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 = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, 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 260
Rule 263
Rule 6091
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*}
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Mathematica [A] time = 0.00, size = 29, normalized size = 1.00 \[ a x+\frac {1}{2} b c \log \left (c^2-x^2\right )+b x \tanh ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 35, normalized size = 1.21 \[ \frac {1}{2} \, b c \log \left (-c^{2} + x^{2}\right ) + \frac {1}{2} \, b x \log \left (-\frac {c + x}{c - x}\right ) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 150, normalized size = 5.17 \[ a x + \frac {{\left (c^{2} {\left (\log \left (\frac {{\left | -c - x \right |}}{{\left | c - x \right |}}\right ) - \log \left ({\left | -\frac {c + x}{c - x} - 1 \right |}\right )\right )} - \frac {c^{2} \log \left (-\frac {\frac {c {\left (\frac {c + x}{{\left (c - x\right )} c} + \frac {1}{c}\right )}}{\frac {c + x}{c - x} - 1} + 1}{\frac {c {\left (\frac {c + x}{{\left (c - x\right )} c} + \frac {1}{c}\right )}}{\frac {c + x}{c - x} - 1} - 1}\right )}{\frac {c + x}{c - x} + 1}\right )} b}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 48, normalized size = 1.66 \[ a x +b x \arctanh \left (\frac {c}{x}\right )-b c \ln \left (\frac {c}{x}\right )+\frac {b c \ln \left (\frac {c}{x}-1\right )}{2}+\frac {b c \ln \left (1+\frac {c}{x}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.31, size = 29, normalized size = 1.00 \[ \frac {1}{2} \, {\left (2 \, x \operatorname {artanh}\left (\frac {c}{x}\right ) + c \log \left (-c^{2} + x^{2}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 27, normalized size = 0.93 \[ a\,x+b\,x\,\mathrm {atanh}\left (\frac {c}{x}\right )+\frac {b\,c\,\ln \left (x^2-c^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.25, size = 24, normalized size = 0.83 \[ a x + b \left (c \log {\left (c - x \right )} + c \operatorname {atanh}{\left (\frac {c}{x} \right )} + x \operatorname {atanh}{\left (\frac {c}{x} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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