Optimal. Leaf size=65 \[ -\frac{d (1+i c x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+i b c^2 d \log (x)-i b c^2 d \log (c x+i)-\frac{b c d}{2 x} \]
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Rubi [A] time = 0.0552368, antiderivative size = 65, 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 = {37, 4872, 12, 77} \[ -\frac{d (1+i c x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+i b c^2 d \log (x)-i b c^2 d \log (c x+i)-\frac{b c d}{2 x} \]
Antiderivative was successfully verified.
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Rule 37
Rule 4872
Rule 12
Rule 77
Rubi steps
\begin{align*} \int \frac{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )}{x^3} \, dx &=-\frac{d (1+i c x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-(b c) \int \frac{d (-i+c x)}{2 x^2 (i+c x)} \, dx\\ &=-\frac{d (1+i c x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{2} (b c d) \int \frac{-i+c x}{x^2 (i+c x)} \, dx\\ &=-\frac{d (1+i c x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{1}{2} (b c d) \int \left (-\frac{1}{x^2}-\frac{2 i c}{x}+\frac{2 i c^2}{i+c x}\right ) \, dx\\ &=-\frac{b c d}{2 x}-\frac{d (1+i c x)^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}+i b c^2 d \log (x)-i b c^2 d \log (i+c x)\\ \end{align*}
Mathematica [C] time = 0.0547187, size = 88, normalized size = 1.35 \[ -\frac{b c d \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )}{2 x}-\frac{d \left (a+b \tan ^{-1}(c x)\right )}{2 x^2}-\frac{i c d \left (a+b \tan ^{-1}(c x)\right )}{x}+\frac{1}{2} i b c^2 d \left (2 \log (x)-\log \left (c^2 x^2+1\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 91, normalized size = 1.4 \begin{align*} -{\frac{da}{2\,{x}^{2}}}-{\frac{idca}{x}}-{\frac{db\arctan \left ( cx \right ) }{2\,{x}^{2}}}-{\frac{idcb\arctan \left ( cx \right ) }{x}}-{\frac{i}{2}}{c}^{2}db\ln \left ({c}^{2}{x}^{2}+1 \right ) -{\frac{b{c}^{2}d\arctan \left ( cx \right ) }{2}}-{\frac{bcd}{2\,x}}+i{c}^{2}db\ln \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49313, size = 101, normalized size = 1.55 \begin{align*} -\frac{1}{2} i \,{\left (c{\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \arctan \left (c x\right )}{x}\right )} b c d - \frac{1}{2} \,{\left ({\left (c \arctan \left (c x\right ) + \frac{1}{x}\right )} c + \frac{\arctan \left (c x\right )}{x^{2}}\right )} b d - \frac{i \, a c d}{x} - \frac{a d}{2 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 3.01925, size = 244, normalized size = 3.75 \begin{align*} \frac{4 i \, b c^{2} d x^{2} \log \left (x\right ) - 3 i \, b c^{2} d x^{2} \log \left (\frac{c x + i}{c}\right ) - i \, b c^{2} d x^{2} \log \left (\frac{c x - i}{c}\right ) +{\left (-4 i \, a - 2 \, b\right )} c d x - 2 \, a d +{\left (2 \, b c d x - i \, b d\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{4 \, x^{2}} \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 = 1.14698, size = 123, normalized size = 1.89 \begin{align*} -\frac{b c^{2} d i x^{2} \log \left (c i x + 1\right ) + 3 \, b c^{2} d i x^{2} \log \left (-c i x + 1\right ) - 4 \, b c^{2} d i x^{2} \log \left (x\right ) + 4 \, b c d i x \arctan \left (c x\right ) + 4 \, a c d i x + 2 \, b c d x + 2 \, b d \arctan \left (c x\right ) + 2 \, a d}{4 \, x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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