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 = 0.02, antiderivative size = 35, normalized size of antiderivative = 1.00, 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 260
Rule 6097
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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.66, size = 48, normalized size = 1.37 \[ -\frac {b x \log \left (-c^{2} + x^{2}\right ) - 2 \, b x \log \relax (x) + b c \log \left (-\frac {c + x}{c - x}\right ) + 2 \, a c}{2 \, c x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 87, normalized size = 2.49 \[ \frac {b \log \left (-\frac {c + x}{c - x} + 1\right ) - b \log \left (-\frac {c + x}{c - x}\right ) - \frac {b \log \left (-\frac {c + x}{c - x}\right )}{\frac {c + x}{c - x} - 1} - \frac {2 \, a}{\frac {c + x}{c - x} - 1}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 37, normalized size = 1.06 \[ -\frac {a}{x}-\frac {b \arctanh \left (\frac {c}{x}\right )}{x}-\frac {b \ln \left (1-\frac {c^{2}}{x^{2}}\right )}{2 c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 37, normalized size = 1.06 \[ -\frac {b {\left (\frac {2 \, c \operatorname {artanh}\left (\frac {c}{x}\right )}{x} + \log \left (-\frac {c^{2}}{x^{2}} + 1\right )\right )}}{2 \, c} - \frac {a}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.74, size = 43, normalized size = 1.23 \[ \frac {b\,x\,\ln \relax (x)-\frac {b\,x\,\ln \left (x^2-c^2\right )}{2}}{c\,x}-\frac {a+b\,\mathrm {atanh}\left (\frac {c}{x}\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.79, size = 39, normalized size = 1.11 \[ \begin {cases} - \frac {a}{x} - \frac {b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{x} + \frac {b \log {\relax (x )}}{c} - \frac {b \log {\left (- c + x \right )}}{c} - \frac {b \operatorname {atanh}{\left (\frac {c}{x} \right )}}{c} & \text {for}\: c \neq 0 \\- \frac {a}{x} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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