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 = 0.02, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, 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 31
Rule 633
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*}
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Mathematica [A] time = 0.01, size = 37, normalized size = 0.80 \[ \frac {d \tanh ^{-1}\left (\frac {c x}{a}\right )}{a c}-\frac {e \log \left (a^2-c^2 x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.77, size = 41, normalized size = 0.89 \[ \frac {{\left (c d - a e\right )} \log \left (c x + a\right ) - {\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 50, normalized size = 1.09 \[ \frac {{\left (c d - a e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a c^{2}} - \frac {{\left (c d + a e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 60, normalized size = 1.30 \[ -\frac {d \ln \left (c x -a \right )}{2 a c}+\frac {d \ln \left (c x +a \right )}{2 a c}-\frac {e \ln \left (c x -a \right )}{2 c^{2}}-\frac {e \ln \left (c x +a \right )}{2 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 46, normalized size = 1.00 \[ \frac {{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a c^{2}} - \frac {{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 45, normalized size = 0.98 \[ -\frac {\ln \left (a+c\,x\right )\,\left (a\,e-c\,d\right )}{2\,a\,c^2}-\frac {\ln \left (a-c\,x\right )\,\left (a\,e+c\,d\right )}{2\,a\,c^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.33, size = 71, normalized size = 1.54 \[ - \frac {\left (a e - c d\right ) \log {\left (x + \frac {a^{2} e - a \left (a e - c d\right )}{c^{2} d} \right )}}{2 a c^{2}} - \frac {\left (a e + c d\right ) \log {\left (x + \frac {a^{2} e - a \left (a e + c d\right )}{c^{2} d} \right )}}{2 a c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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