Optimal. Leaf size=216 \[ -i b d \text {Li}_2\left (1-\frac {2}{i c x+1}\right ) \left (a+b \tan ^{-1}(c x)\right )+i b d \text {Li}_2\left (\frac {2}{i c x+1}-1\right ) \left (a+b \tan ^{-1}(c x)\right )-d \left (a+b \tan ^{-1}(c x)\right )^2+i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 i b d \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 d \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2+b^2 (-d) \text {Li}_2\left (1-\frac {2}{i c x+1}\right )-\frac {1}{2} b^2 d \text {Li}_3\left (1-\frac {2}{i c x+1}\right )+\frac {1}{2} b^2 d \text {Li}_3\left (\frac {2}{i c x+1}-1\right ) \]
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Rubi [A] time = 0.42, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {4876, 4846, 4920, 4854, 2402, 2315, 4850, 4988, 4884, 4994, 6610} \[ -i b d \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+i b d \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+b^2 (-d) \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d \text {PolyLog}\left (3,1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d \text {PolyLog}\left (3,-1+\frac {2}{1+i c x}\right )-d \left (a+b \tan ^{-1}(c x)\right )^2+i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 i b d \log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 d \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )^2 \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2402
Rule 4846
Rule 4850
Rule 4854
Rule 4876
Rule 4884
Rule 4920
Rule 4988
Rule 4994
Rule 6610
Rubi steps
\begin {align*} \int \frac {(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx &=\int \left (i c d \left (a+b \tan ^{-1}(c x)\right )^2+\frac {d \left (a+b \tan ^{-1}(c x)\right )^2}{x}\right ) \, dx\\ &=d \int \frac {\left (a+b \tan ^{-1}(c x)\right )^2}{x} \, dx+(i c d) \int \left (a+b \tan ^{-1}(c x)\right )^2 \, dx\\ &=i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 d \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )-(4 b c d) \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 (2 i b c^2 d\right ) \int \frac {x \left (a+b \tan ^{-1}(c x)\right )}{1+c^2 x^2} \, dx\\ &=-d \left (a+b \tan ^{-1}(c x)\right )^2+i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 d \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+(2 i b c d) \int \frac {a+b \tan ^{-1}(c x)}{i-c x} \, dx+(2 b c d) \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-(2 b c d) \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\\ &=-d \left (a+b \tan ^{-1}(c x)\right )^2+i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 d \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+2 i b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-i b d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+i b d \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 d\right ) \int \frac {\text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (i b^2 c d\right ) \int \frac {\text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx-\left (2 i b^2 c d\right ) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx\\ &=-d \left (a+b \tan ^{-1}(c x)\right )^2+i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 d \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+2 i b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-i b d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+i b d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d \text {Li}_3\left (1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )-\left (2 b^2 d\right ) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )\\ &=-d \left (a+b \tan ^{-1}(c x)\right )^2+i c d x \left (a+b \tan ^{-1}(c x)\right )^2+2 d \left (a+b \tan ^{-1}(c x)\right )^2 \tanh ^{-1}\left (1-\frac {2}{1+i c x}\right )+2 i b d \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )-b^2 d \text {Li}_2\left (1-\frac {2}{1+i c x}\right )-i b d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (1-\frac {2}{1+i c x}\right )+i b d \left (a+b \tan ^{-1}(c x)\right ) \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )-\frac {1}{2} b^2 d \text {Li}_3\left (1-\frac {2}{1+i c x}\right )+\frac {1}{2} b^2 d \text {Li}_3\left (-1+\frac {2}{1+i c x}\right )\\ \end {align*}
