Optimal. Leaf size=29 \[ a x+\frac{1}{2} b c \log \left (c^2-x^2\right )+b x \tanh ^{-1}\left (\frac{c}{x}\right ) \]
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Rubi [A] time = 0.0133345, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {6091, 263, 260} \[ a x+\frac{1}{2} b c \log \left (c^2-x^2\right )+b x \tanh ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 6091
Rule 263
Rule 260
Rubi steps
\begin{align*} \int \left (a+b \tanh ^{-1}\left (\frac{c}{x}\right )\right ) \, dx &=a x+b \int \tanh ^{-1}\left (\frac{c}{x}\right ) \, dx\\ &=a x+b x \tanh ^{-1}\left (\frac{c}{x}\right )+(b c) \int \frac{1}{\left (1-\frac{c^2}{x^2}\right ) x} \, dx\\ &=a x+b x \tanh ^{-1}\left (\frac{c}{x}\right )+(b c) \int \frac{x}{-c^2+x^2} \, dx\\ &=a x+b x \tanh ^{-1}\left (\frac{c}{x}\right )+\frac{1}{2} b c \log \left (c^2-x^2\right )\\ \end{align*}
Mathematica [A] time = 0.0028142, size = 29, normalized size = 1. \[ a x+\frac{1}{2} b c \log \left (c^2-x^2\right )+b x \tanh ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 48, normalized size = 1.7 \begin{align*} ax+bx{\it Artanh} \left ({\frac{c}{x}} \right ) +{\frac{bc}{2}\ln \left ({\frac{c}{x}}-1 \right ) }-bc\ln \left ({\frac{c}{x}} \right ) +{\frac{bc}{2}\ln \left ( 1+{\frac{c}{x}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.949895, size = 39, normalized size = 1.34 \begin{align*} \frac{1}{2} \,{\left (2 \, x \operatorname{artanh}\left (\frac{c}{x}\right ) + c \log \left (-c^{2} + x^{2}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.67863, size = 85, normalized size = 2.93 \begin{align*} \frac{1}{2} \, b c \log \left (-c^{2} + x^{2}\right ) + \frac{1}{2} \, b x \log \left (-\frac{c + x}{c - x}\right ) + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.317447, size = 24, normalized size = 0.83 \begin{align*} a x + b \left (c \log{\left (c - x \right )} + c \operatorname{atanh}{\left (\frac{c}{x} \right )} + x \operatorname{atanh}{\left (\frac{c}{x} \right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13099, size = 57, normalized size = 1.97 \begin{align*} \frac{1}{2} \,{\left (x \log \left (-\frac{\frac{c}{x} + 1}{\frac{c}{x} - 1}\right ) + c \log \left ({\left | -c^{2} + x^{2} \right |}\right )\right )} b + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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