Optimal. Leaf size=136 \[ -\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d^2 \log \left (c^2 x^2+1\right )}{3 c^2}+\frac{3 b d^2 \tan ^{-1}(c x)}{4 c^2}+\frac{1}{12} b c d^2 x^3-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2 \]
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Rubi [A] time = 0.127064, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {43, 4872, 12, 1802, 635, 203, 260} \[ -\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d^2 \log \left (c^2 x^2+1\right )}{3 c^2}+\frac{3 b d^2 \tan ^{-1}(c x)}{4 c^2}+\frac{1}{12} b c d^2 x^3-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2 \]
Antiderivative was successfully verified.
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Rule 43
Rule 4872
Rule 12
Rule 1802
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int x (d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-(b c) \int \frac{d^2 x^2 \left (6+8 i c x-3 c^2 x^2\right )}{12 \left (1+c^2 x^2\right )} \, dx\\ &=\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{12} \left (b c d^2\right ) \int \frac{x^2 \left (6+8 i c x-3 c^2 x^2\right )}{1+c^2 x^2} \, dx\\ &=\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{12} \left (b c d^2\right ) \int \left (\frac{9}{c^2}+\frac{8 i x}{c}-3 x^2+\frac{i (9 i-8 c x)}{c^2 \left (1+c^2 x^2\right )}\right ) \, dx\\ &=-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2+\frac{1}{12} b c d^2 x^3+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )-\frac{\left (i b d^2\right ) \int \frac{9 i-8 c x}{1+c^2 x^2} \, dx}{12 c}\\ &=-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2+\frac{1}{12} b c d^2 x^3+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{1}{3} \left (2 i b d^2\right ) \int \frac{x}{1+c^2 x^2} \, dx+\frac{\left (3 b d^2\right ) \int \frac{1}{1+c^2 x^2} \, dx}{4 c}\\ &=-\frac{3 b d^2 x}{4 c}-\frac{1}{3} i b d^2 x^2+\frac{1}{12} b c d^2 x^3+\frac{3 b d^2 \tan ^{-1}(c x)}{4 c^2}+\frac{1}{2} d^2 x^2 \left (a+b \tan ^{-1}(c x)\right )+\frac{2}{3} i c d^2 x^3 \left (a+b \tan ^{-1}(c x)\right )-\frac{1}{4} c^2 d^2 x^4 \left (a+b \tan ^{-1}(c x)\right )+\frac{i b d^2 \log \left (1+c^2 x^2\right )}{3 c^2}\\ \end{align*}
Mathematica [A] time = 0.0910559, size = 101, normalized size = 0.74 \[ \frac{d^2 \left (c x \left (a c x \left (-3 c^2 x^2+8 i c x+6\right )+b \left (c^2 x^2-4 i c x-9\right )\right )+4 i b \log \left (c^2 x^2+1\right )+b \left (-3 c^4 x^4+8 i c^3 x^3+6 c^2 x^2+9\right ) \tan ^{-1}(c x)\right )}{12 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 141, normalized size = 1. \begin{align*} -{\frac{{c}^{2}{d}^{2}a{x}^{4}}{4}}+{\frac{2\,i}{3}}c{d}^{2}a{x}^{3}+{\frac{{d}^{2}a{x}^{2}}{2}}-{\frac{{c}^{2}{d}^{2}b\arctan \left ( cx \right ){x}^{4}}{4}}+{\frac{2\,i}{3}}c{d}^{2}b\arctan \left ( cx \right ){x}^{3}+{\frac{{d}^{2}b\arctan \left ( cx \right ){x}^{2}}{2}}-{\frac{3\,{d}^{2}bx}{4\,c}}+{\frac{bc{d}^{2}{x}^{3}}{12}}-{\frac{i}{3}}b{d}^{2}{x}^{2}+{\frac{{\frac{i}{3}}{d}^{2}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{3\,b{d}^{2}\arctan \left ( cx \right ) }{4\,{c}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47229, size = 209, normalized size = 1.54 \begin{align*} -\frac{1}{4} \, a c^{2} d^{2} x^{4} + \frac{2}{3} i \, a c d^{2} x^{3} - \frac{1}{12} \,{\left (3 \, x^{4} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{3} - 3 \, x}{c^{4}} + \frac{3 \, \arctan \left (c x\right )}{c^{5}}\right )}\right )} b c^{2} d^{2} + \frac{1}{3} 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^{2} + \frac{1}{2} \, a d^{2} 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^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.79827, size = 333, normalized size = 2.45 \begin{align*} -\frac{6 \, a c^{4} d^{2} x^{4} -{\left (16 i \, a + 2 \, b\right )} c^{3} d^{2} x^{3} - 4 \,{\left (3 \, a - 2 i \, b\right )} c^{2} d^{2} x^{2} + 18 \, b c d^{2} x - 17 i \, b d^{2} \log \left (\frac{c x + i}{c}\right ) + i \, b d^{2} \log \left (\frac{c x - i}{c}\right ) -{\left (-3 i \, b c^{4} d^{2} x^{4} - 8 \, b c^{3} d^{2} x^{3} + 6 i \, b c^{2} d^{2} x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{24 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.71854, size = 202, normalized size = 1.49 \begin{align*} - \frac{a c^{2} d^{2} x^{4}}{4} - \frac{3 b d^{2} x}{4 c} - \frac{i b d^{2} \log{\left (x - \frac{i}{c} \right )}}{24 c^{2}} + \frac{17 i b d^{2} \log{\left (x + \frac{i}{c} \right )}}{24 c^{2}} - x^{3} \left (- \frac{2 i a c d^{2}}{3} - \frac{b c d^{2}}{12}\right ) - x^{2} \left (- \frac{a d^{2}}{2} + \frac{i b d^{2}}{3}\right ) + \left (- \frac{i b c^{2} d^{2} x^{4}}{8} - \frac{b c d^{2} x^{3}}{3} + \frac{i b d^{2} x^{2}}{4}\right ) \log{\left (- i c x + 1 \right )} + \left (\frac{i b c^{2} d^{2} x^{4}}{8} + \frac{b c d^{2} x^{3}}{3} - \frac{i b d^{2} 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 = 1.15196, size = 207, normalized size = 1.52 \begin{align*} -\frac{6 \, b c^{4} d^{2} x^{4} \arctan \left (c x\right ) + 6 \, a c^{4} d^{2} x^{4} - 16 \, b c^{3} d^{2} i x^{3} \arctan \left (c x\right ) - 16 \, a c^{3} d^{2} i x^{3} - 2 \, b c^{3} d^{2} x^{3} + 8 \, b c^{2} d^{2} i x^{2} - 12 \, b c^{2} d^{2} x^{2} \arctan \left (c x\right ) - 12 \, a c^{2} d^{2} x^{2} + 18 \, b c d^{2} x - 17 \, b d^{2} i \log \left (c i x - 1\right ) + b d^{2} i \log \left (-c i x - 1\right )}{24 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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