Optimal. Leaf size=157 \[ -\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac{i b d^3 (-c x+i)^4}{20 c^2}-\frac{b d^3 (-c x+i)^3}{20 c^2}-\frac{3 i b d^3 (-c x+i)^2}{20 c^2}+\frac{6 i b d^3 \log (c x+i)}{5 c^2}-\frac{3 b d^3 x}{5 c} \]
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Rubi [A] time = 0.0968673, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {43, 4872, 12, 77} \[ -\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}+\frac{i b d^3 (-c x+i)^4}{20 c^2}-\frac{b d^3 (-c x+i)^3}{20 c^2}-\frac{3 i b d^3 (-c x+i)^2}{20 c^2}+\frac{6 i b d^3 \log (c x+i)}{5 c^2}-\frac{3 b d^3 x}{5 c} \]
Antiderivative was successfully verified.
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Rule 43
Rule 4872
Rule 12
Rule 77
Rubi steps
\begin{align*} \int x (d+i c d x)^3 \left (a+b \tan ^{-1}(c x)\right ) \, dx &=\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-(b c) \int \frac{d^3 (i-c x)^3 (-1+4 i c x)}{20 c^2 (i+c x)} \, dx\\ &=\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac{\left (b d^3\right ) \int \frac{(i-c x)^3 (-1+4 i c x)}{i+c x} \, dx}{20 c}\\ &=\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}-\frac{\left (b d^3\right ) \int \left (12+4 i (i-c x)^3+6 i (-i+c x)-3 (-i+c x)^2-\frac{24 i}{i+c x}\right ) \, dx}{20 c}\\ &=-\frac{3 b d^3 x}{5 c}-\frac{3 i b d^3 (i-c x)^2}{20 c^2}-\frac{b d^3 (i-c x)^3}{20 c^2}+\frac{i b d^3 (i-c x)^4}{20 c^2}+\frac{d^3 (1+i c x)^4 \left (a+b \tan ^{-1}(c x)\right )}{4 c^2}-\frac{d^3 (1+i c x)^5 \left (a+b \tan ^{-1}(c x)\right )}{5 c^2}+\frac{6 i b d^3 \log (i+c x)}{5 c^2}\\ \end{align*}
Mathematica [A] time = 0.100708, size = 132, normalized size = 0.84 \[ \frac{d^3 \left (c x \left (a c x \left (-4 i c^3 x^3-15 c^2 x^2+20 i c x+10\right )+b \left (i c^3 x^3+5 c^2 x^2-12 i c x-25\right )\right )+12 i b \log \left (c^2 x^2+1\right )+b \left (-4 i c^5 x^5-15 c^4 x^4+20 i c^3 x^3+10 c^2 x^2+25\right ) \tan ^{-1}(c x)\right )}{20 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.027, size = 184, normalized size = 1.2 \begin{align*} -{\frac{i}{5}}{c}^{3}{d}^{3}a{x}^{5}-{\frac{3\,{c}^{2}{d}^{3}a{x}^{4}}{4}}+ic{d}^{3}a{x}^{3}+{\frac{{d}^{3}a{x}^{2}}{2}}-{\frac{i}{5}}{c}^{3}{d}^{3}b\arctan \left ( cx \right ){x}^{5}-{\frac{3\,{c}^{2}{d}^{3}b\arctan \left ( cx \right ){x}^{4}}{4}}+ic{d}^{3}b\arctan \left ( cx \right ){x}^{3}+{\frac{{d}^{3}b\arctan \left ( cx \right ){x}^{2}}{2}}-{\frac{5\,{d}^{3}bx}{4\,c}}+{\frac{i}{20}}{c}^{2}{d}^{3}b{x}^{4}+{\frac{c{d}^{3}b{x}^{3}}{4}}-{\frac{3\,i}{5}}{d}^{3}b{x}^{2}+{\frac{{\frac{3\,i}{5}}{d}^{3}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{{c}^{2}}}+{\frac{5\,b{d}^{3}\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 = 1.48549, size = 300, normalized size = 1.91 \begin{align*} -\frac{1}{5} i \, a c^{3} d^{3} x^{5} - \frac{3}{4} \, a c^{2} d^{3} x^{4} - \frac{1}{20} i \,{\left (4 \, x^{5} \arctan \left (c x\right ) - c{\left (\frac{c^{2} x^{4} - 2 \, x^{2}}{c^{4}} + \frac{2 \, \log \left (c^{2} x^{2} + 1\right )}{c^{6}}\right )}\right )} b c^{3} d^{3} + i \, a c d^{3} x^{3} - \frac{1}{4} \,{\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^{3} + \frac{1}{2} 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^{3} + \frac{1}{2} \, a d^{3} 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^{3} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.76153, size = 404, normalized size = 2.57 \begin{align*} \frac{-8 i \, a c^{5} d^{3} x^{5} - 2 \,{\left (15 \, a - i \, b\right )} c^{4} d^{3} x^{4} +{\left (40 i \, a + 10 \, b\right )} c^{3} d^{3} x^{3} + 4 \,{\left (5 \, a - 6 i \, b\right )} c^{2} d^{3} x^{2} - 50 \, b c d^{3} x + 49 i \, b d^{3} \log \left (\frac{c x + i}{c}\right ) - i \, b d^{3} \log \left (\frac{c x - i}{c}\right ) +{\left (4 \, b c^{5} d^{3} x^{5} - 15 i \, b c^{4} d^{3} x^{4} - 20 \, b c^{3} d^{3} x^{3} + 10 i \, b c^{2} d^{3} x^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{40 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.36243, size = 260, normalized size = 1.66 \begin{align*} - \frac{i a c^{3} d^{3} x^{5}}{5} - \frac{5 b d^{3} x}{4 c} - \frac{i b d^{3} \log{\left (x - \frac{i}{c} \right )}}{40 c^{2}} + \frac{49 i b d^{3} \log{\left (x + \frac{i}{c} \right )}}{40 c^{2}} - x^{4} \left (\frac{3 a c^{2} d^{3}}{4} - \frac{i b c^{2} d^{3}}{20}\right ) - x^{3} \left (- i a c d^{3} - \frac{b c d^{3}}{4}\right ) - x^{2} \left (- \frac{a d^{3}}{2} + \frac{3 i b d^{3}}{5}\right ) + \left (- \frac{b c^{3} d^{3} x^{5}}{10} + \frac{3 i b c^{2} d^{3} x^{4}}{8} + \frac{b c d^{3} x^{3}}{2} - \frac{i b d^{3} x^{2}}{4}\right ) \log{\left (i c x + 1 \right )} + \left (\frac{b c^{3} d^{3} x^{5}}{10} - \frac{3 i b c^{2} d^{3} x^{4}}{8} - \frac{b c d^{3} x^{3}}{2} + \frac{i b d^{3} 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 = 1.17778, size = 266, normalized size = 1.69 \begin{align*} \frac{8 \, b c^{5} d^{3} x^{5} \arctan \left (c x\right ) + 8 \, a c^{5} d^{3} x^{5} - 30 \, b c^{4} d^{3} i x^{4} \arctan \left (c x\right ) - 30 \, a c^{4} d^{3} i x^{4} - 2 \, b c^{4} d^{3} x^{4} + 10 \, b c^{3} d^{3} i x^{3} - 40 \, b c^{3} d^{3} x^{3} \arctan \left (c x\right ) - 40 \, a c^{3} d^{3} x^{3} + 20 \, b c^{2} d^{3} i x^{2} \arctan \left (c x\right ) + 20 \, a c^{2} d^{3} i x^{2} + 24 \, b c^{2} d^{3} x^{2} - 50 \, b c d^{3} i x - 49 \, b d^{3} \log \left (c i x - 1\right ) + b d^{3} \log \left (-c i x - 1\right )}{40 \, c^{2} i} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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