Optimal. Leaf size=173 \[ -3 i b c^2 d^4 \text{PolyLog}(2,-i c x)+3 i b c^2 d^4 \text{PolyLog}(2,i c x)+\frac{1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-\frac{1}{2} b c^3 d^4 x+4 i b c^2 d^4 \log (x)-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{2 x} \]
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Rubi [A] time = 0.199452, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {4876, 4846, 260, 4852, 325, 203, 266, 36, 29, 31, 4848, 2391, 321} \[ -3 i b c^2 d^4 \text{PolyLog}(2,-i c x)+3 i b c^2 d^4 \text{PolyLog}(2,i c x)+\frac{1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-4 i a c^3 d^4 x-6 a c^2 d^4 \log (x)-\frac{1}{2} b c^3 d^4 x+4 i b c^2 d^4 \log (x)-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{2 x} \]
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
Rule 321
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=\int \left (-4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^3}+\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac{6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+c^4 d^4 x \left (a+b \tan ^{-1}(c x)\right )\right ) \, dx\\ &=d^4 \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx+\left (4 i c d^4\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (6 c^2 d^4\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx-\left (4 i c^3 d^4\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx+\left (c^4 d^4\right ) \int x \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=-4 i a c^3 d^4 x-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)+\frac{1}{2} \left (b c d^4\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx-\left (3 i b c^2 d^4\right ) \int \frac{\log (1-i c x)}{x} \, dx+\left (3 i b c^2 d^4\right ) \int \frac{\log (1+i c x)}{x} \, dx+\left (4 i b c^2 d^4\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx-\left (4 i b c^3 d^4\right ) \int \tan ^{-1}(c x) \, dx-\frac{1}{2} \left (b c^5 d^4\right ) \int \frac{x^2}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^4}{2 x}-4 i a c^3 d^4 x-\frac{1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)-3 i b c^2 d^4 \text{Li}_2(-i c x)+3 i b c^2 d^4 \text{Li}_2(i c x)+\left (2 i b c^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )+\left (4 i b c^4 d^4\right ) \int \frac{x}{1+c^2 x^2} \, dx\\ &=-\frac{b c d^4}{2 x}-4 i a c^3 d^4 x-\frac{1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)+2 i b c^2 d^4 \log \left (1+c^2 x^2\right )-3 i b c^2 d^4 \text{Li}_2(-i c x)+3 i b c^2 d^4 \text{Li}_2(i c x)+\left (2 i b c^2 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\left (2 i b c^4 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^4}{2 x}-4 i a c^3 d^4 x-\frac{1}{2} b c^3 d^4 x-4 i b c^3 d^4 x \tan ^{-1}(c x)-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{1}{2} c^4 d^4 x^2 \left (a+b \tan ^{-1}(c x)\right )-6 a c^2 d^4 \log (x)+4 i b c^2 d^4 \log (x)-3 i b c^2 d^4 \text{Li}_2(-i c x)+3 i b c^2 d^4 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [A] time = 0.139878, size = 163, normalized size = 0.94 \[ \frac{d^4 \left (-6 i b c^2 x^2 \text{PolyLog}(2,-i c x)+6 i b c^2 x^2 \text{PolyLog}(2,i c x)+a c^4 x^4-8 i a c^3 x^3-12 a c^2 x^2 \log (x)-8 i a c x-a-b c^3 x^3+8 i b c^2 x^2 \log (c x)+b c^4 x^4 \tan ^{-1}(c x)-8 i b c^3 x^3 \tan ^{-1}(c x)-b c x-8 i b c x \tan ^{-1}(c x)-b \tan ^{-1}(c x)\right )}{2 x^2} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.047, size = 248, normalized size = 1.4 \begin{align*} 3\,i{c}^{2}{d}^{4}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +{\frac{{c}^{4}{d}^{4}a{x}^{2}}{2}}-{\frac{{d}^{4}a}{2\,{x}^{2}}}-{\frac{4\,ic{d}^{4}a}{x}}-6\,{c}^{2}{d}^{4}a\ln \left ( cx \right ) +3\,i{c}^{2}{d}^{4}b{\it dilog} \left ( 1-icx \right ) +{\frac{{c}^{4}{d}^{4}b\arctan \left ( cx \right ){x}^{2}}{2}}-{\frac{b{d}^{4}\arctan \left ( cx \right ) }{2\,{x}^{2}}}-3\,i{c}^{2}{d}^{4}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -6\,{c}^{2}{d}^{4}b\arctan \left ( cx \right ) \ln \left ( cx \right ) -{\frac{b{c}^{3}{d}^{4}x}{2}}-{\frac{{d}^{4}bc}{2\,x}}-4\,ib{c}^{3}{d}^{4}x\arctan \left ( cx \right ) -{\frac{4\,ic{d}^{4}b\arctan \left ( cx \right ) }{x}}-3\,i{c}^{2}{d}^{4}b{\it dilog} \left ( 1+icx \right ) +4\,i{c}^{2}{d}^{4}b\ln \left ( cx \right ) -4\,ia{c}^{3}{d}^{4}x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.16075, size = 350, normalized size = 2.02 \begin{align*} \frac{1}{2} \, a c^{4} d^{4} x^{2} - 4 i \, a c^{3} d^{4} x - \frac{1}{2} \, b c^{3} d^{4} x + \frac{3}{2} \, \pi b c^{2} d^{4} \log \left (c^{2} x^{2} + 1\right ) - 6 \, b c^{2} d^{4} \arctan \left (c x\right ) \log \left (x{\left | c \right |}\right ) - 2 i \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c^{2} d^{4} + 3 i \, b c^{2} d^{4}{\rm Li}_2\left (i \, c x + 1\right ) - 3 i \, b c^{2} d^{4}{\rm Li}_2\left (-i \, c x + 1\right ) - 6 \, a c^{2} d^{4} \log \left (x\right ) - 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^{4} - \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^{4} - \frac{4 i \, a c d^{4}}{x} - \frac{a d^{4}}{2 \, x^{2}} + \frac{1}{2} \,{\left (b c^{4} d^{4} x^{2} + b c^{2} d^{4}{\left (-12 i \, \arctan \left (0, c\right ) + 1\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 \, a c^{4} d^{4} x^{4} - 8 i \, a c^{3} d^{4} x^{3} - 12 \, a c^{2} d^{4} x^{2} + 8 i \, a c d^{4} x + 2 \, a d^{4} +{\left (i \, b c^{4} d^{4} x^{4} + 4 \, b c^{3} d^{4} x^{3} - 6 i \, b c^{2} d^{4} x^{2} - 4 \, b c d^{4} x + i \, b d^{4}\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^{4} \left (\int \frac{a}{x^{3}}\, dx + \int - 4 i a c^{3}\, dx + \int - \frac{6 a c^{2}}{x}\, dx + \int a c^{4} x\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x^{3}}\, dx + \int \frac{4 i a c}{x^{2}}\, dx + \int - 4 i b c^{3} \operatorname{atan}{\left (c x \right )}\, dx + \int - \frac{6 b c^{2} \operatorname{atan}{\left (c x \right )}}{x}\, dx + \int b c^{4} x \operatorname{atan}{\left (c x \right )}\, dx + \int \frac{4 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 )}^{4}{\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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