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Mathematica [A] time = 0.48, size = 272, normalized size = 1.26 \[ d \left (i a^2 c x+a^2 \log (c x)+i a b \left (2 c x \tan ^{-1}(c x)-\log \left (c^2 x^2+1\right )\right )+i a b (\text {Li}_2(-i c x)-\text {Li}_2(i c x))+b^2 \left (\text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+\tan ^{-1}(c x) \left ((1+i c x) \tan ^{-1}(c x)+2 i \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )\right )+b^2 \left (i \tan ^{-1}(c x) \text {Li}_2\left (e^{-2 i \tan ^{-1}(c x)}\right )+i \tan ^{-1}(c x) \text {Li}_2\left (-e^{2 i \tan ^{-1}(c x)}\right )+\frac {1}{2} \text {Li}_3\left (e^{-2 i \tan ^{-1}(c x)}\right )-\frac {1}{2} \text {Li}_3\left (-e^{2 i \tan ^{-1}(c x)}\right )+\frac {2}{3} i \tan ^{-1}(c x)^3+\tan ^{-1}(c x)^2 \log \left (1-e^{-2 i \tan ^{-1}(c x)}\right )-\tan ^{-1}(c x)^2 \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )-\frac {i \pi ^3}{24}\right )\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {4 i \, a^{2} c d x + 4 \, a^{2} d + {\left (-i \, b^{2} c d x - b^{2} d\right )} \log \left (-\frac {c x + i}{c x - i}\right )^{2} - {\left (4 \, a b c d x - 4 i \, a b d\right )} \log \left (-\frac {c x + i}{c x - i}\right )}{4 \, x}, x\right ) \]
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 = 0.79, size = 7034, normalized size = 32.56 \[ \text {output too large to display} \]
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 {1}{4} i \, b^{2} c d x \arctan \left (c x\right )^{2} + 12 i \, b^{2} c^{3} d \int \frac {x^{3} \arctan \left (c x\right )^{2}}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + 4 \, b^{2} c^{3} d \int \frac {x^{3} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + i \, b^{2} c^{3} d \int \frac {x^{3} \log \left (c^{2} x^{2} + 1\right )^{2}}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + 8 \, b^{2} c^{3} d \int \frac {x^{3} \arctan \left (c x\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + 4 i \, b^{2} c^{3} d \int \frac {x^{3} \log \left (c^{2} x^{2} + 1\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} - \frac {1}{4} \, b^{2} c d x \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right ) - \frac {1}{16} i \, b^{2} c d x \log \left (c^{2} x^{2} + 1\right )^{2} + \frac {1}{4} i \, b^{2} d \arctan \left (c x\right )^{3} + 12 \, b^{2} c^{2} d \int \frac {x^{2} \arctan \left (c x\right )^{2}}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} - 4 i \, b^{2} c^{2} d \int \frac {x^{2} \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + 32 \, a b c^{2} d \int \frac {x^{2} \arctan \left (c x\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} - 8 i \, b^{2} c^{2} d \int \frac {x^{2} \arctan \left (c x\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + \frac {1}{96} \, b^{2} d \log \left (c^{2} x^{2} + 1\right )^{3} + i \, a^{2} c d x + 4 \, b^{2} c d \int \frac {x \arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + i \, b^{2} c d \int \frac {x \log \left (c^{2} x^{2} + 1\right )^{2}}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + \frac {1}{16} \, b^{2} d \log \left (c^{2} x^{2} + 1\right )^{2} + i \, {\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} a b d + 12 \, b^{2} d \int \frac {\arctan \left (c x\right )^{2}}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} - 4 i \, b^{2} d \int \frac {\arctan \left (c x\right ) \log \left (c^{2} x^{2} + 1\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + b^{2} d \int \frac {\log \left (c^{2} x^{2} + 1\right )^{2}}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + 32 \, a b d \int \frac {\arctan \left (c x\right )}{16 \, {\left (c^{2} x^{3} + x\right )}}\,{d x} + a^{2} d \log \relax (x) \]
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 \[ \int \frac {{\left (a+b\,\mathrm {atan}\left (c\,x\right )\right )}^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ i d \left (\int a^{2} c\, dx + \int \left (- \frac {i a^{2}}{x}\right )\, dx + \int b^{2} c \operatorname {atan}^{2}{\left (c x \right )}\, dx + \int \left (- \frac {i b^{2} \operatorname {atan}^{2}{\left (c x \right )}}{x}\right )\, dx + \int 2 a b c \operatorname {atan}{\left (c x \right )}\, dx + \int \left (- \frac {2 i a b \operatorname {atan}{\left (c x \right )}}{x}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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