Optimal. Leaf size=139 \[ \frac {3 i b^2 \text {Li}_3\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c d}-\frac {3 b \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d}+\frac {i \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{c d}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{i c x+1}\right )}{4 c d} \]
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Rubi [A] time = 0.23, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {4854, 4884, 4994, 4998, 6610} \[ \frac {3 i b^2 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{2 c d}-\frac {3 b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2}{2 c d}+\frac {3 b^3 \text {PolyLog}\left (4,1-\frac {2}{1+i c x}\right )}{4 c d}+\frac {i \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3}{c d} \]
Antiderivative was successfully verified.
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Rule 4854
Rule 4884
Rule 4994
Rule 4998
Rule 6610
Rubi steps
\begin {align*} \int \frac {\left (a+b \tan ^{-1}(c x)\right )^3}{d+i c d x} \, dx &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {(3 i b) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2 \log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c d}+\frac {\left (3 b^2\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c d}+\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c d}-\frac {\left (3 i b^3\right ) \int \frac {\text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{2 d}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )^3 \log \left (\frac {2}{1+i c x}\right )}{c d}-\frac {3 b \left (a+b \tan ^{-1}(c x)\right )^2 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 c d}+\frac {3 i b^2 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_3\left (1-\frac {2}{1+i c x}\right )}{2 c d}+\frac {3 b^3 \text {Li}_4\left (1-\frac {2}{1+i c x}\right )}{4 c d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 133, normalized size = 0.96 \[ \frac {i \left (4 \log \left (\frac {2 d}{d+i c d x}\right ) \left (a+b \tan ^{-1}(c x)\right )^3+3 i b \left (2 \text {Li}_2\left (\frac {c x+i}{c x-i}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-b \left (2 i \text {Li}_3\left (\frac {c x+i}{c x-i}\right ) \left (a+b \tan ^{-1}(c x)\right )+b \text {Li}_4\left (\frac {c x+i}{c x-i}\right )\right )\right )\right )}{4 c d} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {b^{3} \log \left (-\frac {c x + i}{c x - i}\right )^{3} - 6 i \, a b^{2} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - 12 \, a^{2} b \log \left (-\frac {c x + i}{c x - i}\right ) + 8 i \, a^{3}}{8 \, c d x - 8 i \, d}, x\right ) \]
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 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 2044, normalized size = 14.71 \[ \text {result too large to display} \]
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 \[ -\frac {i \, a^{3} \log \left (i \, c d x + d\right )}{c d} + \frac {16 \, b^{3} \arctan \left (c x\right )^{4} - b^{3} \log \left (c^{2} x^{2} + 1\right )^{4} + {\left (b^{3} c {\left (\frac {4 \, \log \left (c^{2} d x^{2} + d\right ) \log \left (c^{2} x^{2} + 1\right )^{3}}{c^{2} d} + \frac {\frac {4 \, {\left (\log \left (c^{2} x^{2} + 1\right )^{3} + 3 \, \log \left (c^{2} x^{2} + 1\right )^{2} \log \relax (d)\right )} \log \left (c^{2} x^{2} + 1\right )}{c^{2}} - \frac {\log \left (c^{2} x^{2} + 1\right )^{4} + 4 \, \log \left (c^{2} x^{2} + 1\right )^{3} \log \relax (d)}{c^{2}}}{d} - \frac {6 \, {\left (\log \left (c^{2} x^{2} + 1\right )^{2} + 2 \, \log \left (c^{2} x^{2} + 1\right ) \log \relax (d)\right )} \log \left (c^{2} x^{2} + 1\right )^{2}}{c^{2} d}\right )} + \frac {16 \, b^{3} \arctan \left (c x\right )^{4}}{c d} + \frac {128 \, a b^{2} \arctan \left (c x\right )^{3}}{c d} + \frac {192 \, a^{2} b \arctan \left (c x\right )^{2}}{c d}\right )} c d - 4 i \, c d \int \frac {32 \, {\left (b^{3} c x \arctan \left (c x\right )^{3} + 3 \, a b^{2} c x \arctan \left (c x\right )^{2} + 3 \, a^{2} b c x \arctan \left (c x\right )\right )}}{c^{2} d x^{2} + d}\,{d x}}{128 \, c d} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^3}{d+c\,d\,x\,1{}\mathrm {i}} \,d x \]
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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