Optimal. Leaf size=197 \[ -\frac{b c^3 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d x}+\frac{i c^3 \log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{2 b c^3 \log \left (c^2 x^2+1\right )}{3 d}+\frac{i b c^2}{2 d x}-\frac{4 b c^3 \log (x)}{3 d}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{b c}{6 d x^2} \]
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Rubi [A] time = 0.339208, antiderivative size = 197, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {4870, 4852, 266, 44, 325, 203, 36, 29, 31, 4868, 2447} \[ -\frac{b c^3 \text{PolyLog}\left (2,-1+\frac{2}{1+i c x}\right )}{2 d}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d x}+\frac{i c^3 \log \left (2-\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{d}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{2 b c^3 \log \left (c^2 x^2+1\right )}{3 d}+\frac{i b c^2}{2 d x}-\frac{4 b c^3 \log (x)}{3 d}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{b c}{6 d x^2} \]
Antiderivative was successfully verified.
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Rule 4870
Rule 4852
Rule 266
Rule 44
Rule 325
Rule 203
Rule 36
Rule 29
Rule 31
Rule 4868
Rule 2447
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^4 (d+i c d x)} \, dx &=-\left ((i c) \int \frac{a+b \tan ^{-1}(c x)}{x^3 (d+i c d x)} \, dx\right )+\frac{\int \frac{a+b \tan ^{-1}(c x)}{x^4} \, dx}{d}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}-c^2 \int \frac{a+b \tan ^{-1}(c x)}{x^2 (d+i c d x)} \, dx-\frac{(i c) \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx}{d}+\frac{(b c) \int \frac{1}{x^3 \left (1+c^2 x^2\right )} \, dx}{3 d}\\ &=-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}+\left (i c^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x (d+i c d x)} \, dx+\frac{(b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )}{6 d}-\frac{c^2 \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx}{d}-\frac{\left (i b c^2\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx}{2 d}\\ &=\frac{i b c^2}{2 d x}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d x}+\frac{i c^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{d}+\frac{(b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^2}-\frac{c^2}{x}+\frac{c^4}{1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d}-\frac{\left (b c^3\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx}{d}+\frac{\left (i b c^4\right ) \int \frac{1}{1+c^2 x^2} \, dx}{2 d}-\frac{\left (i b c^4\right ) \int \frac{\log \left (2-\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{d}\\ &=-\frac{b c}{6 d x^2}+\frac{i b c^2}{2 d x}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac{b c^3 \log (x)}{3 d}+\frac{b c^3 \log \left (1+c^2 x^2\right )}{6 d}+\frac{i c^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{b c^3 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c}{6 d x^2}+\frac{i b c^2}{2 d x}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac{b c^3 \log (x)}{3 d}+\frac{b c^3 \log \left (1+c^2 x^2\right )}{6 d}+\frac{i c^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{b c^3 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}-\frac{\left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )}{2 d}+\frac{\left (b c^5\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )}{2 d}\\ &=-\frac{b c}{6 d x^2}+\frac{i b c^2}{2 d x}+\frac{i b c^3 \tan ^{-1}(c x)}{2 d}-\frac{a+b \tan ^{-1}(c x)}{3 d x^3}+\frac{i c \left (a+b \tan ^{-1}(c x)\right )}{2 d x^2}+\frac{c^2 \left (a+b \tan ^{-1}(c x)\right )}{d x}-\frac{4 b c^3 \log (x)}{3 d}+\frac{2 b c^3 \log \left (1+c^2 x^2\right )}{3 d}+\frac{i c^3 \left (a+b \tan ^{-1}(c x)\right ) \log \left (2-\frac{2}{1+i c x}\right )}{d}-\frac{b c^3 \text{Li}_2\left (-1+\frac{2}{1+i c x}\right )}{2 d}\\ \end{align*}
Mathematica [C] time = 0.136916, size = 254, normalized size = 1.29 \[ \frac{3 i b c^2 x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )-3 b c^3 x^3 \text{PolyLog}(2,-i c x)+3 b c^3 x^3 \text{PolyLog}(2,i c x)-3 b c^3 x^3 \text{PolyLog}\left (2,\frac{c x+i}{c x-i}\right )+6 a c^2 x^2+6 i a c^3 x^3 \log (x)+6 i a c^3 x^3 \log \left (\frac{2 i}{-c x+i}\right )+3 i a c x-2 a-8 b c^3 x^3 \log (x)+4 b c^3 x^3 \log \left (c^2 x^2+1\right )+6 b c^2 x^2 \tan ^{-1}(c x)+6 i b c^3 x^3 \log \left (\frac{2 i}{-c x+i}\right ) \tan ^{-1}(c x)-b c x+3 i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)}{6 d x^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.055, size = 369, normalized size = 1.9 \begin{align*}{\frac{i{c}^{3}b\arctan \left ( cx \right ) \ln \left ( cx \right ) }{d}}+{\frac{a{c}^{3}\arctan \left ( cx \right ) }{d}}-{\frac{a}{3\,d{x}^{3}}}+{\frac{{\frac{i}{2}}cb\arctan \left ( cx \right ) }{d{x}^{2}}}+{\frac{{\frac{i}{2}}b{c}^{3}\arctan \left ( cx \right ) }{d}}+{\frac{{c}^{2}a}{dx}}-{\frac{i{c}^{3}b\arctan \left ( cx \right ) \ln \left ( cx-i \right ) }{d}}-{\frac{b\arctan \left ( cx \right ) }{3\,d{x}^{3}}}+{\frac{i{c}^{3}a\ln \left ( cx \right ) }{d}}+{\frac{{\frac{i}{2}}ca}{d{x}^{2}}}+{\frac{{c}^{2}b\arctan \left ( cx \right ) }{dx}}-{\frac{{c}^{3}b\ln \left ( cx-i \right ) \ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{2\,d}}-{\frac{{c}^{3}b{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{2\,d}}+{\frac{{c}^{3}b \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{4\,d}}-{\frac{{c}^{3}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) }{2\,d}}+{\frac{{c}^{3}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) }{2\,d}}+{\frac{{c}^{3}b{\it dilog} \left ( 1-icx \right ) }{2\,d}}-{\frac{{c}^{3}b{\it dilog} \left ( 1+icx \right ) }{2\,d}}+{\frac{2\,b{c}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,d}}+{\frac{{\frac{i}{2}}b{c}^{2}}{dx}}-{\frac{{\frac{i}{2}}{c}^{3}a\ln \left ({c}^{2}{x}^{2}+1 \right ) }{d}}-{\frac{bc}{6\,d{x}^{2}}}-{\frac{4\,{c}^{3}b\ln \left ( cx \right ) }{3\,d}} \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*} -\frac{1}{6} \,{\left (\frac{6 i \, c^{3} \log \left (i \, c x + 1\right )}{d} - \frac{6 i \, c^{3} \log \left (x\right )}{d} - \frac{6 \, c^{2} x^{2} + 3 i \, c x - 2}{d x^{3}}\right )} a +{\left (-i \, c \int \frac{\arctan \left (c x\right )}{c^{2} d x^{5} + d x^{3}}\,{d x} + \int \frac{\arctan \left (c x\right )}{c^{2} d x^{6} + d x^{4}}\,{d x}\right )} b \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{b \log \left (-\frac{c x + i}{c x - i}\right ) - 2 i \, a}{2 \,{\left (c d x^{5} - i \, d x^{4}\right )}}, 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 )} x^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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