Optimal. Leaf size=34 \[ \frac{b \log \left (\frac{c^2}{x^2}+1\right )}{2 c}-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{x} \]
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Rubi [A] time = 0.0190498, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {5033, 260} \[ \frac{b \log \left (\frac{c^2}{x^2}+1\right )}{2 c}-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5033
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{x^2} \, dx &=-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{x}-(b c) \int \frac{1}{\left (1+\frac{c^2}{x^2}\right ) x^3} \, dx\\ &=-\frac{a+b \tan ^{-1}\left (\frac{c}{x}\right )}{x}+\frac{b \log \left (1+\frac{c^2}{x^2}\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.0080562, size = 37, normalized size = 1.09 \[ -\frac{a}{x}+\frac{b \log \left (\frac{c^2}{x^2}+1\right )}{2 c}-\frac{b \tan ^{-1}\left (\frac{c}{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.021, size = 36, normalized size = 1.1 \begin{align*} -{\frac{a}{x}}-{\frac{b}{x}\arctan \left ({\frac{c}{x}} \right ) }+{\frac{b}{2\,c}\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 = 1.02608, size = 51, normalized size = 1.5 \begin{align*} -\frac{b{\left (\frac{2 \, c \arctan \left (\frac{c}{x}\right )}{x} - \log \left (\frac{c^{2}}{x^{2}} + 1\right )\right )}}{2 \, c} - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.22687, size = 103, normalized size = 3.03 \begin{align*} -\frac{2 \, b c \arctan \left (\frac{c}{x}\right ) - b x \log \left (c^{2} + x^{2}\right ) + 2 \, b x \log \left (x\right ) + 2 \, a c}{2 \, c x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.976839, size = 36, normalized size = 1.06 \begin{align*} \begin{cases} - \frac{a}{x} - \frac{b \operatorname{atan}{\left (\frac{c}{x} \right )}}{x} - \frac{b \log{\left (x \right )}}{c} + \frac{b \log{\left (c^{2} + x^{2} \right )}}{2 c} & \text{for}\: c \neq 0 \\- \frac{a}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.11594, size = 54, normalized size = 1.59 \begin{align*} -\frac{b{\left (\frac{2 \, c \arctan \left (\frac{c}{x}\right )}{x} - \log \left (\frac{c^{2}}{x^{2}} + 1\right )\right )} + \frac{2 \, a c}{x}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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