Optimal. Leaf size=91 \[ \frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d \log \left (c^2 x^2+1\right )}{6 c^2}+\frac{b d \tan ^{-1}(c x)}{2 c^2}-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2 \]
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Rubi [A] time = 0.0773123, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {43, 4872, 12, 801, 635, 203, 260} \[ \frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d \log \left (c^2 x^2+1\right )}{6 c^2}+\frac{b d \tan ^{-1}(c x)}{2 c^2}-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 4872
Rule 12
Rule 801
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int x (d+i c d x) \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{d x^2 (3+2 i c x)}{6+6 c^2 x^2} \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c d) \int \frac{x^2 (3+2 i c x)}{6+6 c^2 x^2} \, dx\\ &=\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )-(b c d) \int \left (\frac{1}{2 c^2}+\frac{i x}{3 c}+\frac{i (3 i-2 c x)}{c^2 \left (6+6 c^2 x^2\right )}\right ) \, dx\\ &=-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{(i b d) \int \frac{3 i-2 c x}{6+6 c^2 x^2} \, dx}{c}\\ &=-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )+(2 i b d) \int \frac{x}{6+6 c^2 x^2} \, dx+\frac{(3 b d) \int \frac{1}{6+6 c^2 x^2} \, dx}{c}\\ &=-\frac{b d x}{2 c}-\frac{1}{6} i b d x^2+\frac{b d \tan ^{-1}(c x)}{2 c^2}+\frac{1}{2} d x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} i c d x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d \log \left (1+c^2 x^2\right )}{6 c^2}\\ \end{align*}
Mathematica [A] time = 0.0468314, size = 76, normalized size = 0.84 \[ \frac{d \left (c x (a c x (3+2 i c x)+b (-3-i c x))+i b \log \left (c^2 x^2+1\right )+b \left (2 i c^3 x^3+3 c^2 x^2+3\right ) \tan ^{-1}(c x)\right )}{6 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 87, normalized size = 1. \begin{align*}{\frac{i}{3}}cda{x}^{3}+{\frac{da{x}^{2}}{2}}+{\frac{i}{3}}cdb\arctan \left ( cx \right ){x}^{3}+{\frac{db\arctan \left ( cx \right ){x}^{2}}{2}}-{\frac{i}{6}}bd{x}^{2}-{\frac{dbx}{2\,c}}+{\frac{{\frac{i}{6}}db\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{db\arctan \left ( cx \right ) }{2\,{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.46907, size = 119, normalized size = 1.31 \begin{align*} \frac{1}{3} i \, a c d x^{3} + \frac{1}{6} i \,{\left (2 \, x^{3} \arctan \left (c x\right ) - c{\left (\frac{x^{2}}{c^{2}} - \frac{\log \left (c^{2} x^{2} + 1\right )}{c^{4}}\right )}\right )} b c d + \frac{1}{2} \, a d x^{2} + \frac{1}{2} \,{\left (x^{2} \arctan \left (c x\right ) - c{\left (\frac{x}{c^{2}} - \frac{\arctan \left (c x\right )}{c^{3}}\right )}\right )} b d \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.80572, size = 243, normalized size = 2.67 \begin{align*} \frac{4 i \, a c^{3} d x^{3} + 2 \,{\left (3 \, a - i \, b\right )} c^{2} d x^{2} - 6 \, b c d x + 5 i \, b d \log \left (\frac{c x + i}{c}\right ) - i \, b d \log \left (\frac{c x - i}{c}\right ) -{\left (2 \, b c^{3} d x^{3} - 3 i \, b c^{2} d x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{12 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.7801, size = 128, normalized size = 1.41 \begin{align*} \frac{i a c d x^{3}}{3} - \frac{b d x}{2 c} - \frac{i b d \log{\left (x - \frac{i}{c} \right )}}{12 c^{2}} + \frac{5 i b d \log{\left (x + \frac{i}{c} \right )}}{12 c^{2}} + x^{2} \left (\frac{a d}{2} - \frac{i b d}{6}\right ) + \left (- \frac{b c d x^{3}}{6} + \frac{i b d x^{2}}{4}\right ) \log{\left (- i c x + 1 \right )} + \left (\frac{b c d x^{3}}{6} - \frac{i b d x^{2}}{4}\right ) \log{\left (i c x + 1 \right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1604, size = 134, normalized size = 1.47 \begin{align*} -\frac{4 \, b c^{3} d x^{3} \arctan \left (c x\right ) + 4 \, a c^{3} d x^{3} - 6 \, b c^{2} d i x^{2} \arctan \left (c x\right ) - 6 \, a c^{2} d i x^{2} - 2 \, b c^{2} d x^{2} + 6 \, b c d i x + 5 \, b d \log \left (c i x - 1\right ) - b d \log \left (-c i x - 1\right )}{12 \, c^{2} i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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