Optimal. Leaf size=304 \[ \frac {b \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^3}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (-c x+i)}+\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (-c x+i)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (-c x+i)}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (-c x+i)^2}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac {i \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}-\frac {i b^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )}{2 c^3 d^3}+\frac {13 b^2}{16 c^3 d^3 (-c x+i)}-\frac {i b^2}{16 c^3 d^3 (-c x+i)^2}-\frac {13 b^2 \tan ^{-1}(c x)}{16 c^3 d^3} \]
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Rubi [A] time = 0.56, antiderivative size = 304, normalized size of antiderivative = 1.00, number of steps used = 26, number of rules used = 10, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {4876, 4864, 4862, 627, 44, 203, 4884, 4854, 4994, 6610} \[ \frac {b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^3 d^3}-\frac {i b^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (-c x+i)}+\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (-c x+i)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (-c x+i)}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (-c x+i)^2}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}-\frac {i \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3}+\frac {13 b^2}{16 c^3 d^3 (-c x+i)}-\frac {i b^2}{16 c^3 d^3 (-c x+i)^2}-\frac {13 b^2 \tan ^{-1}(c x)}{16 c^3 d^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 4854
Rule 4862
Rule 4864
Rule 4876
Rule 4884
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {x^2 \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^2 d^3 (-i+c x)^3}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^3 (-i+c x)^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{c^2 d^3 (-i+c x)}\right ) \, dx\\ &=-\frac {i \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^3} \, dx}{c^2 d^3}+\frac {i \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{-i+c x} \, dx}{c^2 d^3}-\frac {2 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{(-i+c x)^2} \, dx}{c^2 d^3}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {(i 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^2 d^3}+\frac {(2 i 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^2 d^3}-\frac {(4 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^2 d^3}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {(i b) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{4 c^2 d^3}+\frac {(i b) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{4 c^2 d^3}+\frac {(2 i b) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{c^2 d^3}-\frac {(2 i b) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx}{c^2 d^3}-\frac {b \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^3} \, dx}{2 c^2 d^3}-\frac {b^2 \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^2 d^3}\\ &=\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)^2}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {i b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}-\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{4 c^2 d^3}+\frac {\left (2 i b^2\right ) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{c^2 d^3}-\frac {b^2 \int \frac {1}{(-i+c x)^2 \left (1+c^2 x^2\right )} \, dx}{4 c^2 d^3}\\ &=\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)^2}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {i b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}-\frac {\left (i b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{4 c^2 d^3}+\frac {\left (2 i b^2\right ) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{c^2 d^3}-\frac {b^2 \int \frac {1}{(-i+c x)^3 (i+c x)} \, dx}{4 c^2 d^3}\\ &=\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)^2}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {i b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}-\frac {\left (i b^2\right ) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{4 c^2 d^3}+\frac {\left (2 i 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^2 d^3}-\frac {b^2 \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^2 d^3}\\ &=-\frac {i b^2}{16 c^3 d^3 (i-c x)^2}+\frac {13 b^2}{16 c^3 d^3 (i-c x)}+\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)^2}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {i b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}+\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{16 c^2 d^3}+\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{8 c^2 d^3}-\frac {b^2 \int \frac {1}{1+c^2 x^2} \, dx}{c^2 d^3}\\ &=-\frac {i b^2}{16 c^3 d^3 (i-c x)^2}+\frac {13 b^2}{16 c^3 d^3 (i-c x)}-\frac {13 b^2 \tan ^{-1}(c x)}{16 c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)^2}+\frac {7 i b \left (a+b \tan ^{-1}(c x)\right )}{4 c^3 d^3 (i-c x)}-\frac {7 i \left (a+b \tan ^{-1}(c x)\right )^2}{8 c^3 d^3}+\frac {i \left (a+b \tan ^{-1}(c x)\right )^2}{2 c^3 d^3 (i-c x)^2}-\frac {2 \left (a+b \tan ^{-1}(c x)\right )^2}{c^3 d^3 (i-c x)}-\frac {i \left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{c^3 d^3}+\frac {b \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{c^3 d^3}-\frac {i b^2 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c^3 d^3}\\ \end {align*}
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Mathematica [A] time = 1.26, size = 431, normalized size = 1.42 \[ \frac {96 i a^2 \log \left (c^2 x^2+1\right )+\frac {384 a^2}{c x-i}+\frac {96 i a^2}{(c x-i)^2}-192 a^2 \tan ^{-1}(c x)-12 a b \left (16 \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+32 \tan ^{-1}(c x)^2+12 i \sin \left (2 \tan ^{-1}(c x)\right )-i \sin \left (4 \tan ^{-1}(c x)\right )-12 \cos \left (2 \tan ^{-1}(c x)\right )+\cos \left (4 \tan ^{-1}(c x)\right )+4 \tan ^{-1}(c x) \left (8 i \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-6 \sin \left (2 \tan ^{-1}(c x)\right )+\sin \left (4 \tan ^{-1}(c x)\right )-6 i \cos \left (2 \tan ^{-1}(c x)\right )+i \cos \left (4 \tan ^{-1}(c x)\right )\right )\right )-b^2 \left (192 \tan ^{-1}(c x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+96 i \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )+128 \tan ^{-1}(c x)^3+192 i \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-144 \tan ^{-1}(c x)^2 \sin \left (2 \tan ^{-1}(c x)\right )+24 \tan ^{-1}(c x)^2 \sin \left (4 \tan ^{-1}(c x)\right )+144 i \tan ^{-1}(c x) \sin \left (2 \tan ^{-1}(c x)\right )-12 i \tan ^{-1}(c x) \sin \left (4 \tan ^{-1}(c x)\right )+72 \sin \left (2 \tan ^{-1}(c x)\right )-3 \sin \left (4 \tan ^{-1}(c x)\right )-144 i \tan ^{-1}(c x)^2 \cos \left (2 \tan ^{-1}(c x)\right )+24 i \tan ^{-1}(c x)^2 \cos \left (4 \tan ^{-1}(c x)\right )-144 \tan ^{-1}(c x) \cos \left (2 \tan ^{-1}(c x)\right )+12 \tan ^{-1}(c x) \cos \left (4 \tan ^{-1}(c x)\right )+72 i \cos \left (2 \tan ^{-1}(c x)\right )-3 i \cos \left (4 \tan ^{-1}(c x)\right )\right )}{192 c^3 d^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {-i \, b^{2} x^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 4 \, a b x^{2} \log \left (-\frac {c x + i}{c x - i}\right ) + 4 i \, a^{2} x^{2}}{4 \, c^{3} d^{3} x^{3} - 12 i \, c^{2} d^{3} x^{2} - 12 \, c d^{3} x + 4 i \, d^{3}}, x\right ) \]
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 = 0.47, size = 1276, normalized size = 4.20 \[ \text {result too large to display} \]
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 \[ \int \frac {x^2\,{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2}{{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3} \,d x \]
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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