Optimal. Leaf size=100 \[ -\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {i c \log \left (2-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}+\frac {b c \text {Li}_2\left (\frac {2}{i c x+1}-1\right )}{2 d}+\frac {b c \log (x)}{d} \]
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Rubi [A] time = 0.15, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {4870, 4852, 266, 36, 29, 31, 4868, 2447} \[ \frac {b c \text {PolyLog}\left (2,-1+\frac {2}{1+i c x}\right )}{2 d}-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {i c \log \left (2-\frac {2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac {b c \log \left (c^2 x^2+1\right )}{2 d}+\frac {b c \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 266
Rule 2447
Rule 4852
Rule 4868
Rule 4870
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)} \, dx &=-\left ((i c) \int \frac {a+b \tan ^{-1}(c x)}{x (d+i c d x)} \, dx\right )+\frac {\int \frac {a+b \tan ^{-1}(c x)}{x^2} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {(b c) \int \frac {1}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac {\left (i b c^2\right ) \int \frac {\log \left (2-\frac {2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}+\frac {(b c) \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,x^2\right )}{2 d}-\frac {\left (b c^3\right ) \operatorname {Subst}\left (\int \frac {1}{1+c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac {a+b \tan ^{-1}(c x)}{d x}+\frac {b c \log (x)}{d}-\frac {b c \log \left (1+c^2 x^2\right )}{2 d}-\frac {i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac {2}{1+i c x}\right )}{d}+\frac {b c \text {Li}_2\left (-1+\frac {2}{1+i c x}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 149, normalized size = 1.49 \[ -\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {i c \log \left (\frac {2 i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}-\frac {i a c \log (x)}{d}+\frac {b c \left (2 \log (x)-\log \left (c^2 x^2+1\right )\right )}{2 d}+\frac {b c \text {Li}_2(-i c x)}{2 d}-\frac {b c \text {Li}_2(i c x)}{2 d}+\frac {b c \text {Li}_2\left (-\frac {c x+i}{i-c x}\right )}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \log \left (-\frac {c x + i}{c x - i}\right ) - 2 i \, a}{2 \, c d x^{3} - 2 i \, d x^{2}}, x\right ) \]
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 [B] time = 0.07, size = 252, normalized size = 2.52 \[ -\frac {a}{d x}-\frac {i c a \ln \left (c x \right )}{d}-\frac {i c b \arctan \left (c x \right ) \ln \left (c x \right )}{d}-\frac {c a \arctan \left (c x \right )}{d}-\frac {b \arctan \left (c x \right )}{d x}+\frac {i c a \ln \left (c^{2} x^{2}+1\right )}{2 d}+\frac {i c b \arctan \left (c x \right ) \ln \left (c x -i\right )}{d}+\frac {c b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 d}-\frac {c b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 d}+\frac {c b \dilog \left (i c x +1\right )}{2 d}-\frac {c b \dilog \left (-i c x +1\right )}{2 d}+\frac {c b \ln \left (c x -i\right ) \ln \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}+\frac {c b \dilog \left (-\frac {i \left (c x +i\right )}{2}\right )}{2 d}-\frac {c b \ln \left (c x -i\right )^{2}}{4 d}+\frac {c b \ln \left (c x \right )}{d}-\frac {b c \ln \left (c^{2} x^{2}+1\right )}{2 d} \]
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 \[ {\left (-i \, c \int \frac {\arctan \left (c x\right )}{c^{2} d x^{3} + d x}\,{d x} + \int \frac {\arctan \left (c x\right )}{c^{2} d x^{4} + d x^{2}}\,{d x}\right )} b + a {\left (\frac {i \, c \log \left (i \, c x + 1\right )}{d} - \frac {i \, c \log \relax (x)}{d} - \frac {1}{d x}\right )} \]
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 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x^2\,\left (d+c\,d\,x\,1{}\mathrm {i}\right )} \,d x \]
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 \[ - \frac {i \left (\int \frac {a}{c x^{3} - i x^{2}}\, dx + \int \frac {b \operatorname {atan}{\left (c x \right )}}{c x^{3} - i x^{2}}\, dx\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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