Optimal. Leaf size=87 \[ -\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i b c^2 d^2}{x}-\frac{4}{3} b c^3 d^2 \log (x)+\frac{4}{3} b c^3 d^2 \log (c x+i)-\frac{b c d^2}{6 x^2} \]
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Rubi [A] time = 0.0818295, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {37, 4872, 12, 88} \[ -\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i b c^2 d^2}{x}-\frac{4}{3} b c^3 d^2 \log (x)+\frac{4}{3} b c^3 d^2 \log (c x+i)-\frac{b c d^2}{6 x^2} \]
Antiderivative was successfully verified.
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Rule 37
Rule 4872
Rule 12
Rule 88
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )}{x^4} \, dx &=-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{i d^2 (i-c x)^2}{3 x^3 (i+c x)} \, dx\\ &=-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{3} \left (i b c d^2\right ) \int \frac{(i-c x)^2}{x^3 (i+c x)} \, dx\\ &=-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{3} \left (i b c d^2\right ) \int \left (\frac{i}{x^3}-\frac{3 c}{x^2}-\frac{4 i c^2}{x}+\frac{4 i c^3}{i+c x}\right ) \, dx\\ &=-\frac{b c d^2}{6 x^2}-\frac{i b c^2 d^2}{x}-\frac{d^2 (1+i c x)^3 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{4}{3} b c^3 d^2 \log (x)+\frac{4}{3} b c^3 d^2 \log (i+c x)\\ \end{align*}
Mathematica [C] time = 0.0903624, size = 114, normalized size = 1.31 \[ -\frac{d^2 \left (6 i b c^2 x^2 \text{Hypergeometric2F1}\left (-\frac{1}{2},1,\frac{1}{2},-c^2 x^2\right )-6 a c^2 x^2+6 i a c x+2 a+8 b c^3 x^3 \log (x)-4 b c^3 x^3 \log \left (c^2 x^2+1\right )+2 b \left (-3 c^2 x^2+3 i c x+1\right ) \tan ^{-1}(c x)+b c x\right )}{6 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.036, size = 145, normalized size = 1.7 \begin{align*}{\frac{-ic{d}^{2}a}{{x}^{2}}}+{\frac{{c}^{2}{d}^{2}a}{x}}-{\frac{{d}^{2}a}{3\,{x}^{3}}}-{\frac{ic{d}^{2}b\arctan \left ( cx \right ) }{{x}^{2}}}+{\frac{b{c}^{2}{d}^{2}\arctan \left ( cx \right ) }{x}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) }{3\,{x}^{3}}}+{\frac{2\,{c}^{3}{d}^{2}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{3}}-i{c}^{3}{d}^{2}b\arctan \left ( cx \right ) -{\frac{ib{c}^{2}{d}^{2}}{x}}-{\frac{bc{d}^{2}}{6\,{x}^{2}}}-{\frac{4\,{c}^{3}{d}^{2}b\ln \left ( cx \right ) }{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47882, size = 194, normalized size = 2.23 \begin{align*} \frac{1}{2} \,{\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^{2} d^{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^{2} + \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^{2} + \frac{a c^{2} d^{2}}{x} - \frac{i \, a c d^{2}}{x^{2}} - \frac{a d^{2}}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.74852, size = 316, normalized size = 3.63 \begin{align*} -\frac{8 \, b c^{3} d^{2} x^{3} \log \left (x\right ) - 7 \, b c^{3} d^{2} x^{3} \log \left (\frac{c x + i}{c}\right ) - b c^{3} d^{2} x^{3} \log \left (\frac{c x - i}{c}\right ) - 6 \,{\left (a - i \, b\right )} c^{2} d^{2} x^{2} -{\left (-6 i \, a - b\right )} c d^{2} x + 2 \, a d^{2} -{\left (3 i \, b c^{2} d^{2} x^{2} + 3 \, b c d^{2} x - i \, b d^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{6 \, 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 = 1.22176, size = 194, normalized size = 2.23 \begin{align*} \frac{7 \, b c^{3} d^{2} x^{3} \log \left (c x + i\right ) + b c^{3} d^{2} x^{3} \log \left (c x - i\right ) - 8 \, b c^{3} d^{2} x^{3} \log \left (x\right ) - 6 \, b c^{2} d^{2} i x^{2} + 6 \, b c^{2} d^{2} x^{2} \arctan \left (c x\right ) + 6 \, a c^{2} d^{2} x^{2} - 6 \, b c d^{2} i x \arctan \left (c x\right ) - 6 \, a c d^{2} i x - b c d^{2} x - 2 \, b d^{2} \arctan \left (c x\right ) - 2 \, a d^{2}}{6 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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