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 = 0.0116355, antiderivative size = 27, 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 = {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 5027
Rule 263
Rule 260
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*}
Mathematica [A] time = 0.002662, size = 27, normalized size = 1. \[ a x+\frac{1}{2} b c \log \left (c^2+x^2\right )+b x \tan ^{-1}\left (\frac{c}{x}\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.029, size = 38, normalized size = 1.4 \begin{align*} ax+bx\arctan \left ({\frac{c}{x}} \right ) +{\frac{bc}{2}\ln \left ( 1+{\frac{{c}^{2}}{{x}^{2}}} \right ) }-bc\ln \left ({\frac{c}{x}} \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.987169, size = 36, normalized size = 1.33 \begin{align*} \frac{1}{2} \,{\left (2 \, x \arctan \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 = 2.12627, size = 65, normalized size = 2.41 \begin{align*} b x \arctan \left (\frac{c}{x}\right ) + \frac{1}{2} \, b c \log \left (c^{2} + x^{2}\right ) + a x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.220325, size = 22, normalized size = 0.81 \begin{align*} a x + b \left (\frac{c \log{\left (c^{2} + x^{2} \right )}}{2} + x \operatorname{atan}{\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 = 1.12084, size = 36, normalized size = 1.33 \begin{align*} \frac{1}{2} \,{\left (2 \, x \arctan \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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