Optimal. Leaf size=14 \[ \log \left (-5+e^x-\frac {1}{e x}\right ) \]
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Rubi [F]
time = 0.49, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1+e^{1+x} x^2}{-x-5 e x^2+e^{1+x} x^2} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {1+x+5 e x^2}{x \left (-1-5 e x+e^{1+x} x\right )}\right ) \, dx\\ &=x+\int \frac {1+x+5 e x^2}{x \left (-1-5 e x+e^{1+x} x\right )} \, dx\\ &=x+\int \left (\frac {1}{-1-5 e x+e^{1+x} x}+\frac {1}{x \left (-1-5 e x+e^{1+x} x\right )}+\frac {5 e x}{-1-5 e x+e^{1+x} x}\right ) \, dx\\ &=x+(5 e) \int \frac {x}{-1-5 e x+e^{1+x} x} \, dx+\int \frac {1}{-1-5 e x+e^{1+x} x} \, dx+\int \frac {1}{x \left (-1-5 e x+e^{1+x} x\right )} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.53, size = 20, normalized size = 1.43 \begin {gather*} -\log (x)+\log \left (1+5 e x-e^{1+x} x\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.08, size = 19, normalized size = 1.36
method | result | size |
risch | \(\ln \left ({\mathrm e}^{x}-\frac {\left (5 x \,{\mathrm e}+1\right ) {\mathrm e}^{-1}}{x}\right )\) | \(19\) |
norman | \(-\ln \left (x \right )+\ln \left (x \,{\mathrm e} \,{\mathrm e}^{x}-5 x \,{\mathrm e}-1\right )\) | \(20\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 22, normalized size = 1.57 \begin {gather*} \log \left (-\frac {{\left (5 \, x e - x e^{\left (x + 1\right )} + 1\right )} e^{\left (-1\right )}}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 20, normalized size = 1.43 \begin {gather*} \log \left (-\frac {5 \, x e - x e^{\left (x + 1\right )} + 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 19, normalized size = 1.36 \begin {gather*} \log {\left (e^{x} + \frac {- 5 e x - 1}{e x} \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 19, normalized size = 1.36 \begin {gather*} \log \left (-5 \, x e + x e^{\left (x + 1\right )} - 1\right ) - \log \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 19, normalized size = 1.36 \begin {gather*} \ln \left (x\,\mathrm {e}\,{\mathrm {e}}^x-5\,x\,\mathrm {e}-1\right )-\ln \left (x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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