Optimal. Leaf size=35 \[ -\frac{a+b \tan ^{-1}(c x)}{x}-\frac{1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x) \]
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Rubi [A] time = 0.0230129, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {4852, 266, 36, 29, 31} \[ -\frac{a+b \tan ^{-1}(c x)}{x}-\frac{1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x) \]
Antiderivative was successfully verified.
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Rule 4852
Rule 266
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{a+b \tan ^{-1}(c x)}{x^2} \, dx &=-\frac{a+b \tan ^{-1}(c x)}{x}+(b c) \int \frac{1}{x \left (1+c^2 x^2\right )} \, dx\\ &=-\frac{a+b \tan ^{-1}(c x)}{x}+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{1}{x \left (1+c^2 x\right )} \, dx,x,x^2\right )\\ &=-\frac{a+b \tan ^{-1}(c x)}{x}+\frac{1}{2} (b c) \operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,x^2\right )-\frac{1}{2} \left (b c^3\right ) \operatorname{Subst}\left (\int \frac{1}{1+c^2 x} \, dx,x,x^2\right )\\ &=-\frac{a+b \tan ^{-1}(c x)}{x}+b c \log (x)-\frac{1}{2} b c \log \left (1+c^2 x^2\right )\\ \end{align*}
Mathematica [A] time = 0.002613, size = 38, normalized size = 1.09 \[ -\frac{a}{x}-\frac{1}{2} b c \log \left (c^2 x^2+1\right )+b c \log (x)-\frac{b \tan ^{-1}(c x)}{x} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 39, normalized size = 1.1 \begin{align*} -{\frac{a}{x}}-{\frac{b\arctan \left ( cx \right ) }{x}}-{\frac{bc\ln \left ({c}^{2}{x}^{2}+1 \right ) }{2}}+cb\ln \left ( cx \right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.969497, size = 53, normalized size = 1.51 \begin{align*} -\frac{1}{2} \,{\left (c{\left (\log \left (c^{2} x^{2} + 1\right ) - \log \left (x^{2}\right )\right )} + \frac{2 \, \arctan \left (c x\right )}{x}\right )} b - \frac{a}{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.49191, size = 100, normalized size = 2.86 \begin{align*} -\frac{b c x \log \left (c^{2} x^{2} + 1\right ) - 2 \, b c x \log \left (x\right ) + 2 \, b \arctan \left (c x\right ) + 2 \, a}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.88915, size = 37, normalized size = 1.06 \begin{align*} \begin{cases} - \frac{a}{x} + b c \log{\left (x \right )} - \frac{b c \log{\left (x^{2} + \frac{1}{c^{2}} \right )}}{2} - \frac{b \operatorname{atan}{\left (c x \right )}}{x} & \text{for}\: c \neq 0 \\- \frac{a}{x} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.32274, size = 50, normalized size = 1.43 \begin{align*} -\frac{b c x \log \left (c^{2} x^{2} + 1\right ) - 2 \, b c x \log \left (x\right ) + 2 \, b \arctan \left (c x\right ) + 2 \, a}{2 \, x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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