Optimal. Leaf size=196 \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c^4 d}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}+\frac{\log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d}-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac{i a x}{c^3 d}+\frac{i b x^2}{6 c^2 d}-\frac{2 i b \log \left (c^2 x^2+1\right )}{3 c^4 d}-\frac{b x}{2 c^3 d}+\frac{i b x \tan ^{-1}(c x)}{c^3 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d} \]
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Rubi [A] time = 0.286152, antiderivative size = 196, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.478, Rules used = {4866, 4852, 266, 43, 321, 203, 4846, 260, 4854, 2402, 2315} \[ \frac{i b \text{PolyLog}\left (2,1-\frac{2}{1+i c x}\right )}{2 c^4 d}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}+\frac{\log \left (\frac{2}{1+i c x}\right ) \left (a+b \tan ^{-1}(c x)\right )}{c^4 d}-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac{i a x}{c^3 d}+\frac{i b x^2}{6 c^2 d}-\frac{2 i b \log \left (c^2 x^2+1\right )}{3 c^4 d}-\frac{b x}{2 c^3 d}+\frac{i b x \tan ^{-1}(c x)}{c^3 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d} \]
Antiderivative was successfully verified.
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Rule 4866
Rule 4852
Rule 266
Rule 43
Rule 321
Rule 203
Rule 4846
Rule 260
Rule 4854
Rule 2402
Rule 2315
Rubi steps
\begin{align*} \int \frac{x^3 \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx &=\frac{i \int \frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx}{c}-\frac{i \int x^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c d}\\ &=-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}-\frac{\int \frac{x \left (a+b \tan ^{-1}(c x)\right )}{d+i c d x} \, dx}{c^2}+\frac{(i b) \int \frac{x^3}{1+c^2 x^2} \, dx}{3 d}+\frac{\int x \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^2 d}\\ &=\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}-\frac{i \int \frac{a+b \tan ^{-1}(c x)}{d+i c d x} \, dx}{c^3}+\frac{(i b) \operatorname{Subst}\left (\int \frac{x}{1+c^2 x} \, dx,x,x^2\right )}{6 d}+\frac{i \int \left (a+b \tan ^{-1}(c x)\right ) \, dx}{c^3 d}-\frac{b \int \frac{x^2}{1+c^2 x^2} \, dx}{2 c d}\\ &=\frac{i a x}{c^3 d}-\frac{b x}{2 c^3 d}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^4 d}+\frac{(i b) \operatorname{Subst}\left (\int \left (\frac{1}{c^2}-\frac{1}{c^2 \left (1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{6 d}+\frac{(i b) \int \tan ^{-1}(c x) \, dx}{c^3 d}+\frac{b \int \frac{1}{1+c^2 x^2} \, dx}{2 c^3 d}-\frac{b \int \frac{\log \left (\frac{2}{1+i c x}\right )}{1+c^2 x^2} \, dx}{c^3 d}\\ &=\frac{i a x}{c^3 d}-\frac{b x}{2 c^3 d}+\frac{i b x^2}{6 c^2 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d}+\frac{i b x \tan ^{-1}(c x)}{c^3 d}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^4 d}-\frac{i b \log \left (1+c^2 x^2\right )}{6 c^4 d}+\frac{(i b) \operatorname{Subst}\left (\int \frac{\log (2 x)}{1-2 x} \, dx,x,\frac{1}{1+i c x}\right )}{c^4 d}-\frac{(i b) \int \frac{x}{1+c^2 x^2} \, dx}{c^2 d}\\ &=\frac{i a x}{c^3 d}-\frac{b x}{2 c^3 d}+\frac{i b x^2}{6 c^2 d}+\frac{b \tan ^{-1}(c x)}{2 c^4 d}+\frac{i b x \tan ^{-1}(c x)}{c^3 d}+\frac{x^2 \left (a+b \tan ^{-1}(c x)\right )}{2 c^2 d}-\frac{i x^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c d}+\frac{\left (a+b \tan ^{-1}(c x)\right ) \log \left (\frac{2}{1+i c x}\right )}{c^4 d}-\frac{2 i b \log \left (1+c^2 x^2\right )}{3 c^4 d}+\frac{i b \text{Li}_2\left (1-\frac{2}{1+i c x}\right )}{2 c^4 d}\\ \end{align*}
