Optimal. Leaf size=77 \[ -\frac{1}{2} b c d \text{PolyLog}(2,-i c x)+\frac{1}{2} b c d \text{PolyLog}(2,i c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x}+i a c d \log (x)-\frac{1}{2} b c d \log \left (c^2 x^2+1\right )+b c d \log (x) \]
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Rubi [A] time = 0.100228, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {4876, 4852, 266, 36, 29, 31, 4848, 2391} \[ -\frac{1}{2} b c d \text{PolyLog}(2,-i c x)+\frac{1}{2} b c d \text{PolyLog}(2,i c x)-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x}+i a c d \log (x)-\frac{1}{2} b c d \log \left (c^2 x^2+1\right )+b c d \log (x) \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx+(i c d) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x}+i a c d \log (x)-\frac{1}{2} (b c d) \int \frac{\log (1-i c x)}{x} \, dx+\frac{1}{2} (b c d) \int \frac{\log (1+i c x)}{x} \, dx+(b c d) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x}+i a c d \log (x)-\frac{1}{2} b c d \text{Li}_2(-i c x)+\frac{1}{2} b c d \text{Li}_2(i c x)+\frac{1}{2} (b c d) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x}+i a c d \log (x)-\frac{1}{2} b c d \text{Li}_2(-i c x)+\frac{1}{2} b c d \text{Li}_2(i c x)+\frac{1}{2} (b c d) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (b c^3 d\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{x}+i a c d \log (x)+b c d \log (x)-\frac{1}{2} b c d \log \left (1+c^2 x^2\right )-\frac{1}{2} b c d \text{Li}_2(-i c x)+\frac{1}{2} b c d \text{Li}_2(i c x)\\ \end{align*}
Mathematica [A] time = 0.048796, size = 75, normalized size = 0.97 \[ \frac{d \left (-b c x \text{PolyLog}(2,-i c x)+b c x \text{PolyLog}(2,i c x)+2 i a c x \log (x)-2 a-b c x \log \left (c^2 x^2+1\right )+2 b c x \log (x)-2 b \tan ^{-1}(c x)\right )}{2 x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 127, normalized size = 1.7 \begin{align*} -{\frac{da}{x}}+icda\ln \left ( cx \right ) -{\frac{db\arctan \left ( cx \right ) }{x}}+icdb\arctan \left ( cx \right ) \ln \left ( cx \right ) -{\frac{cdb\ln \left ( cx \right ) \ln \left ( 1+icx \right ) }{2}}+{\frac{cdb\ln \left ( cx \right ) \ln \left ( 1-icx \right ) }{2}}-{\frac{cdb{\it dilog} \left ( 1+icx \right ) }{2}}+{\frac{cdb{\it dilog} \left ( 1-icx \right ) }{2}}-{\frac{bcd\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2}}+cdb\ln \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} i \, b c d \int \frac{\arctan \left (c x\right )}{x}\,{d x} + i \, a c d \log \left (x\right ) - \frac{1}{2} \,{\left (c{\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \arctan \left (c x\right )}{x}\right )} b d - \frac{a d}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{2 i \, a c d x + 2 \, a d -{\left (b c d x - i \, b d\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{2 \, x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} d \left (\int \frac{a}{x^{2}}\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{i a c}{x}\, dx + \int \frac{i b c \operatorname{atan}{\left (c x \right )}}{x}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (i \, c d x + d\right )}{\left (b \arctan \left (c x\right ) + a\right )}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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