Optimal. Leaf size=181 \[ -\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d}+\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}+\frac {a \log (x)}{d}+\frac {i b \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d}+\frac {i b \text {Li}_2(-i c x)}{2 d}-\frac {i b \text {Li}_2(i c x)}{2 d}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d} \]
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Rubi [A] time = 0.19, antiderivative size = 181, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {4876, 4848, 2391, 4856, 2402, 2315, 2447} \[ \frac {i b \text {PolyLog}\left (2,1-\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{2 d}+\frac {i b \text {PolyLog}(2,-i c x)}{2 d}-\frac {i b \text {PolyLog}(2,i c x)}{2 d}-\frac {i b \text {PolyLog}\left (2,1-\frac {2}{1-i c x}\right )}{2 d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(1-i c x) (c d+i e)}\right )}{d}+\frac {\log \left (\frac {2}{1-i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}+\frac {a \log (x)}{d} \]
Antiderivative was successfully verified.
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Rule 2315
Rule 2391
Rule 2402
Rule 2447
Rule 4848
Rule 4856
Rule 4876
Rubi steps
\begin {align*} \int \frac {a+b \tan ^{-1}(c x)}{x (d+e x)} \, dx &=\int \left (\frac {a+b \tan ^{-1}(c x)}{d x}-\frac {e \left (a+b \tan ^{-1}(c x)\right )}{d (d+e x)}\right ) \, dx\\ &=\frac {\int \frac {a+b \tan ^{-1}(c x)}{x} \, dx}{d}-\frac {e \int \frac {a+b \tan ^{-1}(c x)}{d+e x} \, dx}{d}\\ &=\frac {a \log (x)}{d}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d}+\frac {(i b) \int \frac {\log (1-i c x)}{x} \, dx}{2 d}-\frac {(i b) \int \frac {\log (1+i c x)}{x} \, dx}{2 d}-\frac {(b c) \int \frac {\log \left (\frac {2}{1-i c x}\right )}{1+c^2 x^2} \, dx}{d}+\frac {(b c) \int \frac {\log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{1+c^2 x^2} \, dx}{d}\\ &=\frac {a \log (x)}{d}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d}+\frac {i b \text {Li}_2(-i c x)}{2 d}-\frac {i b \text {Li}_2(i c x)}{2 d}+\frac {i b \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d}-\frac {(i b) \operatorname {Subst}\left (\int \frac {\log (2 x)}{1-2 x} \, dx,x,\frac {1}{1-i c x}\right )}{d}\\ &=\frac {a \log (x)}{d}+\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2}{1-i c x}\right )}{d}-\frac {\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{d}+\frac {i b \text {Li}_2(-i c x)}{2 d}-\frac {i b \text {Li}_2(i c x)}{2 d}-\frac {i b \text {Li}_2\left (1-\frac {2}{1-i c x}\right )}{2 d}+\frac {i b \text {Li}_2\left (1-\frac {2 c (d+e x)}{(c d+i e) (1-i c x)}\right )}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 169, normalized size = 0.93 \[ \frac {-2 a \log (d+e x)+2 a \log (x)-i b \text {Li}_2\left (\frac {e (1-i c x)}{i c d+e}\right )+i b \text {Li}_2\left (-\frac {e (c x-i)}{c d+i e}\right )-i b \log (1-i c x) \log \left (\frac {c (d+e x)}{c d-i e}\right )+i b \log (1+i c x) \log \left (\frac {c (d+e x)}{c d+i e}\right )+i b \text {Li}_2(-i c x)-i b \text {Li}_2(i c x)}{2 d} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {b \arctan \left (c x\right ) + a}{e x^{2} + d x}, x\right ) \]
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 = 0.06, size = 260, normalized size = 1.44 \[ \frac {a \ln \left (c x \right )}{d}-\frac {a \ln \left (c e x +d c \right )}{d}+\frac {b \arctan \left (c x \right ) \ln \left (c x \right )}{d}-\frac {b \arctan \left (c x \right ) \ln \left (c e x +d c \right )}{d}+\frac {i b \ln \left (c x \right ) \ln \left (i c x +1\right )}{2 d}-\frac {i b \ln \left (c x \right ) \ln \left (-i c x +1\right )}{2 d}+\frac {i b \dilog \left (i c x +1\right )}{2 d}-\frac {i b \dilog \left (-i c x +1\right )}{2 d}-\frac {i b \ln \left (c e x +d c \right ) \ln \left (\frac {-c e x +i e}{d c +i e}\right )}{2 d}+\frac {i b \ln \left (c e x +d c \right ) \ln \left (\frac {c e x +i e}{-d c +i e}\right )}{2 d}-\frac {i b \dilog \left (\frac {-c e x +i e}{d c +i e}\right )}{2 d}+\frac {i b \dilog \left (\frac {c e x +i e}{-d c +i e}\right )}{2 d} \]
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 \[ -a {\left (\frac {\log \left (e x + d\right )}{d} - \frac {\log \relax (x)}{d}\right )} + 2 \, b \int \frac {\arctan \left (c x\right )}{2 \, {\left (e x^{2} + d x\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {a+b\,\mathrm {atan}\left (c\,x\right )}{x\,\left (d+e\,x\right )} \,d x \]
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 \[ \int \frac {a + b \operatorname {atan}{\left (c x \right )}}{x \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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