Optimal. Leaf size=55 \[ -\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{b \log \left (c^2+x^2\right )}{6 c^3}+\frac{b \log (x)}{3 c^3}+\frac{b}{6 c x^2} \]
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Rubi [A] time = 0.0362819, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {5033, 263, 266, 44} \[ -\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{b \log \left (c^2+x^2\right )}{6 c^3}+\frac{b \log (x)}{3 c^3}+\frac{b}{6 c x^2} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 263
Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{x^4} \, dx &=-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{3} (b c) \int \frac{1}{\left (1+\frac{c^2}{x^2}\right ) x^5} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{3} (b c) \int \frac{1}{x^3 \left (c^2+x^2\right )} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (c^2+x\right )} \, dx,x,x^2\right )\\ &=-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}-\frac{1}{6} (b c) \operatorname{Subst}\left (\int \left (\frac{1}{c^2 x^2}-\frac{1}{c^4 x}+\frac{1}{c^4 \left (c^2+x\right )}\right ) \, dx,x,x^2\right )\\ &=\frac{b}{6 c x^2}-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3}+\frac{b \log (x)}{3 c^3}-\frac{b \log \left (c^2+x^2\right )}{6 c^3}\\ \end{align*}
Mathematica [A] time = 0.0099862, size = 60, normalized size = 1.09 \[ -\frac{a}{3 x^3}-\frac{b \log \left (c^2+x^2\right )}{6 c^3}+\frac{b \log (x)}{3 c^3}+\frac{b}{6 c x^2}-\frac{b \tan ^{-1}\left (\frac{c}{x}\right )}{3 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.026, size = 45, normalized size = 0.8 \begin{align*} -{\frac{a}{3\,{x}^{3}}}-{\frac{b}{3\,{x}^{3}}\arctan \left ({\frac{c}{x}} \right ) }+{\frac{b}{6\,c{x}^{2}}}-{\frac{b}{6\,{c}^{3}}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.02616, size = 73, normalized size = 1.33 \begin{align*} -\frac{1}{6} \,{\left (c{\left (\frac{\log \left (c^{2} + x^{2}\right )}{c^{4}} - \frac{\log \left (x^{2}\right )}{c^{4}} - \frac{1}{c^{2} x^{2}}\right )} + \frac{2 \, \arctan \left (\frac{c}{x}\right )}{x^{3}}\right )} b - \frac{a}{3 \, x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.23219, size = 132, normalized size = 2.4 \begin{align*} -\frac{2 \, b c^{3} \arctan \left (\frac{c}{x}\right ) + b x^{3} \log \left (c^{2} + x^{2}\right ) - 2 \, b x^{3} \log \left (x\right ) + 2 \, a c^{3} - b c^{2} x}{6 \, c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.63721, size = 60, normalized size = 1.09 \begin{align*} \begin{cases} - \frac{a}{3 x^{3}} - \frac{b \operatorname{atan}{\left (\frac{c}{x} \right )}}{3 x^{3}} + \frac{b}{6 c x^{2}} + \frac{b \log{\left (x \right )}}{3 c^{3}} - \frac{b \log{\left (c^{2} + x^{2} \right )}}{6 c^{3}} & \text{for}\: c \neq 0 \\- \frac{a}{3 x^{3}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13462, size = 74, normalized size = 1.35 \begin{align*} -\frac{2 \, b c^{3} \arctan \left (\frac{c}{x}\right ) + b x^{3} \log \left (c^{2} + x^{2}\right ) - 2 \, b x^{3} \log \left (x\right ) + 2 \, a c^{3} - b c^{2} x}{6 \, c^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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