Optimal. Leaf size=28 \[ \log \left (\frac {2+e^x}{x}\right )-x \left (\left (x+x^2\right )^2+\log \left (\frac {16 x}{25}\right )\right ) \]
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Rubi [A]
time = 0.18, antiderivative size = 35, normalized size of antiderivative = 1.25, number of steps
used = 11, number of rules used = 7, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {6874, 2320,
36, 29, 31, 14, 2332} \begin {gather*} -x^5-2 x^4-x^3-x \log \left (\frac {16 x}{25}\right )+\log \left (e^x+2\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 29
Rule 31
Rule 36
Rule 2320
Rule 2332
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {2}{2+e^x}+\frac {-1-3 x^3-8 x^4-5 x^5-x \log \left (\frac {16 x}{25}\right )}{x}\right ) \, dx\\ &=-\left (2 \int \frac {1}{2+e^x} \, dx\right )+\int \frac {-1-3 x^3-8 x^4-5 x^5-x \log \left (\frac {16 x}{25}\right )}{x} \, dx\\ &=-\left (2 \text {Subst}\left (\int \frac {1}{x (2+x)} \, dx,x,e^x\right )\right )+\int \left (\frac {-1-3 x^3-8 x^4-5 x^5}{x}-\log \left (\frac {16 x}{25}\right )\right ) \, dx\\ &=\int \frac {-1-3 x^3-8 x^4-5 x^5}{x} \, dx-\int \log \left (\frac {16 x}{25}\right ) \, dx-\text {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\text {Subst}\left (\int \frac {1}{2+x} \, dx,x,e^x\right )\\ &=\log \left (2+e^x\right )-x \log \left (\frac {16 x}{25}\right )+\int \left (-\frac {1}{x}-3 x^2-8 x^3-5 x^4\right ) \, dx\\ &=-x^3-2 x^4-x^5+\log \left (2+e^x\right )-x \log \left (\frac {16 x}{25}\right )-\log (x)\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.08, size = 35, normalized size = 1.25 \begin {gather*} -x^3-2 x^4-x^5+\log \left (2+e^x\right )-x \log \left (\frac {16 x}{25}\right )-\log (x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 33, normalized size = 1.18
method | result | size |
risch | \(-\ln \left (\frac {16 x}{25}\right ) x -x^{5}-2 x^{4}-x^{3}-\ln \left (x \right )+\ln \left ({\mathrm e}^{x}+2\right )\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.54, size = 38, normalized size = 1.36 \begin {gather*} -x^{5} - 2 \, x^{4} - x^{3} + 2 \, x {\left (\log \left (5\right ) - 2 \, \log \left (2\right )\right )} - {\left (x + 1\right )} \log \left (x\right ) + \log \left (e^{x} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.31, size = 32, normalized size = 1.14 \begin {gather*} -x^{5} - 2 \, x^{4} - x^{3} - x \log \left (\frac {16}{25} \, x\right ) - \log \left (x\right ) + \log \left (e^{x} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.09, size = 29, normalized size = 1.04 \begin {gather*} - x^{5} - 2 x^{4} - x^{3} - x \log {\left (\frac {16 x}{25} \right )} - \log {\left (x \right )} + \log {\left (e^{x} + 2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.39, size = 34, normalized size = 1.21 \begin {gather*} -x^{5} - 2 \, x^{4} - x^{3} - x \log \left (\frac {16}{25} \, x\right ) - \log \left (\frac {1}{25} \, x\right ) + \log \left (e^{x} + 2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.36, size = 32, normalized size = 1.14 \begin {gather*} \ln \left ({\mathrm {e}}^x+2\right )-\ln \left (x\right )-x\,\ln \left (\frac {16\,x}{25}\right )-x^3-2\,x^4-x^5 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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