Optimal. Leaf size=383 \[ \frac {3 i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (-c x+i)}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (-c x+i)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (-c x+i)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (-c x+i)^2}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {2 i b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3}+\frac {3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{c^4 d^3}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 c^4 d^3}+\frac {21 i b^2}{16 c^4 d^3 (-c x+i)}+\frac {b^2}{16 c^4 d^3 (-c x+i)^2}-\frac {21 i b^2 \tan ^{-1}(c x)}{16 c^4 d^3} \]
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Rubi [A] time = 0.67, antiderivative size = 383, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 14, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.560, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4864, 4862, 627, 44, 203, 4884, 4994, 6610} \[ \frac {3 i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3}-\frac {b^2 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (-c x+i)}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (-c x+i)^2}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (-c x+i)}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (-c x+i)^2}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {2 i b \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d^3}+\frac {3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac {21 i b^2}{16 c^4 d^3 (-c x+i)}+\frac {b^2}{16 c^4 d^3 (-c x+i)^2}-\frac {21 i b^2 \tan ^{-1}(c x)}{16 c^4 d^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 2315
Rule 2402
Rule 4846
Rule 4854
Rule 4862
Rule 4864
Rule 4876
Rule 4884
Rule 4920
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \tan ^{-1}(c x)\right )^2}{(d+i c d x)^3} \, dx &=\int \left (\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}+\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (-i+c x)^3}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (-i+c x)^2}-\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (-i+c x)}\right ) \, dx\\ &=\frac {i \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx}{c^3 d^3}-\frac {(3 i) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{c^3 d^3}+\frac {\int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^3} \, dx}{c^3 d^3}-\frac {3 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{c^3 d^3}\\ &=\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {(6 i b) \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^3 d^3}+\frac {b \int \left (-\frac {i \left (a+b \tan ^{-1}(c x)\right )}{2 (-i+c x)^3}+\frac {a+b \tan ^{-1}(c x)}{4 (-i+c x)^2}-\frac {a+b \tan ^{-1}(c x)}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{c^3 d^3}-\frac {(6 b) \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^3 d^3}-\frac {(2 i b) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {(i b) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{2 c^3 d^3}+\frac {(2 i b) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx}{c^3 d^3}+\frac {b \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{4 c^3 d^3}-\frac {b \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{4 c^3 d^3}-\frac {(3 b) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c^3 d^3}+\frac {(3 b) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^3 d^3}-\frac {\left (3 i b^2\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^3}\\ &=\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)^2}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {2 i b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 c^3 d^3}-\frac {\left (2 i b^2\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d^3}+\frac {b^2 \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 c^3 d^3}-\frac {\left (3 b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^3 d^3}\\ &=\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)^2}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {2 i b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {\left (2 b^2\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{c^4 d^3}-\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{4 c^3 d^3}+\frac {b^2 \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{4 c^3 d^3}-\frac {\left (3 b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^3 d^3}\\ &=\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)^2}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {2 i b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}-\frac {\left (i b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^3}+\frac {1}{4 (-i+c x)^2}-\frac {1}{4 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^3 d^3}+\frac {b^2 \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^3 d^3}-\frac {\left (3 b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{c^3 d^3}\\ &=\frac {b^2}{16 c^4 d^3 (i-c x)^2}+\frac {21 i b^2}{16 c^4 d^3 (i-c x)}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)^2}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {2 i b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}+\frac {\left (i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{16 c^3 d^3}+\frac {\left (i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{8 c^3 d^3}-\frac {\left (3 i b^2\right ) \int \frac {1}{1+c^2 x^2} \, dx}{2 c^3 d^3}\\ &=\frac {b^2}{16 c^4 d^3 (i-c x)^2}+\frac {21 i b^2}{16 c^4 d^3 (i-c x)}-\frac {21 i b^2 \tan ^{-1}(c x)}{16 c^4 d^3}+\frac {i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)^2}-\frac {11 b \left (a+b \tan ^{-1}(c x)\right )}{4 c^4 d^3 (i-c x)}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^4 d^3}+\frac {i x \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {\left (a+b \tan ^{-1}(c x)\right )^2}{2 c^4 d^3 (i-c x)^2}-\frac {3 i \left (a+b \tan ^{-1}(c x)\right )^2}{c^4 d^3 (i-c x)}+\frac {2 i b \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^4 d^3}-\frac {b^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 i b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^4 d^3}+\frac {3 b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^4 d^3}\\ \end {align*}
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Mathematica [A] time = 1.66, size = 507, normalized size = 1.32 \[ \frac {-96 a^2 \log \left (c^2 x^2+1\right )+64 i a^2 c x+\frac {192 i a^2}{c x-i}-\frac {32 a^2}{(c x-i)^2}-192 i a^2 \tan ^{-1}(c x)+4 i a b \left (-16 \log \left (c^2 x^2+1\right )-48 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-96 \tan ^{-1}(c x)^2-20 i \sin \left (2 \tan ^{-1}(c x)\right )+i \sin \left (4 \tan ^{-1}(c x)\right )+20 \cos \left (2 \tan ^{-1}(c x)\right )-\cos \left (4 \tan ^{-1}(c x)\right )+4 \tan ^{-1}(c x) \left (8 c x-24 i \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+10 \sin \left (2 \tan ^{-1}(c x)\right )-\sin \left (4 \tan ^{-1}(c x)\right )+10 i \cos \left (2 \tan ^{-1}(c x)\right )-i \cos \left (4 \tan ^{-1}(c x)\right )\right )\right )+i b^2 \left (-64 \left (3 \tan ^{-1}(c x)+i\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-96 i \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )-128 \tan ^{-1}(c x)^3+64 c x \tan ^{-1}(c x)^2-64 i \tan ^{-1}(c x)^2-192 i \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+128 \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )+80 \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )-8 \tan ^{-1}(c x)^2 \sin \left (4 \tan ^{-1}(c x)\right )-80 i \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )+4 i \tan ^{-1}(c x) \sin \left (4 \tan ^{-1}(c x)\right )-40 \sin \left (2 \tan ^{-1}(c x)\right )+\sin \left (4 \tan ^{-1}(c x)\right )+80 i \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )-8 i \tan ^{-1}(c x)^2 \cos \left (4 \tan ^{-1}(c x)\right )+80 \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )-4 \tan ^{-1}(c x) \cos \left (4 \tan ^{-1}(c x)\right )-40 i \cos \left (2 \tan ^{-1}(c x)\right )+i \cos \left (4 \tan ^{-1}(c x)\right )\right )}{64 c^4 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {-i \, b^{2} x^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 4 \, a b x^{3} \log \left (-\frac {c x + i}{c x - i}\right ) + 4 i \, a^{2} x^{3}}{4 \, c^{3} d^{3} x^{3} - 12 i \, c^{2} d^{3} x^{2} - 12 \, c d^{3} x + 4 i \, d^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.93, size = 5012, normalized size = 13.09 \[ \text {output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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