Optimal. Leaf size=35 \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\frac{b \log \left (1-\frac{c^2}{x^2}\right )}{2 c} \]
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Rubi [A] time = 0.0208549, antiderivative size = 35, 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 = {6097, 260} \[ -\frac{a+b \tanh ^{-1}\left (\frac{c}{x}\right )}{x}-\frac{b \log \left (1-\frac{c^2}{x^2}\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 6097
Rule 260
Rubi steps
\begin{align*} \int \frac{a+b \tanh ^{-1}\left (\frac{c}{x}\right )}{x^2} \, dx &=-\frac{a+b \tanh ^{-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 \tanh ^{-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 = 0.0082962, size = 38, normalized size = 1.09 \[ -\frac{a}{x}-\frac{b \log \left (1-\frac{c^2}{x^2}\right )}{2 c}-\frac{b \tanh ^{-1}\left (\frac{c}{x}\right )}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 37, normalized size = 1.1 \begin{align*} -{\frac{a}{x}}-{\frac{b}{x}{\it Artanh} \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 = 0.956861, size = 50, normalized size = 1.43 \begin{align*} -\frac{b{\left (\frac{2 \, c \operatorname{artanh}\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 = 1.69369, size = 115, normalized size = 3.29 \begin{align*} -\frac{b x \log \left (-c^{2} + x^{2}\right ) - 2 \, b x \log \left (x\right ) + b c \log \left (-\frac{c + x}{c - 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 = 1.8518, size = 39, normalized size = 1.11 \begin{align*} \begin{cases} - \frac{a}{x} - \frac{b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{x} + \frac{b \log{\left (x \right )}}{c} - \frac{b \log{\left (- c + x \right )}}{c} - \frac{b \operatorname{atanh}{\left (\frac{c}{x} \right )}}{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 = 1.13314, size = 66, normalized size = 1.89 \begin{align*} -\frac{1}{2} \, b{\left (\frac{\log \left (-\frac{\frac{c}{x} + 1}{\frac{c}{x} - 1}\right )}{x} + \frac{\log \left ({\left | \frac{c^{2}}{x^{2}} - 1 \right |}\right )}{c}\right )} - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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