3}+\frac {\left (b^2 d^2\right ) \int \tan ^{-1}(c x) \, dx}{3 c^3}-\frac {\left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{2 c^2}-\frac {1}{30} \left (b^2 c^2 d^2\right ) \text {Subst}\left (\int \left (-\frac {1}{c^4}+\frac {x}{c^2}+\frac {1}{c^4 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {5 a b d^2 x}{6 c^3}-\frac {3 i b^2 d^2 x}{5 c^3}+\frac {7 b^2 d^2 x^2}{60 c^2}+\frac {i b^2 d^2 x^3}{15 c}-\frac {1}{60} b^2 d^2 x^4+\frac {3 i b^2 d^2 \tan ^{-1}(c x)}{5 c^4}+\frac {5 b^2 d^2 x \tan ^{-1}(c x)}{6 c^3}+\frac {2 i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac {5 b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{18 c}-\frac {1}{5} i b d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} b c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {49 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{60 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^4}-\frac {11 b^2 d^2 \log \left (1+c^2 x^2\right )}{30 c^4}+\frac {1}{18} \left (b^2 d^2\right ) \text {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 {\left (4 b^2 d^2\right ) \text {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{5 c^4}-\frac {\left (b^2 d^2\right ) \int \frac {x}{1+c^2 x^2} \, dx}{3 c^2}\\ &=\frac {5 a b d^2 x}{6 c^3}-\frac {3 i b^2 d^2 x}{5 c^3}+\frac {31 b^2 d^2 x^2}{180 c^2}+\frac {i b^2 d^2 x^3}{15 c}-\frac {1}{60} b^2 d^2 x^4+\frac {3 i b^2 d^2 \tan ^{-1}(c x)}{5 c^4}+\frac {5 b^2 d^2 x \tan ^{-1}(c x)}{6 c^3}+\frac {2 i b d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac {5 b d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )}{18 c}-\frac {1}{5} i b d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac {1}{15} b c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )-\frac {49 d^2 \left (a+b \tan ^{-1}(c x)\right )^2}{60 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {2}{5} i c d^2 x^5 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{6} c^2 d^2 x^6 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {4 i b d^2 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{5 c^4}-\frac {53 b^2 d^2 \log \left (1+c^2 x^2\right )}{90 c^4}-\frac {2 b^2 d^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{5 c^4}\\ \end {align*}
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Mathematica [A]
time = 0.74, size = 342, normalized size = 0.92 \begin {gather*} \frac {d^2 \left (108 i a b+34 b^2+150 a b c x-108 i b^2 c x+72 i a b c^2 x^2+31 b^2 c^2 x^2-50 a b c^3 x^3+12 i b^2 c^3 x^3+45 a^2 c^4 x^4-36 i a b c^4 x^4-3 b^2 c^4 x^4+72 i a^2 c^5 x^5+12 a b c^5 x^5-30 a^2 c^6 x^6-3 b^2 \left (1-15 c^4 x^4-24 i c^5 x^5+10 c^6 x^6\right ) \text {ArcTan}(c x)^2+2 b \text {ArcTan}(c x) \left (b \left (54 i+75 c x+36 i c^2 x^2-25 c^3 x^3-18 i c^4 x^4+6 c^5 x^5\right )+a \left (-75+45 c^4 x^4+72 i c^5 x^5-30 c^6 x^6\right )+72 i b \log \left (1+e^{2 i \text {ArcTan}(c x)}\right )\right )-72 i a b \log \left (1+c^2 x^2\right )-106 b^2 \log \left (1+c^2 x^2\right )+72 b^2 \text {PolyLog}\left (2,-e^{2 i \text {ArcTan}(c x)}\right )\right )}{180 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 624, normalized size = 1.67 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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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