Optimal. Leaf size=27 \[ a x+\frac {1}{2} b c \log \left (c^2+x^2\right )+b x \tan ^{-1}\left (\frac {c}{x}\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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 = {5027, 263, 260} \[ a x+\frac {1}{2} b c \log \left (c^2+x^2\right )+b x \tan ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
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Rule 260
Rule 263
Rule 5027
Rubi steps
\begin {align*} \int \left (a+b \tan ^{-1}\left (\frac {c}{x}\right )\right ) \, dx &=a x+b \int \tan ^{-1}\left (\frac {c}{x}\right ) \, dx\\ &=a x+b x \tan ^{-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 \tan ^{-1}\left (\frac {c}{x}\right )+(b c) \int \frac {x}{c^2+x^2} \, dx\\ &=a x+b x \tan ^{-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 = 27, normalized size = 1.00 \[ a x+\frac {1}{2} b c \log \left (c^2+x^2\right )+b x \tan ^{-1}\left (\frac {c}{x}\right ) \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 25, normalized size = 0.93 \[ b x \arctan \left (\frac {c}{x}\right ) + \frac {1}{2} \, b c \log \left (c^{2} + x^{2}\right ) + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.76, size = 46, normalized size = 1.70 \[ a x + \frac {{\left (c^{2} {\left (\log \left (\frac {c^{2}}{x^{2}} + 1\right ) - \log \left (\frac {c^{2}}{x^{2}}\right )\right )} + 2 \, c x \arctan \left (\frac {c}{x}\right )\right )} b}{2 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 38, normalized size = 1.41 \[ a x +b x \arctan \left (\frac {c}{x}\right )-b c \ln \left (\frac {c}{x}\right )+\frac {b c \ln \left (1+\frac {c^{2}}{x^{2}}\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 27, normalized size = 1.00 \[ \frac {1}{2} \, {\left (2 \, x \arctan \left (\frac {c}{x}\right ) + c \log \left (c^{2} + x^{2}\right )\right )} b + a x \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.30, size = 25, normalized size = 0.93 \[ a\,x+b\,x\,\mathrm {atan}\left (\frac {c}{x}\right )+\frac {b\,c\,\ln \left (c^2+x^2\right )}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.18, size = 22, normalized size = 0.81 \[ a x + b \left (\frac {c \log {\left (c^{2} + x^{2} \right )}}{2} + x \operatorname {atan}{\left (\frac {c}{x} \right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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