Optimal. Leaf size=201 \[ 2 b c^3 d^4 \text{PolyLog}(2,-i c x)-2 b c^3 d^4 \text{PolyLog}(2,i c x)+\frac{6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+a c^4 d^4 x-4 i a c^3 d^4 \log (x)+\frac{8}{3} b c^3 d^4 \log \left (c^2 x^2+1\right )-\frac{2 i b c^2 d^4}{x}-\frac{19}{3} b c^3 d^4 \log (x)-2 i b c^3 d^4 \tan ^{-1}(c x)+b c^4 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{6 x^2} \]
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Rubi [A] time = 0.217667, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 13, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.565, Rules used = {4876, 4846, 260, 4852, 266, 44, 325, 203, 36, 29, 31, 4848, 2391} \[ 2 b c^3 d^4 \text{PolyLog}(2,-i c x)-2 b c^3 d^4 \text{PolyLog}(2,i c x)+\frac{6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}-\frac{2 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+a c^4 d^4 x-4 i a c^3 d^4 \log (x)+\frac{8}{3} b c^3 d^4 \log \left (c^2 x^2+1\right )-\frac{2 i b c^2 d^4}{x}-\frac{19}{3} b c^3 d^4 \log (x)-2 i b c^3 d^4 \tan ^{-1}(c x)+b c^4 d^4 x \tan ^{-1}(c x)-\frac{b c d^4}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 4876
Rule 4846
Rule 260
Rule 4852
Rule 266
Rule 44
Rule 325
Rule 203
Rule 36
Rule 29
Rule 31
Rule 4848
Rule 2391
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx &=\int \left (c^4 d^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^4}+\frac{4 i c d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^3}-\frac{6 c^2 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x^2}-\frac{4 i c^3 d^4 \left (a+b \tan ^{-1}(c x)\right )}{x}\right ) \, dx\\ &=d^4 \int \frac{a+b \tan ^{-1}(c x)}{x^4} \, dx+\left (4 i c d^4\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^3} \, dx-\left (6 c^2 d^4\right ) \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx-\left (4 i c^3 d^4\right ) \int \frac{a+b \tan ^{-1}(c x)}{x} \, dx+\left (c^4 d^4\right ) \int \left (a+b \tan ^{-1}(c x)\right ) \, dx\\ &=a c^4 d^4 x-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{2 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}-4 i a c^3 d^4 \log (x)+\frac{1}{3} \left (b c d^4\right ) \int \frac{1}{x^3 \left (1+c^2 x^2\right )} \, dx+\left (2 i b c^2 d^4\right ) \int \frac{1}{x^2 \left (1+c^2 x^2\right )} \, dx+\left (2 b c^3 d^4\right ) \int \frac{\log (1-i c x)}{x} \, dx-\left (2 b c^3 d^4\right ) \int \frac{\log (1+i c x)}{x} \, dx-\left (6 b c^3 d^4\right ) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx+\left (b c^4 d^4\right ) \int \tan ^{-1}(c x) \, dx\\ &=-\frac{2 i b c^2 d^4}{x}+a c^4 d^4 x+b c^4 d^4 x \tan ^{-1}(c x)-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{2 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}-4 i a c^3 d^4 \log (x)+2 b c^3 d^4 \text{Li}_2(-i c x)-2 b c^3 d^4 \text{Li}_2(i c x)+\frac{1}{6} \left (b c d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (1+c^2 x\right )} \, dx,x,x^2\right )-\left (3 b c^3 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )-\left (2 i b c^4 d^4\right ) \int \frac{1}{1+c^2 x^2} \, dx-\left (b c^5 d^4\right ) \int \frac{x}{1+c^2 x^2} \, dx\\ &=-\frac{2 i b c^2 d^4}{x}+a c^4 d^4 x-2 i b c^3 d^4 \tan ^{-1}(c x)+b c^4 d^4 x \tan ^{-1}(c x)-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{2 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}-4 i a c^3 d^4 \log (x)-\frac{1}{2} b c^3 d^4 \log \left (1+c^2 x^2\right )+2 b c^3 d^4 \text{Li}_2(-i c x)-2 b c^3 d^4 \text{Li}_2(i c x)+\frac{1}{6} \left (b c d^4\right ) \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 )-\left (3 b c^3 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )+\left (3 b c^5 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{b c d^4}{6 x^2}-\frac{2 i b c^2 d^4}{x}+a c^4 d^4 x-2 i b c^3 d^4 \tan ^{-1}(c x)+b c^4 d^4 x \tan ^{-1}(c x)-\frac{d^4 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{2 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}-4 i a c^3 d^4 \log (x)-\frac{19}{3} b c^3 d^4 \log (x)+\frac{8}{3} b c^3 d^4 \log \left (1+c^2 x^2\right )+2 b c^3 d^4 \text{Li}_2(-i c x)-2 b c^3 d^4 \text{Li}_2(i c x)\\ \end{align*}
Mathematica [A] time = 0.147416, size = 193, normalized size = 0.96 \[ \frac{d^4 \left (12 b c^3 x^3 \text{PolyLog}(2,-i c x)-12 b c^3 x^3 \text{PolyLog}(2,i c x)+6 a c^4 x^4+36 a c^2 x^2-24 i a c^3 x^3 \log (x)-12 i a c x-2 a-12 i b c^2 x^2-38 b c^3 x^3 \log (c x)+16 b c^3 x^3 \log \left (c^2 x^2+1\right )+6 b c^4 x^4 \tan ^{-1}(c x)-12 i b c^3 x^3 \tan ^{-1}(c x)+36 b c^2 x^2 \tan ^{-1}(c x)-b c x-12 i b c x \tan ^{-1}(c x)-2 b \tan ^{-1}(c x)\right )}{6 x^3} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.047, size = 277, normalized size = 1.4 \begin{align*} a{c}^{4}{d}^{4}x-4\,i{c}^{3}{d}^{4}a\ln \left ( cx \right ) +6\,{\frac{{c}^{2}{d}^{4}a}{x}}-{\frac{{d}^{4}a}{3\,{x}^{3}}}-{\frac{2\,ic{d}^{4}b\arctan \left ( cx \right ) }{{x}^{2}}}+b{c}^{4}{d}^{4}x\arctan \left ( cx \right ) -{\frac{2\,ib{c}^{2}{d}^{4}}{x}}+6\,{\frac{{c}^{2}{d}^{4}b\arctan \left ( cx \right ) }{x}}-{\frac{b{d}^{4}\arctan \left ( cx \right ) }{3\,{x}^{3}}}-4\,i{c}^{3}{d}^{4}b\arctan \left ( cx \right ) \ln \left ( cx \right ) +2\,{c}^{3}{d}^{4}b\ln \left ( cx \right ) \ln \left ( 1+icx \right ) -2\,{c}^{3}{d}^{4}b\ln \left ( cx \right ) \ln \left ( 1-icx \right ) +2\,{c}^{3}{d}^{4}b{\it dilog} \left ( 1+icx \right ) -2\,{c}^{3}{d}^{4}b{\it dilog} \left ( 1-icx \right ) +{\frac{8\,b{c}^{3}{d}^{4}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3}}-{\frac{2\,ic{d}^{4}a}{{x}^{2}}}-{\frac{{d}^{4}bc}{6\,{x}^{2}}}-2\,ib{c}^{3}{d}^{4}\arctan \left ( cx \right ) -{\frac{19\,{c}^{3}{d}^{4}b\ln \left ( cx \right ) }{3}} \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*} a c^{4} d^{4} x + \frac{1}{2} \,{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b c^{3} d^{4} - 4 i \, b c^{3} d^{4} \int \frac{\arctan \left (c x\right )}{x}\,{d x} - 4 i \, a c^{3} d^{4} \log \left (x\right ) + 3 \,{\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^{2} d^{4} - 2 i \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b c d^{4} + \frac{1}{6} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d^{4} + \frac{6 \, a c^{2} d^{4}}{x} - \frac{2 i \, a c d^{4}}{x^{2}} - \frac{a d^{4}}{3 \, x^{3}} \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^{4}}, 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 a c^{4}\, dx + \int \frac{a}{x^{4}}\, dx + \int - \frac{6 a c^{2}}{x^{2}}\, dx + \int b c^{4} \operatorname{atan}{\left (c x \right )}\, dx + \int \frac{b \operatorname{atan}{\left (c x \right )}}{x^{4}}\, dx + \int \frac{4 i a c}{x^{3}}\, dx + \int - \frac{4 i a c^{3}}{x}\, dx + \int - \frac{6 b c^{2} \operatorname{atan}{\left (c x \right )}}{x^{2}}\, dx + \int \frac{4 i b c \operatorname{atan}{\left (c x \right )}}{x^{3}}\, dx + \int - \frac{4 i b c^{3} \operatorname{atan}{\left (c x \right )}}{x}\, 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^{4}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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