Optimal. Leaf size=83 \[ -\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}-\frac{2}{3} i b d^2 x \]
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Rubi [A] time = 0.0456393, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4862, 627, 43} \[ -\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}-\frac{2}{3} i b d^2 x \]
Antiderivative was successfully verified.
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Rule 4862
Rule 627
Rule 43
Rubi steps
\begin{align*} \int (d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(i b) \int \frac{(d+i c d x)^3}{1+c^2 x^2} \, dx}{3 d}\\ &=-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(i b) \int \frac{(d+i c d x)^2}{\frac{1}{d}-\frac{i c x}{d}} \, dx}{3 d}\\ &=-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}+\frac{(i b) \int \left (-2 d^3+\frac{4 d^2}{\frac{1}{d}-\frac{i c x}{d}}-d^2 (d+i c d x)\right ) \, dx}{3 d}\\ &=-\frac{2}{3} i b d^2 x-\frac{b d^2 (1+i c x)^2}{6 c}-\frac{i d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 c}-\frac{4 b d^2 \log (1-i c x)}{3 c}\\ \end{align*}
Mathematica [A] time = 0.0385873, size = 57, normalized size = 0.69 \[ \frac{1}{3} d^2 \left (-\frac{(c x-i)^3 \left (a+b \tan ^{-1}(c x)\right )}{c}+\frac{1}{2} b x (c x-6 i)-\frac{4 b \log (c x+i)}{c}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.028, size = 133, normalized size = 1.6 \begin{align*} -{\frac{{c}^{2}{x}^{3}a{d}^{2}}{3}}+ic{x}^{2}a{d}^{2}+ax{d}^{2}-{\frac{{\frac{i}{3}}{d}^{2}a}{c}}-{\frac{{c}^{2}{d}^{2}b\arctan \left ( cx \right ){x}^{3}}{3}}+ic{d}^{2}b\arctan \left ( cx \right ){x}^{2}+{d}^{2}bx\arctan \left ( cx \right ) +{\frac{i{d}^{2}b\arctan \left ( cx \right ) }{c}}-i{d}^{2}bx+{\frac{c{d}^{2}b{x}^{2}}{6}}-{\frac{2\,{d}^{2}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.49232, size = 186, normalized size = 2.24 \begin{align*} -\frac{1}{3} \, a c^{2} d^{2} x^{3} - \frac{1}{6} \,{\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^{2} d^{2} + i \, a c d^{2} x^{2} + i \,{\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 c d^{2} + a d^{2} x + \frac{{\left (2 \, c x \arctan \left (c x\right ) - \log \left (c^{2} x^{2} + 1\right )\right )} b d^{2}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.68781, size = 279, normalized size = 3.36 \begin{align*} -\frac{2 \, a c^{3} d^{2} x^{3} -{\left (6 i \, a + b\right )} c^{2} d^{2} x^{2} - 6 \,{\left (a - i \, b\right )} c d^{2} x + 7 \, b d^{2} \log \left (\frac{c x + i}{c}\right ) + b d^{2} \log \left (\frac{c x - i}{c}\right ) -{\left (-i \, b c^{3} d^{2} x^{3} - 3 \, b c^{2} d^{2} x^{2} + 3 i \, b c d^{2} x\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.60729, size = 165, normalized size = 1.99 \begin{align*} - \frac{a c^{2} d^{2} x^{3}}{3} - \frac{b d^{2} \left (\frac{\log{\left (x - \frac{i}{c} \right )}}{6} + \frac{7 \log{\left (x + \frac{i}{c} \right )}}{6}\right )}{c} - x^{2} \left (- i a c d^{2} - \frac{b c d^{2}}{6}\right ) - x \left (- a d^{2} + i b d^{2}\right ) + \left (- \frac{i b c^{2} d^{2} x^{3}}{6} - \frac{b c d^{2} x^{2}}{2} + \frac{i b d^{2} x}{2}\right ) \log{\left (- i c x + 1 \right )} + \left (\frac{i b c^{2} d^{2} x^{3}}{6} + \frac{b c d^{2} x^{2}}{2} - \frac{i b d^{2} x}{2}\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 = 1.15826, size = 176, normalized size = 2.12 \begin{align*} -\frac{2 \, b c^{3} d^{2} x^{3} \arctan \left (c x\right ) + 2 \, a c^{3} d^{2} x^{3} - 6 \, b c^{2} d^{2} i x^{2} \arctan \left (c x\right ) - 6 \, a c^{2} d^{2} i x^{2} - b c^{2} d^{2} x^{2} + 6 \, b c d^{2} i x - 6 \, b c d^{2} x \arctan \left (c x\right ) - 6 \, a c d^{2} x + 7 \, b d^{2} \log \left (c x + i\right ) + b d^{2} \log \left (c x - i\right )}{6 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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