Optimal. Leaf size=162 \[ -\frac{3}{2} b c d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} b c d^3 \text{PolyLog}(2,i c x)-\frac{1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 x+3 i a c d^3 \log (x)+b c d^3 \log \left (c^2 x^2+1\right )+\frac{1}{2} i b c^2 d^3 x-3 b c^2 d^3 x \tan ^{-1}(c x)+b c d^3 \log (x)-\frac{1}{2} i b c d^3 \tan ^{-1}(c x) \]
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Rubi [A] time = 0.173482, antiderivative size = 162, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 12, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.522, Rules used = {4876, 4846, 260, 4852, 266, 36, 29, 31, 4848, 2391, 321, 203} \[ -\frac{3}{2} b c d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} b c d^3 \text{PolyLog}(2,i c x)-\frac{1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 x+3 i a c d^3 \log (x)+b c d^3 \log \left (c^2 x^2+1\right )+\frac{1}{2} i b c^2 d^3 x-3 b c^2 d^3 x \tan ^{-1}(c x)+b c d^3 \log (x)-\frac{1}{2} i b c d^3 \tan ^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4846
Rule 260
Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rule 321
Rule 203
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )}{x^2} \, dx &=\int \left (-3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^2}+\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i c^3 d^3 x \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^3 \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx+\left (3 i c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx-\left (3 c^2 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx-\left (i c^3 d^3\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-3 a c^2 d^3 x-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 i a c d^3 \log (x)+\left (b c d^3\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx-\frac{1}{2} \left (3 b c d^3\right ) \int \frac{\log (1-i c x)}{x} \, dx+\frac{1}{2} \left (3 b c d^3\right ) \int \frac{\log (1+i c x)}{x} \, dx-\left (3 b c^2 d^3\right ) \int \tan ^{-1}(c x) \, dx+\frac{1}{2} \left (i b c^4 d^3\right ) \int \frac{x^2}{1+c^2 x^2} \, dx\\ &=-3 a c^2 d^3 x+\frac{1}{2} i b c^2 d^3 x-3 b c^2 d^3 x \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 i a c d^3 \log (x)-\frac{3}{2} b c d^3 \text{Li}_2(-i c x)+\frac{3}{2} b c d^3 \text{Li}_2(i c x)+\frac{1}{2} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\frac{1}{2} \left (i b c^2 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx+\left (3 b c^3 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx\\ &=-3 a c^2 d^3 x+\frac{1}{2} i b c^2 d^3 x-\frac{1}{2} i b c d^3 \tan ^{-1}(c x)-3 b c^2 d^3 x \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 i a c d^3 \log (x)+\frac{3}{2} b c d^3 \log \left (1+c^2 x^2\right )-\frac{3}{2} b c d^3 \text{Li}_2(-i c x)+\frac{3}{2} b c d^3 \text{Li}_2(i c x)+\frac{1}{2} \left (b c d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (b c^3 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-3 a c^2 d^3 x+\frac{1}{2} i b c^2 d^3 x-\frac{1}{2} i b c d^3 \tan ^{-1}(c x)-3 b c^2 d^3 x \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{1}{2} i c^3 d^3 x^2 \left (a+b \tan ^{-1}(c x)\right )+3 i a c d^3 \log (x)+b c d^3 \log (x)+b c d^3 \log \left (1+c^2 x^2\right )-\frac{3}{2} b c d^3 \text{Li}_2(-i c x)+\frac{3}{2} b c d^3 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [A] time = 0.115984, size = 150, normalized size = 0.93 \[ \frac{d^3 \left (-3 b c x \text{PolyLog}(2,-i c x)+3 b c x \text{PolyLog}(2,i c x)-i a c^3 x^3-6 a c^2 x^2+6 i a c x \log (x)-2 a+i b c^2 x^2+2 b c x \log \left (c^2 x^2+1\right )-i b c^3 x^3 \tan ^{-1}(c x)-6 b c^2 x^2 \tan ^{-1}(c x)+2 b c x \log (c x)-i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)\right )}{2 x} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.046, size = 223, normalized size = 1.4 \begin{align*} -3\,a{c}^{2}{d}^{3}x+{\frac{i}{2}}b{c}^{2}{d}^{3}x-{\frac{{d}^{3}a}{x}}+3\,ic{d}^{3}a\ln \left ( cx \right ) -3\,b{c}^{2}{d}^{3}x\arctan \left ( cx \right ) -{\frac{i}{2}}bc{d}^{3}\arctan \left ( cx \right ) -{\frac{b{d}^{3}\arctan \left ( cx \right ) }{x}}+3\,ic{d}^{3}b\arctan \left ( cx \right ) \ln \left ( cx \right ) -{\frac{3\,c{d}^{3}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) }{2}}+{\frac{3\,c{d}^{3}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) }{2}}-{\frac{3\,c{d}^{3}b{\it dilog} \left ( 1+icx \right ) }{2}}+{\frac{3\,c{d}^{3}b{\it dilog} \left ( 1-icx \right ) }{2}}-{\frac{i}{2}}{d}^{3}b\arctan \left ( cx \right ){c}^{3}{x}^{2}+bc{d}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) -{\frac{i}{2}}{d}^{3}a{c}^{3}{x}^{2}+c{d}^{3}b\ln \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.17171, size = 281, normalized size = 1.73 \begin{align*} -\frac{1}{2} i \, a c^{3} d^{3} x^{2} - 3 \, a c^{2} d^{3} x + \frac{1}{2} i \, b c^{2} d^{3} x - \frac{3}{4} i \, \pi b c d^{3} \log \left (c^{2} x^{2} + 1\right ) + 3 i \, b c d^{3} \arctan \left (c x\right ) \log \left (x{\left | c \right |}\right ) - \frac{3}{2} \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c d^{3} + \frac{3}{2} \, b c d^{3}{\rm Li}_2\left (i \, c x + 1\right ) - \frac{3}{2} \, b c d^{3}{\rm Li}_2\left (-i \, c x + 1\right ) + 3 i \, a c d^{3} \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^{3} - \frac{a d^{3}}{x} - \frac{1}{4} \,{\left (2 i \, b c^{3} d^{3} x^{2} + b c d^{3}{\left (12 \, \arctan \left (0, c\right ) + 2 i\right )}\right )} \arctan \left (c 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{-2 i \, a c^{3} d^{3} x^{3} - 6 \, a c^{2} d^{3} x^{2} + 6 i \, a c d^{3} x + 2 \, a d^{3} +{\left (b c^{3} d^{3} x^{3} - 3 i \, b c^{2} d^{3} x^{2} - 3 \, b c d^{3} x + i \, b d^{3}\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^{3} \left (\int - 3 a c^{2}\, dx + \int \frac{a}{x^{2}}\, dx + \int - 3 b c^{2} \operatorname{atan}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{3 i a c}{x}\, dx + \int - i a c^{3} x\, dx + \int \frac{3 i b c \operatorname{atan}{\left (c x \right )}}{x}\, dx + \int - i b c^{3} x \operatorname{atan}{\left (c x \right )}\, 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 )}^{3}{\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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