Mathematica [A] time = 0.442145, size = 166, normalized size = 0.85 \[ -\frac{i \left (3 b \text{PolyLog}\left (2,-e^{2 i \tan ^{-1}(c x)}\right )+\tan ^{-1}(c x) \left (6 a+b \left (2 c^3 x^3+3 i c^2 x^2-6 c x+3 i\right )+6 i b \log \left (1+e^{2 i \tan ^{-1}(c x)}\right )\right )+2 a c^3 x^3+3 i a c^2 x^2-3 i a \log \left (c^2 x^2+1\right )-6 a c x-b c^2 x^2+4 b \log \left (c^2 x^2+1\right )-3 i b c x+6 b \tan ^{-1}(c x)^2-b\right )}{6 c^4 d} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.053, size = 353, normalized size = 1.8 \begin{align*}{\frac{-{\frac{i}{3}}a{x}^{3}}{dc}}+{\frac{{\frac{i}{6}}b{x}^{2}}{{c}^{2}d}}+{\frac{a{x}^{2}}{2\,{c}^{2}d}}-{\frac{a\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2\,d{c}^{4}}}-{\frac{ia\arctan \left ( cx \right ) }{d{c}^{4}}}+{\frac{iax}{d{c}^{3}}}+{\frac{{\frac{i}{2}}b{\it dilog} \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) }{d{c}^{4}}}+{\frac{b{x}^{2}\arctan \left ( cx \right ) }{2\,{c}^{2}d}}-{\frac{b\arctan \left ( cx \right ) \ln \left ( cx-i \right ) }{d{c}^{4}}}-{\frac{{\frac{i}{3}}b\arctan \left ( cx \right ){x}^{3}}{dc}}+{\frac{ibx\arctan \left ( cx \right ) }{d{c}^{3}}}+{\frac{{\frac{i}{2}}b\ln \left ( -{\frac{i}{2}} \left ( cx+i \right ) \right ) \ln \left ( cx-i \right ) }{d{c}^{4}}}-{\frac{bx}{2\,d{c}^{3}}}-{\frac{{\frac{i}{4}}b \left ( \ln \left ( cx-i \right ) \right ) ^{2}}{d{c}^{4}}}-{\frac{{\frac{11\,i}{24}}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{d{c}^{4}}}+{\frac{{\frac{2\,i}{3}}b}{d{c}^{4}}}+{\frac{5\,b}{24\,d{c}^{4}}\arctan \left ({\frac{cx}{2}} \right ) }-{\frac{5\,b}{24\,d{c}^{4}}\arctan \left ({\frac{{c}^{3}{x}^{3}}{6}}+{\frac{7\,cx}{6}} \right ) }-{\frac{5\,b}{12\,d{c}^{4}}\arctan \left ({\frac{cx}{2}}-{\frac{i}{2}} \right ) }-{\frac{{\frac{5\,i}{48}}b\ln \left ({c}^{4}{x}^{4}+10\,{c}^{2}{x}^{2}+9 \right ) }{d{c}^{4}}}+{\frac{11\,b\arctan \left ( cx \right ) }{12\,d{c}^{4}}} \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} \, a{\left (\frac{i \,{\left (2 \, c^{2} x^{3} + 3 i \, c x^{2} - 6 \, x\right )}}{c^{3} d} + \frac{6 \, \log \left (i \, c x + 1\right )}{c^{4} d}\right )} - \frac{-\frac{1}{2} \,{\left (-12 i \,{\left (2 \,{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{7} d} + \frac{3 \, \arctan \left (c x\right )}{c^{8} d}\right )} \arctan \left (c x\right ) - \frac{c^{2} x^{2} + 3 \, \arctan \left (c x\right )^{2} - 4 \, \log \left (c^{2} x^{2} + 1\right )}{c^{8} d}\right )} c^{8} d - 36 \, c^{8} d \int \frac{x^{4} \log \left (c^{2} x^{2} + 1\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 9 i \,{\left (2 \,{\left (\frac{x^{2}}{c^{5} d} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{7} d}\right )} \log \left (c^{2} x^{2} + 1\right ) - \frac{2 \, c^{2} x^{2} - \log \left (c^{2} x^{2} + 1\right )^{2} - 2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{7} d}\right )} c^{7} d + 72 \, c^{7} d \int \frac{x^{3} \arctan \left (c x\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 72 \, c^{5} d \int \frac{x \arctan \left (c x\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 8 \, c^{3} x^{3} + 36 \, c^{4} d \int \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{5} d x^{2} + c^{3} d}\,{d x} - 6 i \, c^{2} x^{2} + 60 \, c x - 2 \,{\left (12 i \, c^{3} x^{3} - 18 \, c^{2} x^{2} - 36 i \, c x + 30\right )} \arctan \left (c x\right ) - 36 i \, \arctan \left (c x\right )^{2} + 2 \,{\left (6 \, c^{3} x^{3} + 9 i \, c^{2} x^{2} - 18 \, c x + 3 i\right )} \log \left (c^{2} x^{2} + 1\right ) - 9 i \, \log \left (c^{2} x^{2} + 1\right )^{2} - 36 i \, \log \left (12 \, c^{5} d x^{2} + 12 \, c^{3} d\right )\right )} b}{72 \, c^{4} d} \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 x^{3} \log \left (-\frac{c x + i}{c x - i}\right ) - 2 i \, a x^{3}}{2 \, c d x - 2 i \, d}, 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{{\left (b \arctan \left (c x\right ) + a\right )} x^{3}}{i \, c d x + d}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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