Optimal. Leaf size=180 \[ -\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,i c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^2 d^3 \log \left (c^2 x^2+1\right )+3 i b c^2 d^3 \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{b c d^3}{2 x} \]
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Rubi [A] time = 0.178091, antiderivative size = 180, 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, 325, 203, 266, 36, 29, 31, 4848, 2391} \[ -\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,-i c x)+\frac{3}{2} i b c^2 d^3 \text{PolyLog}(2,i c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-i a c^3 d^3 x-3 a c^2 d^3 \log (x)-i b c^2 d^3 \log \left (c^2 x^2+1\right )+3 i b c^2 d^3 \log (x)-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{b c d^3}{2 x} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4846
Rule 260
Rule 4852
Rule 325
Rule 203
Rule 266
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (-i c^3 d^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac{3 c^2 d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^3 \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (3 i c d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (3 c^2 d^3\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx-\left (i c^3 d^3\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-i a c^3 d^3 x-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)+\frac{1}{2} \left (b c d^3\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx-\frac{1}{2} \left (3 i b c^2 d^3\right ) \int \frac{\log (1-i c x)}{x} \, dx+\frac{1}{2} \left (3 i b c^2 d^3\right ) \int \frac{\log (1+i c x)}{x} \, dx+\left (3 i b c^2 d^3\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx-\left (i b c^3 d^3\right ) \int \tan ^{-1}(c x) \, dx\\ &=-\frac{b c d^3}{2 x}-i a c^3 d^3 x-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)-\frac{3}{2} i b c^2 d^3 \text{Li}_2(-i c x)+\frac{3}{2} i b c^2 d^3 \text{Li}_2(i c x)+\frac{1}{2} \left (3 i b c^2 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 (b c^3 d^3\right ) \int \frac{1}{1+c^2 x^2} \, dx+\left (i b c^4 d^3\right ) \int \frac{x}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^3}{2 x}-i a c^3 d^3 x-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)+\frac{1}{2} i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac{3}{2} i b c^2 d^3 \text{Li}_2(-i c x)+\frac{3}{2} i b c^2 d^3 \text{Li}_2(i c x)+\frac{1}{2} \left (3 i b c^2 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (3 i b c^4 d^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^3}{2 x}-i a c^3 d^3 x-\frac{1}{2} b c^2 d^3 \tan ^{-1}(c x)-i b c^3 d^3 x \tan ^{-1}(c x)-\frac{d^3 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{3 i c d^3 \left (a+b \tan ^{-1}(c x)\right )}{x}-3 a c^2 d^3 \log (x)+3 i b c^2 d^3 \log (x)-i b c^2 d^3 \log \left (1+c^2 x^2\right )-\frac{3}{2} i b c^2 d^3 \text{Li}_2(-i c x)+\frac{3}{2} i b c^2 d^3 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [A] time = 0.121194, size = 164, normalized size = 0.91 \[ -\frac{i d^3 \left (3 b c^2 x^2 \text{PolyLog}(2,-i c x)-3 b c^2 x^2 \text{PolyLog}(2,i c x)+2 a c^3 x^3-6 i a c^2 x^2 \log (x)+6 a c x-i a-6 b c^2 x^2 \log (c x)+2 b c^2 x^2 \log \left (c^2 x^2+1\right )+2 b c^3 x^3 \tan ^{-1}(c x)-i b c^2 x^2 \tan ^{-1}(c x)-i b c x+6 b c x \tan ^{-1}(c x)-i b \tan ^{-1}(c x)\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.047, size = 243, normalized size = 1.4 \begin{align*} 3\,i{c}^{2}{d}^{3}b\ln \left ( cx \right ) -{\frac{{d}^{3}a}{2\,{x}^{2}}}+{\frac{3\,i}{2}}{c}^{2}{d}^{3}b{\it dilog} \left ( 1-icx \right ) -3\,{c}^{2}{d}^{3}a\ln \left ( cx \right ) -{\frac{3\,ic{d}^{3}b\arctan \left ( cx \right ) }{x}}-{\frac{b{d}^{3}\arctan \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{3\,i}{2}}{c}^{2}{d}^{3}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -3\,{c}^{2}{d}^{3}b\arctan \left ( cx \right ) \ln \left ( cx \right ) -ib{c}^{2}{d}^{3}\ln \left ({c}^{2}{x}^{2}+1 \right ) -ib{c}^{3}{d}^{3}x\arctan \left ( cx \right ) -{\frac{3\,i}{2}}{c}^{2}{d}^{3}b{\it dilog} \left ( 1+icx \right ) -{\frac{3\,ic{d}^{3}a}{x}}+{\frac{3\,i}{2}}{c}^{2}{d}^{3}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) -{\frac{b{c}^{2}{d}^{3}\arctan \left ( cx \right ) }{2}}-{\frac{bc{d}^{3}}{2\,x}}-ia{c}^{3}{d}^{3}x \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 \, a c^{3} d^{3} x - \frac{1}{2} i \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c^{2} d^{3} - 3 \, b c^{2} d^{3} \int \frac{\arctan \left (c x\right )}{x}\,{d x} - 3 \, a c^{2} d^{3} \log \left (x\right ) - \frac{3}{2} i \,{\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 c d^{3} - \frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b d^{3} - \frac{3 i \, a c d^{3}}{x} - \frac{a d^{3}}{2 \, x^{2}} \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^{3}}, 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 \frac{a}{x^{3}}\, dx + \int - i a c^{3}\, dx + \int - \frac{3 a c^{2}}{x}\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{3 i a c}{x^{2}}\, dx + \int - i b c^{3} \operatorname{atan}{\left (c x \right )}\, dx + \int - \frac{3 b c^{2} \operatorname{atan}{\left (c x \right )}}{x}\, dx + \int \frac{3 i b c \operatorname{atan}{\left (c x \right )}}{x^{2}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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