Optimal. Leaf size=64 \[ -\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{b c^3}{10 x^2}-\frac{1}{10} b c^5 \log \left (c^2 x^2+1\right )+\frac{1}{5} b c^5 \log (x)-\frac{b c}{20 x^4} \]
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Rubi [A] time = 0.0382962, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {4852, 266, 44} \[ -\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{b c^3}{10 x^2}-\frac{1}{10} b c^5 \log \left (c^2 x^2+1\right )+\frac{1}{5} b c^5 \log (x)-\frac{b c}{20 x^4} \]
Antiderivative was successfully verified.
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Rule 4852
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^6} \, dx &=-\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{1}{5} (b c) \int \frac{1}{x^5 \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{1}{10} (b c) \operatorname{Subst}\left (\int \frac{1}{x^3 \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{1}{10} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{x^3}-\frac{c^2}{x^2}+\frac{c^4}{x}-\frac{c^6}{1+c^2 x}\right ) \, dx,x,x^2\right )\\ &=-\frac{b c}{20 x^4}+\frac{b c^3}{10 x^2}-\frac{a+b \tan ^{-1}(c x)}{5 x^5}+\frac{1}{5} b c^5 \log (x)-\frac{1}{10} b c^5 \log \left (1+c^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0168336, size = 64, normalized size = 1. \[ -\frac{a}{5 x^5}+\frac{1}{10} b c \left (\frac{c^2}{x^2}-c^4 \log \left (c^2 x^2+1\right )+2 c^4 \log (x)-\frac{1}{2 x^4}\right )-\frac{b \tan ^{-1}(c x)}{5 x^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 60, normalized size = 0.9 \begin{align*} -{\frac{a}{5\,{x}^{5}}}-{\frac{b\arctan \left ( cx \right ) }{5\,{x}^{5}}}-{\frac{b{c}^{5}\ln \left ({c}^{2}{x}^{2}+1 \right ) }{10}}-{\frac{bc}{20\,{x}^{4}}}+{\frac{{c}^{5}b\ln \left ( cx \right ) }{5}}+{\frac{b{c}^{3}}{10\,{x}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.975209, size = 84, normalized size = 1.31 \begin{align*} -\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 - \frac{a}{5 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.52753, size = 147, normalized size = 2.3 \begin{align*} -\frac{2 \, b c^{5} x^{5} \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{5} x^{5} \log \left (x\right ) - 2 \, b c^{3} x^{3} + b c x + 4 \, b \arctan \left (c x\right ) + 4 \, a}{20 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.5915, size = 71, normalized size = 1.11 \begin{align*} \begin{cases} - \frac{a}{5 x^{5}} + \frac{b c^{5} \log{\left (x \right )}}{5} - \frac{b c^{5} \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{10} + \frac{b c^{3}}{10 x^{2}} - \frac{b c}{20 x^{4}} - \frac{b \operatorname{atan}{\left (c x \right )}}{5 x^{5}} & \text{for}\: c \neq 0 \\- \frac{a}{5 x^{5}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40601, size = 80, normalized size = 1.25 \begin{align*} -\frac{2 \, b c^{5} x^{5} \log \left (c^{2} x^{2} + 1\right ) - 4 \, b c^{5} x^{5} \log \left (x\right ) - 2 \, b c^{3} x^{3} + b c x + 4 \, b \arctan \left (c x\right ) + 4 \, a}{20 \, x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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