Optimal. Leaf size=171 \[ \frac{c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{4 b c^3 d^2}{15 x^2}-\frac{i b c^2 d^2}{6 x^3}+\frac{i b c^4 d^2}{2 x}+\frac{8}{15} b c^5 d^2 \log (x)-\frac{1}{60} b c^5 d^2 \log (-c x+i)-\frac{31}{60} b c^5 d^2 \log (c x+i)-\frac{b c d^2}{20 x^4} \]
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Rubi [A] time = 0.158001, antiderivative size = 171, 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 = {43, 4872, 12, 1802} \[ \frac{c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}+\frac{4 b c^3 d^2}{15 x^2}-\frac{i b c^2 d^2}{6 x^3}+\frac{i b c^4 d^2}{2 x}+\frac{8}{15} b c^5 d^2 \log (x)-\frac{1}{60} b c^5 d^2 \log (-c x+i)-\frac{31}{60} b c^5 d^2 \log (c x+i)-\frac{b c d^2}{20 x^4} \]
Antiderivative was successfully verified.
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Rule 43
Rule 4872
Rule 12
Rule 1802
Rubi steps
\begin{align*} \int \frac{(d+i c d x)^2 \left (a+b \tan ^{-1}(c x)\right )}{x^6} \, dx &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac{c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-(b c) \int \frac{d^2 \left (-6-15 i c x+10 c^2 x^2\right )}{30 x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac{c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{30} \left (b c d^2\right ) \int \frac{-6-15 i c x+10 c^2 x^2}{x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac{c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}-\frac{1}{30} \left (b c d^2\right ) \int \left (-\frac{6}{x^5}-\frac{15 i c}{x^4}+\frac{16 c^2}{x^3}+\frac{15 i c^3}{x^2}-\frac{16 c^4}{x}+\frac{c^5}{2 (-i+c x)}+\frac{31 c^5}{2 (i+c x)}\right ) \, dx\\ &=-\frac{b c d^2}{20 x^4}-\frac{i b c^2 d^2}{6 x^3}+\frac{4 b c^3 d^2}{15 x^2}+\frac{i b c^4 d^2}{2 x}-\frac{d^2 \left (a+b \tan ^{-1}(c x)\right )}{5 x^5}-\frac{i c d^2 \left (a+b \tan ^{-1}(c x)\right )}{2 x^4}+\frac{c^2 d^2 \left (a+b \tan ^{-1}(c x)\right )}{3 x^3}+\frac{8}{15} b c^5 d^2 \log (x)-\frac{1}{60} b c^5 d^2 \log (i-c x)-\frac{31}{60} b c^5 d^2 \log (i+c x)\\ \end{align*}
Mathematica [C] time = 0.0862631, size = 124, normalized size = 0.73 \[ \frac{d^2 \left (-10 i b c^2 x^2 \text{Hypergeometric2F1}\left (-\frac{3}{2},1,-\frac{1}{2},-c^2 x^2\right )+20 a c^2 x^2-30 i a c x-12 a+16 b c^3 x^3+32 b c^5 x^5 \log (x)-16 b c^5 x^5 \log \left (c^2 x^2+1\right )+2 b \left (10 c^2 x^2-15 i c x-6\right ) \tan ^{-1}(c x)-3 b c x\right )}{60 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.037, size = 172, normalized size = 1. \begin{align*}{\frac{-{\frac{i}{2}}c{d}^{2}a}{{x}^{4}}}-{\frac{{d}^{2}a}{5\,{x}^{5}}}+{\frac{{c}^{2}{d}^{2}a}{3\,{x}^{3}}}-{\frac{{\frac{i}{2}}c{d}^{2}b\arctan \left ( cx \right ) }{{x}^{4}}}-{\frac{b{d}^{2}\arctan \left ( cx \right ) }{5\,{x}^{5}}}+{\frac{b{c}^{2}{d}^{2}\arctan \left ( cx \right ) }{3\,{x}^{3}}}-{\frac{4\,{c}^{5}{d}^{2}b\ln \left ({c}^{2}{x}^{2}+1 \right ) }{15}}+{\frac{i}{2}}{c}^{5}{d}^{2}b\arctan \left ( cx \right ) -{\frac{{\frac{i}{6}}b{c}^{2}{d}^{2}}{{x}^{3}}}+{\frac{{\frac{i}{2}}b{c}^{4}{d}^{2}}{x}}-{\frac{bc{d}^{2}}{20\,{x}^{4}}}+{\frac{4\,b{c}^{3}{d}^{2}}{15\,{x}^{2}}}+{\frac{8\,{c}^{5}{d}^{2}b\ln \left ( cx \right ) }{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.48874, size = 247, normalized size = 1.44 \begin{align*} -\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 c^{2} d^{2} + \frac{1}{6} i \,{\left ({\left (3 \, c^{3} \arctan \left (c x\right ) + \frac{3 \, c^{2} x^{2} - 1}{x^{3}}\right )} c - \frac{3 \, \arctan \left (c x\right )}{x^{4}}\right )} b c d^{2} - \frac{1}{20} \,{\left ({\left (2 \, c^{4} \log \left (c^{2} x^{2} + 1\right ) - 2 \, c^{4} \log \left (x^{2}\right ) - \frac{2 \, c^{2} x^{2} - 1}{x^{4}}\right )} c + \frac{4 \, \arctan \left (c x\right )}{x^{5}}\right )} b d^{2} + \frac{a c^{2} d^{2}}{3 \, x^{3}} - \frac{i \, a c d^{2}}{2 \, x^{4}} - \frac{a d^{2}}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.89968, size = 387, normalized size = 2.26 \begin{align*} \frac{32 \, b c^{5} d^{2} x^{5} \log \left (x\right ) - 31 \, b c^{5} d^{2} x^{5} \log \left (\frac{c x + i}{c}\right ) - b c^{5} d^{2} x^{5} \log \left (\frac{c x - i}{c}\right ) + 30 i \, b c^{4} d^{2} x^{4} + 16 \, b c^{3} d^{2} x^{3} + 10 \,{\left (2 \, a - i \, b\right )} c^{2} d^{2} x^{2} +{\left (-30 i \, a - 3 \, b\right )} c d^{2} x - 12 \, a d^{2} +{\left (10 i \, b c^{2} d^{2} x^{2} + 15 \, b c d^{2} x - 6 i \, b d^{2}\right )} \log \left (-\frac{c x + i}{c x - i}\right )}{60 \, x^{5}} \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 = 1.38457, size = 228, normalized size = 1.33 \begin{align*} -\frac{31 \, b c^{5} d^{2} x^{5} \log \left (c x + i\right ) + b c^{5} d^{2} x^{5} \log \left (c x - i\right ) - 32 \, b c^{5} d^{2} x^{5} \log \left (x\right ) - 30 \, b c^{4} d^{2} i x^{4} - 16 \, b c^{3} d^{2} x^{3} + 10 \, b c^{2} d^{2} i x^{2} - 20 \, b c^{2} d^{2} x^{2} \arctan \left (c x\right ) - 20 \, a c^{2} d^{2} x^{2} + 30 \, b c d^{2} i x \arctan \left (c x\right ) + 30 \, a c d^{2} i x + 3 \, b c d^{2} x + 12 \, b d^{2} \arctan \left (c x\right ) + 12 \, a d^{2}}{60 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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