Optimal. Leaf size=402 \[ -\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+i a b c^2 d^3 x-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2+3 b c d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )-3 b c d^3 \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) \left (a+b \tan ^{-1}(c x)\right )-\frac {9}{2} i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-6 b c d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 b c d^3 \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+6 i c d^3 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i b^2 c d^3 \log \left (c^2 x^2+1\right )+i b^2 c^2 d^3 x \tan ^{-1}(c x)-i b^2 c d^3 \text {Li}_2\left (\frac {2}{1-i c x}-1\right )-3 i b^2 c d^3 \text {Li}_2\left (1-\frac {2}{i c x+1}\right )-\frac {3}{2} i b^2 c d^3 \text {Li}_3\left (1-\frac {2}{i c x+1}\right )+\frac {3}{2} i b^2 c d^3 \text {Li}_3\left (\frac {2}{i c x+1}-1\right ) \]
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Rubi [A] time = 0.74, antiderivative size = 402, normalized size of antiderivative = 1.00, number of steps used = 23, number of rules used = 17, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.680, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4852, 4924, 4868, 2447, 4850, 4988, 4884, 4994, 6610, 4916, 260} \[ 3 b c d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-3 b c d^3 \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )-i b^2 c d^3 \text {PolyLog}\left (2,-1+\frac {2}{1-i c x}\right )-3 i b^2 c d^3 \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-\frac {3}{2} i b^2 c d^3 \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b^2 c d^3 \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )-\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+i a b c^2 d^3 x-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac {9}{2} i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-6 b c d^3 \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 b c d^3 \log \left (2-\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+6 i c d^3 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i b^2 c d^3 \log \left (c^2 x^2+1\right )+i b^2 c^2 d^3 x \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 260
Rule 2315
Rule 2402
Rule 2447
Rule 4846
Rule 4850
Rule 4852
Rule 4854
Rule 4868
Rule 4876
Rule 4884
Rule 4916
Rule 4920
Rule 4924
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx &=\int \left (-3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )^2+\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x^2}+\frac {3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2\right ) \, dx\\ &=d^3 \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x^2} \, dx+\left (3 i c d^3\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx-\left (3 c^2 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx-\left (i c^3 d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+6 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+\left (2 b c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x \left (1+c^2 x^2\right )} \, dx-\left (12 i b c^2 d^3\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (6 b c^3 d^3\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx+\left (i b c^4 d^3\right ) \int \frac {x^2 \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-4 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+6 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+\left (2 i b c d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{x (i+c x)} \, dx+\left (i b c^2 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (i b c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{1+c^2 x^2} \, dx+\left (6 i b c^2 d^3\right ) \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-\left (6 i b c^2 d^3\right ) \int \frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (6 b c^2 d^3\right ) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx\\ &=i a b c^2 d^3 x-\frac {9}{2} i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+6 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-6 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+2 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )+3 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )+\left (i b^2 c^2 d^3\right ) \int \tan ^{-1}(c x) \, dx-\left (2 b^2 c^2 d^3\right ) \int \frac {\log \left (2-\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx-\left (3 b^2 c^2 d^3\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (3 b^2 c^2 d^3\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx+\left (6 b^2 c^2 d^3\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=i a b c^2 d^3 x+i b^2 c^2 d^3 x \tan ^{-1}(c x)-\frac {9}{2} i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+6 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-6 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )+2 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^3 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )+3 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )-\frac {3}{2} i b^2 c d^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b^2 c d^3 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )-\left (6 i b^2 c d^3\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )-\left (i b^2 c^3 d^3\right ) \int \frac {x}{1+c^2 x^2} \, dx\\ &=i a b c^2 d^3 x+i b^2 c^2 d^3 x \tan ^{-1}(c x)-\frac {9}{2} i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2-\frac {d^3 \left (a+b \tan ^{-1}(c x)\right )^2}{x}-3 c^2 d^3 x \left (a+b \tan ^{-1}(c x)\right )^2-\frac {1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )^2+6 i c d^3 \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-6 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-\frac {1}{2} i b^2 c d^3 \log \left (1+c^2 x^2\right )+2 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1-i c x}\right )-i b^2 c d^3 \text {Li}_2\left (-1+\frac {2}{1-i c x}\right )-3 i b^2 c d^3 \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+3 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-3 b c d^3 \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )-\frac {3}{2} i b^2 c d^3 \text {Li}_3\left (1-\frac {2}{1+i c x}\right )+\frac {3}{2} i b^2 c d^3 \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.63, size = 512, normalized size = 1.27 \[ \frac {d^3 \left (-4 i a^2 c^3 x^3-24 a^2 c^2 x^2+24 i a^2 c x \log (x)-8 a^2-8 i a b c^3 x^3 \tan ^{-1}(c x)+8 i a b c^2 x^2+16 a b c x \log \left (c^2 x^2+1\right )-48 a b c^2 x^2 \tan ^{-1}(c x)-24 a b c x \text {Li}_2(-i c x)+24 a b c x \text {Li}_2(i c x)+16 a b c x \log (c x)-8 i a b c x \tan ^{-1}(c x)-16 a b \tan ^{-1}(c x)-4 i b^2 c^3 x^3 \tan ^{-1}(c x)^2-4 i b^2 c x \log \left (c^2 x^2+1\right )-24 b^2 c^2 x^2 \tan ^{-1}(c x)^2+8 i b^2 c^2 x^2 \tan ^{-1}(c x)-24 b^2 c x \tan ^{-1}(c x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(c x)}\right )-24 b^2 c x \left (\tan ^{-1}(c x)-i\right ) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )-8 i b^2 c x \text {Li}_2\left (e^{2 i \tan ^{-1}(c x)}\right )+12 i b^2 c x \text {Li}_3\left (e^{-2 i \tan ^{-1}(c x)}\right )-12 i b^2 c x \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )+\pi ^3 b^2 c x-16 b^2 c x \tan ^{-1}(c x)^3+12 i b^2 c x \tan ^{-1}(c x)^2-8 b^2 \tan ^{-1}(c x)^2+24 i b^2 c x \tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )+16 b^2 c x \tan ^{-1}(c x) \log \left (1-e^{2 i \tan ^{-1}(c x)}\right )-24 i b^2 c x \tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-48 b^2 c x \tan ^{-1}(c x) \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )}{8 x} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {-4 i \, a^{2} c^{3} d^{3} x^{3} - 12 \, a^{2} c^{2} d^{3} x^{2} + 12 i \, a^{2} c d^{3} x + 4 \, a^{2} d^{3} + {\left (i \, b^{2} c^{3} d^{3} x^{3} + 3 \, b^{2} c^{2} d^{3} x^{2} - 3 i \, b^{2} c d^{3} x - b^{2} d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} + {\left (4 \, a b c^{3} d^{3} x^{3} - 12 i \, a b c^{2} d^{3} x^{2} - 12 \, a b c d^{3} x + 4 i \, a b d^{3}\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{4 \, x^{2}}, x\right ) \]
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 = 8.25, size = 1739, normalized size = 4.33 \[ \text {result too large to display} \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^3}{x^2} \,d x \]
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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