Optimal. Leaf size=194 \[ \frac{b c \text{PolyLog}(2,-i c x)}{d^2}-\frac{b c \text{PolyLog}(2,i c x)}{d^2}+\frac{b c \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{d^2}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{2 i c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{2 i a c \log (x)}{d^2}-\frac{b c \log \left (c^2 x^2+1\right )}{2 d^2}-\frac{i b c}{2 d^2 (-c x+i)}+\frac{b c \log (x)}{d^2}+\frac{i b c \tan ^{-1}(c x)}{2 d^2} \]
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Rubi [A] time = 0.241161, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 18, number of rules used = 15, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.652, Rules used = {4876, 4852, 266, 36, 29, 31, 4848, 2391, 4862, 627, 44, 203, 4854, 2402, 2315} \[ \frac{b c \text{PolyLog}(2,-i c x)}{d^2}-\frac{b c \text{PolyLog}(2,i c x)}{d^2}+\frac{b c \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{d^2}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-c x+i)}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}-\frac{2 i c \log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d^2}-\frac{2 i a c \log (x)}{d^2}-\frac{b c \log \left (c^2 x^2+1\right )}{2 d^2}-\frac{i b c}{2 d^2 (-c x+i)}+\frac{b c \log (x)}{d^2}+\frac{i b c \tan ^{-1}(c x)}{2 d^2} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rule 4862
Rule 627
Rule 44
Rule 203
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)^2} \, dx &=\int \left (\frac{a+b \tan ^{-1}(c x)}{d^2 x^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right )}{d^2 x}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d^2 (-i+c x)^2}+\frac{2 i c^2 \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^2} \, dx}{d^2}-\frac{(2 i c) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx}{d^2}+\frac{\left (2 i c^2\right ) \int \frac{a+b \tan ^{-1}(c x)}{-i+c x} \, dx}{d^2}+\frac{c^2 \int \frac{a+b \tan ^{-1}(c x)}{(-i+c x)^2} \, dx}{d^2}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{2 i a c \log (x)}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{(b c) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx}{d^2}+\frac{(b c) \int \frac{\log (1-i c x)}{x} \, dx}{d^2}-\frac{(b c) \int \frac{\log (1+i c x)}{x} \, dx}{d^2}+\frac{\left (2 i b c^2\right ) \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d^2}+\frac{\left (b c^2\right ) \int \frac{1}{(-i+c x) \left (1+c^2 x^2\right )} \, dx}{d^2}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{2 i a c \log (x)}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{b c \text{Li}_2(-i c x)}{d^2}-\frac{b c \text{Li}_2(i c x)}{d^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d^2}+\frac{(2 b c) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{d^2}+\frac{\left (b c^2\right ) \int \frac{1}{(-i+c x)^2 (i+c x)} \, dx}{d^2}\\ &=-\frac{a+b \tan ^{-1}(c x)}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{2 i a c \log (x)}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{d^2}+\frac{b c \text{Li}_2(-i c x)}{d^2}-\frac{b c \text{Li}_2(i c x)}{d^2}+\frac{b c \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d^2}+\frac{\left (b c^2\right ) \int \left (-\frac{i}{2 (-i+c x)^2}+\frac{i}{2 \left (1+c^2 x^2\right )}\right ) \, dx}{d^2}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{2 d^2}\\ &=-\frac{i b c}{2 d^2 (i-c x)}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{2 i a c \log (x)}{d^2}+\frac{b c \log (x)}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{d^2}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{b c \text{Li}_2(-i c x)}{d^2}-\frac{b c \text{Li}_2(i c x)}{d^2}+\frac{b c \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{d^2}+\frac{\left (i b c^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d^2}\\ &=-\frac{i b c}{2 d^2 (i-c x)}+\frac{i b c \tan ^{-1}(c x)}{2 d^2}-\frac{a+b \tan ^{-1}(c x)}{d^2 x}+\frac{c \left (a+b \tan ^{-1}(c x)\right )}{d^2 (i-c x)}-\frac{2 i a c \log (x)}{d^2}+\frac{b c \log (x)}{d^2}-\frac{2 i c \left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{d^2}-\frac{b c \log \left (1+c^2 x^2\right )}{2 d^2}+\frac{b c \text{Li}_2(-i c x)}{d^2}-\frac{b c \text{Li}_2(i c x)}{d^2}+\frac{b c \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{d^2}\\ \end{align*}
Mathematica [A] time = 0.287197, size = 165, normalized size = 0.85 \[ -\frac{-2 b c \text{PolyLog}(2,-i c x)+2 b c \text{PolyLog}(2,i c x)-2 b c \text{PolyLog}\left (2,\frac{c x+i}{c x-i}\right )+\frac{2 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{2 c \left (a+b \tan ^{-1}(c x)\right )}{c x-i}+4 i c \log \left (\frac{2 i}{-c x+i}\right ) \left (a+b \tan ^{-1}(c x)\right )+4 i a c \log (x)+b c \left (\log \left (c^2 x^2+1\right )-2 \log (x)\right )+i b c \left (-\tan ^{-1}(c x)+\frac{1}{-c x+i}\right )}{2 d^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 340, normalized size = 1.8 \begin{align*} -{\frac{ca}{{d}^{2} \left ( cx-i \right ) }}-2\,{\frac{ca\arctan \left ( cx \right ) }{{d}^{2}}}+{\frac{ica\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{d}^{2}}}-{\frac{a}{{d}^{2}x}}-{\frac{2\,icb\arctan \left ( cx \right ) \ln \left ( cx \right ) }{{d}^{2}}}-{\frac{bc\arctan \left ( cx \right ) }{{d}^{2} \left ( cx-i \right ) }}+{\frac{{\frac{i}{2}}bc\arctan \left ( cx \right ) }{{d}^{2}}}-{\frac{b\arctan \left ( cx \right ) }{{d}^{2}x}}+{\frac{2\,icb\arctan \left ( cx \right ) \ln \left ( cx-i \right ) }{{d}^{2}}}-{\frac{bc\ln \left ( -i \left ( -cx+i \right ) \right ) \ln \left ( -icx \right ) }{{d}^{2}}}+{\frac{bc\ln \left ( -i \left ( -cx+i \right ) \right ) \ln \left ( cx \right ) }{{d}^{2}}}-{\frac{bc{\it dilog} \left ( -icx \right ) }{{d}^{2}}}-{\frac{bc{\it dilog} \left ( -i \left ( cx+i \right ) \right ) }{{d}^{2}}}-{\frac{cb\ln \left ( cx \right ) \ln \left ( -i \left ( cx+i \right ) \right ) }{{d}^{2}}}+{\frac{bc{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{{d}^{2}}}+{\frac{bc\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{{d}^{2}}}-{\frac{bc \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{2\,{d}^{2}}}-{\frac{2\,ica\ln \left ( cx \right ) }{{d}^{2}}}-{\frac{bc\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,{d}^{2}}}+{\frac{{\frac{i}{2}}bc}{{d}^{2} \left ( cx-i \right ) }}+{\frac{cb\ln \left ( cx \right ) }{{d}^{2}}} \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*}{\left (-2 i \, c \int \frac{\arctan \left (c x\right )}{c^{4} d^{2} x^{5} + 2 \, c^{2} d^{2} x^{3} + d^{2} x}\,{d x} - \int \frac{{\left (c^{2} x^{2} - 1\right )} \arctan \left (c x\right )}{c^{4} d^{2} x^{6} + 2 \, c^{2} d^{2} x^{4} + d^{2} x^{2}}\,{d x}\right )} b - a{\left (\frac{c}{c d^{2} x - i \, d^{2}} - \frac{2 i \, c \log \left (c x - i\right )}{d^{2}} + \frac{2 i \, c \log \left (x\right )}{d^{2}} + \frac{1}{d^{2} x}\right )} \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{-i \, b \log \left (-\frac{c x + i}{c x - i}\right ) - 2 \, a}{2 \, c^{2} d^{2} x^{4} - 4 i \, c d^{2} x^{3} - 2 \, d^{2} x^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \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{b \arctan \left (c x\right ) + a}{{\left (i \, c d x + d\right )}^{2} x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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