Optimal. Leaf size=150 \[ \frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {a \log (x)}{d^2}+\frac {i b \text {Li}_2(-i c x)}{2 d^2}-\frac {i b \text {Li}_2(i c x)}{2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{i c x+1}\right )}{2 d^2}+\frac {b}{2 d^2 (-c x+i)}-\frac {b \tan ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.19, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 10, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.435, Rules used = {4876, 4848, 2391, 4862, 627, 44, 203, 4854, 2402, 2315} \[ \frac {i b \text {PolyLog}(2,-i c x)}{2 d^2}-\frac {i b \text {PolyLog}(2,i c x)}{2 d^2}+\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}+\frac {\log \left (\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}+\frac {a \log (x)}{d^2}+\frac {b}{2 d^2 (-c x+i)}-\frac {b \tan ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 44
Rule 203
Rule 627
Rule 2315
Rule 2391
Rule 2402
Rule 4848
Rule 4854
Rule 4862
Rule 4876
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x (d+i c d x)^2} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d^2 x}+\frac {i c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-i+c x)^2}-\frac {c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-i+c x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x} \, dx}{d^2}+\frac {(i c) \int \frac {a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{d^2}-\frac {c \int \frac {a+b \tan ^{-1}(c x)}{-i+c x} \, dx}{d^2}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {(i b) \int \frac {\log (1-i c x)}{x} \, dx}{2 d^2}-\frac {(i b) \int \frac {\log (1+i c x)}{x} \, dx}{2 d^2}+\frac {(i b c) \int \frac {1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2}-\frac {(b c) \int \frac {\log \left (\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \text {Li}_2(-i c x)}{2 d^2}-\frac {i b \text {Li}_2(i c x)}{2 d^2}+\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1+i c x}\right )}{d^2}+\frac {(i b c) \int \frac {1}{(-i+c x)^2 (i+c x)} \, dx}{d^2}\\ &=\frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \text {Li}_2(-i c x)}{2 d^2}-\frac {i b \text {Li}_2(i c x)}{2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 d^2}+\frac {(i b c) \int \left (-\frac {i}{2 (-i+c x)^2}+\frac {i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}\\ &=\frac {b}{2 d^2 (i-c x)}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \text {Li}_2(-i c x)}{2 d^2}-\frac {i b \text {Li}_2(i c x)}{2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 d^2}-\frac {(b c) \int \frac {1}{1+c^2 x^2} \, dx}{2 d^2}\\ &=\frac {b}{2 d^2 (i-c x)}-\frac {b \tan ^{-1}(c x)}{2 d^2}+\frac {i \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}+\frac {a \log (x)}{d^2}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1+i c x}\right )}{d^2}+\frac {i b \text {Li}_2(-i c x)}{2 d^2}-\frac {i b \text {Li}_2(i c x)}{2 d^2}+\frac {i b \text {Li}_2\left (1-\frac {2}{1+i c x}\right )}{2 d^2}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 128, normalized size = 0.85 \[ \frac {-\frac {2 i \left (a+b \tan ^{-1}(c x)\right )}{c x-i}+2 \log \left (\frac {2 i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )+2 a \log (x)+i b \text {Li}_2(-i c x)-i b \text {Li}_2(i c x)+i b \text {Li}_2\left (\frac {c x+i}{c x-i}\right )+b \left (-\tan ^{-1}(c x)+\frac {1}{-c x+i}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {-i \, b \log \left (-\frac {c x + i}{c x - i}\right ) - 2 \, a}{2 \, {\left (c^{2} d^{2} x^{3} - 2 i \, c d^{2} x^{2} - d^{2} x\right )}}, x\right ) \]
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 [A] time = 0.08, size = 251, normalized size = 1.67 \[ \frac {a \ln \left (c x \right )}{d^{2}}-\frac {i b \arctan \left (c x \right )}{d^{2} \left (c x -i\right )}-\frac {a \ln \left (c^{2} x^{2}+1\right )}{2 d^{2}}+\frac {i b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d^{2}}+\frac {b \ln \left (c x \right ) \arctan \left (c x \right )}{d^{2}}+\frac {i b \dilog \left (i c x +1\right )}{2 d^{2}}-\frac {b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d^{2}}+\frac {i b \ln \left (-\frac {i \left (c x +i\right )}{2}\right ) \ln \left (c x -i\right )}{2 d^{2}}+\frac {i b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 d^{2}}-\frac {i a \arctan \left (c x \right )}{d^{2}}-\frac {i b \ln \left (c x -i\right )^{2}}{4 d^{2}}-\frac {b \arctan \left (c x \right )}{2 d^{2}}-\frac {b}{2 d^{2} \left (c x -i\right )}-\frac {i b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 d^{2}}-\frac {i b \dilog \left (-i c x +1\right )}{2 d^{2}}-\frac {i a}{d^{2} \left (c x -i\right )} \]
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 \[ {\left (-2 i \, c \int \frac {\arctan \left (c x\right )}{c^{4} d^{2} x^{4} + 2 \, c^{2} d^{2} x^{2} + d^{2}}\,{d x} - \int \frac {{\left (c^{2} x^{2} - 1\right )} \arctan \left (c x\right )}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x}\right )} b + a {\left (-\frac {i}{c d^{2} x - i \, d^{2}} - \frac {\log \left (c x - i\right )}{d^{2}} + \frac {\log \relax (x)}{d^{2}}\right )} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,{\left (d+c\,d\,x\,1{}\mathrm {i}\right )}^2} \,d x \]
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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