Optimal. Leaf size=56 \[ -\frac{(a e+c d) \log (a-c x)}{2 a^2 c}-\frac{(c d-a e) \log (a+c x)}{2 a^2 c}+\frac{d \log (x)}{a^2} \]
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Rubi [A] time = 0.0488401, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {801} \[ -\frac{(a e+c d) \log (a-c x)}{2 a^2 c}-\frac{(c d-a e) \log (a+c x)}{2 a^2 c}+\frac{d \log (x)}{a^2} \]
Antiderivative was successfully verified.
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Rule 801
Rubi steps
\begin{align*} \int \frac{d+e x}{x \left (a^2-c^2 x^2\right )} \, dx &=\int \left (\frac{d}{a^2 x}-\frac{-c d-a e}{2 a^2 (a-c x)}+\frac{-c d+a e}{2 a^2 (a+c x)}\right ) \, dx\\ &=\frac{d \log (x)}{a^2}-\frac{(c d+a e) \log (a-c x)}{2 a^2 c}-\frac{(c d-a e) \log (a+c x)}{2 a^2 c}\\ \end{align*}
Mathematica [A] time = 0.0129546, size = 44, normalized size = 0.79 \[ -\frac{d \log \left (a^2-c^2 x^2\right )}{2 a^2}+\frac{d \log (x)}{a^2}+\frac{e \tanh ^{-1}\left (\frac{c x}{a}\right )}{a c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 67, normalized size = 1.2 \begin{align*}{\frac{d\ln \left ( x \right ) }{{a}^{2}}}+{\frac{\ln \left ( cx+a \right ) e}{2\,ac}}-{\frac{\ln \left ( cx+a \right ) d}{2\,{a}^{2}}}-{\frac{\ln \left ( cx-a \right ) e}{2\,ac}}-{\frac{\ln \left ( cx-a \right ) d}{2\,{a}^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.09237, size = 72, normalized size = 1.29 \begin{align*} \frac{d \log \left (x\right )}{a^{2}} - \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{2} c} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.57148, size = 111, normalized size = 1.98 \begin{align*} \frac{2 \, c d \log \left (x\right ) -{\left (c d - a e\right )} \log \left (c x + a\right ) -{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.65768, size = 194, normalized size = 3.46 \begin{align*} \frac{d \log{\left (x \right )}}{a^{2}} + \frac{\left (a e - c d\right ) \log{\left (x + \frac{- 2 a^{2} d e^{2} + \frac{a^{2} e^{2} \left (a e - c d\right )}{c} - 6 c^{2} d^{3} - 3 c d^{2} \left (a e - c d\right ) + 3 d \left (a e - c d\right )^{2}}{a^{2} e^{3} - 9 c^{2} d^{2} e} \right )}}{2 a^{2} c} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{- 2 a^{2} d e^{2} - \frac{a^{2} e^{2} \left (a e + c d\right )}{c} - 6 c^{2} d^{3} + 3 c d^{2} \left (a e + c d\right ) + 3 d \left (a e + c d\right )^{2}}{a^{2} e^{3} - 9 c^{2} d^{2} e} \right )}}{2 a^{2} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15739, size = 86, normalized size = 1.54 \begin{align*} \frac{d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac{{\left (c^{2} d - a c e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{2} c^{2}} - \frac{{\left (c^{2} d + a c e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{2} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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