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 = 0.05, antiderivative size = 56, normalized size of antiderivative = 1.00, 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*}
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Mathematica [A] time = 0.01, 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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fricas [A] time = 0.91, size = 48, normalized size = 0.86 \[ \frac {2 \, c d \log \relax (x) - {\left (c d - a e\right )} \log \left (c x + a\right ) - {\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 64, normalized size = 1.14 \[ \frac {d \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {{\left (c^{2} d - a c e\right )} \log \left ({\left | c x + a \right |}\right )}{2 \, a^{2} c^{2}} - \frac {{\left (c^{2} d + a c e\right )} \log \left ({\left | c x - a \right |}\right )}{2 \, a^{2} c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 67, normalized size = 1.20 \[ -\frac {e \ln \left (c x -a \right )}{2 a c}+\frac {e \ln \left (c x +a \right )}{2 a c}+\frac {d \ln \relax (x )}{a^{2}}-\frac {d \ln \left (c x -a \right )}{2 a^{2}}-\frac {d \ln \left (c x +a \right )}{2 a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.55, size = 53, normalized size = 0.95 \[ \frac {d \log \relax (x)}{a^{2}} - \frac {{\left (c d - a e\right )} \log \left (c x + a\right )}{2 \, a^{2} c} - \frac {{\left (c d + a e\right )} \log \left (c x - a\right )}{2 \, a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 52, normalized size = 0.93 \[ \frac {d\,\ln \relax (x)}{a^2}+\frac {\ln \left (a+c\,x\right )\,\left (a\,e-c\,d\right )}{2\,a^2\,c}-\frac {\ln \left (a-c\,x\right )\,\left (a\,e+c\,d\right )}{2\,a^2\,c} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.59, size = 194, normalized size = 3.46 \[ \frac {d \log {\relax (x )}}{a^{2}} + \frac {\left (a e - c d\right ) \log {\left (x + \frac {- 2 a^{2} d e^{2} + \frac {a^{2} e^{2} \left (a e - c d\right )}{c} - 6 c^{2} d^{3} - 3 c d^{2} \left (a e - c d\right ) + 3 d \left (a e - c d\right )^{2}}{a^{2} e^{3} - 9 c^{2} d^{2} e} \right )}}{2 a^{2} c} - \frac {\left (a e + c d\right ) \log {\left (x + \frac {- 2 a^{2} d e^{2} - \frac {a^{2} e^{2} \left (a e + c d\right )}{c} - 6 c^{2} d^{3} + 3 c d^{2} \left (a e + c d\right ) + 3 d \left (a e + c d\right )^{2}}{a^{2} e^{3} - 9 c^{2} d^{2} e} \right )}}{2 a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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