Optimal. Leaf size=106 \[ -\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i b c^2 d}{2 x}-\frac{1}{3} b c^3 d \log (x)-\frac{1}{12} b c^3 d \log (-c x+i)+\frac{5}{12} b c^3 d \log (c x+i)-\frac{b c d}{6 x^2} \]
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Rubi [A] time = 0.0908906, antiderivative size = 106, 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, 801} \[ -\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i b c^2 d}{2 x}-\frac{1}{3} b c^3 d \log (x)-\frac{1}{12} b c^3 d \log (-c x+i)+\frac{5}{12} b c^3 d \log (c x+i)-\frac{b c d}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 4872
Rule 12
Rule 801
Rubi steps
\begin{align*} \int \frac{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-(b c) \int \frac{d (-2-3 i c x)}{6 x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{6} (b c d) \int \frac{-2-3 i c x}{x^3 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{6} (b c d) \int \left (-\frac{2}{x^3}-\frac{3 i c}{x^2}+\frac{2 c^2}{x}+\frac{c^3}{2 (-i+c x)}-\frac{5 c^3}{2 (i+c x)}\right ) \, dx\\ &=-\frac{b c d}{6 x^2}-\frac{i b c^2 d}{2 x}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{3} b c^3 d \log (x)-\frac{1}{12} b c^3 d \log (i-c x)+\frac{5}{12} b c^3 d \log (i+c x)\\ \end{align*}
Mathematica [C] time = 0.0520788, size = 94, normalized size = 0.89 \[ -\frac{d \left (3 i b c^2 x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )+3 i a c x+2 a+2 b c^3 x^3 \log (x)-b c^3 x^3 \log \left (c^2 x^2+1\right )+b c x+b (2+3 i c x) \tan ^{-1}(c x)\right )}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 101, normalized size = 1. \begin{align*}{\frac{-{\frac{i}{2}}cda}{{x}^{2}}}-{\frac{da}{3\,{x}^{3}}}-{\frac{{\frac{i}{2}}cdb\arctan \left ( cx \right ) }{{x}^{2}}}-{\frac{db\arctan \left ( cx \right ) }{3\,{x}^{3}}}+{\frac{{c}^{3}db\ln \left ({c}^{2}{x}^{2}+1 \right ) }{6}}-{\frac{i}{2}}{c}^{3}db\arctan \left ( cx \right ) -{\frac{{\frac{i}{2}}{c}^{2}bd}{x}}-{\frac{bcd}{6\,{x}^{2}}}-{\frac{{c}^{3}db\ln \left ( cx \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48848, size = 117, normalized size = 1.1 \begin{align*} -\frac{1}{2} i \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b c d + \frac{1}{6} \,{\left ({\left (c^{2} \log \left (c^{2} x^{2} + 1\right ) - c^{2} \log \left (x^{2}\right ) - \frac{1}{x^{2}}\right )} c - \frac{2 \, \arctan \left (c x\right )}{x^{3}}\right )} b d - \frac{i \, a c d}{2 \, x^{2}} - \frac{a d}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.95781, size = 266, normalized size = 2.51 \begin{align*} -\frac{4 \, b c^{3} d x^{3} \log \left (x\right ) - 5 \, b c^{3} d x^{3} \log \left (\frac{c x + i}{c}\right ) + b c^{3} d x^{3} \log \left (\frac{c x - i}{c}\right ) + 6 i \, b c^{2} d x^{2} -{\left (-6 i \, a - 2 \, b\right )} c d x + 4 \, a d -{\left (3 \, b c d x - 2 i \, b d\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19741, size = 134, normalized size = 1.26 \begin{align*} \frac{5 \, b c^{3} d x^{3} \log \left (c x + i\right ) - b c^{3} d x^{3} \log \left (c x - i\right ) - 4 \, b c^{3} d x^{3} \log \left (x\right ) - 6 \, b c^{2} d i x^{2} - 6 \, b c d i x \arctan \left (c x\right ) - 6 \, a c d i x - 2 \, b c d x - 4 \, b d \arctan \left (c x\right ) - 4 \, a d}{12 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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