Optimal. Leaf size=46 \[ \frac{\left (\frac{c d}{a}-e\right ) \log (a+c x)}{2 c^2}-\frac{\left (\frac{c d}{a}+e\right ) \log (a-c x)}{2 c^2} \]
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Rubi [A] time = 0.0231256, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {633, 31} \[ \frac{\left (\frac{c d}{a}-e\right ) \log (a+c x)}{2 c^2}-\frac{\left (\frac{c d}{a}+e\right ) \log (a-c x)}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 633
Rule 31
Rubi steps
\begin{align*} \int \frac{d+e x}{a^2-c^2 x^2} \, dx &=\frac{1}{2} \left (-\frac{c d}{a}+e\right ) \int \frac{1}{-a c-c^2 x} \, dx+\frac{1}{2} \left (\frac{c d}{a}+e\right ) \int \frac{1}{a c-c^2 x} \, dx\\ &=-\frac{\left (\frac{c d}{a}+e\right ) \log (a-c x)}{2 c^2}+\frac{\left (\frac{c d}{a}-e\right ) \log (a+c x)}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0069652, size = 37, normalized size = 0.8 \[ \frac{d \tanh ^{-1}\left (\frac{c x}{a}\right )}{a c}-\frac{e \log \left (a^2-c^2 x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 60, normalized size = 1.3 \begin{align*} -{\frac{\ln \left ( cx+a \right ) e}{2\,{c}^{2}}}+{\frac{\ln \left ( cx+a \right ) d}{2\,ac}}-{\frac{\ln \left ( cx-a \right ) e}{2\,{c}^{2}}}-{\frac{\ln \left ( cx-a \right ) d}{2\,ac}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.997951, size = 62, normalized size = 1.35 \begin{align*} \frac{{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a c^{2}} - \frac{{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.47897, size = 90, normalized size = 1.96 \begin{align*} \frac{{\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 c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.31694, size = 71, normalized size = 1.54 \begin{align*} - \frac{\left (a e - c d\right ) \log{\left (x + \frac{a^{2} e - a \left (a e - c d\right )}{c^{2} d} \right )}}{2 a c^{2}} - \frac{\left (a e + c d\right ) \log{\left (x + \frac{a^{2} e - a \left (a e + c d\right )}{c^{2} d} \right )}}{2 a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14586, size = 68, normalized size = 1.48 \begin{align*} \frac{{\left (c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a c^{2}} - \frac{{\left (c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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