Optimal. Leaf size=27 \[ \frac {5-x}{4+\frac {e^x}{2}+x+\log \left (4-e^4+x\right )} \]
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Rubi [F] time = 6.59, 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 {164-36 e^4+32 x+e^x \left (48+4 x-2 x^2+e^4 (-12+2 x)\right )+\left (16-4 e^4+4 x\right ) \log \left (4-e^4+x\right )}{-256+e^{2 x} \left (-4+e^4-x\right )-192 x-48 x^2-4 x^3+e^4 \left (64+32 x+4 x^2\right )+e^x \left (-64-32 x-4 x^2+e^4 (16+4 x)\right )+\left (-128+e^x \left (-16+4 e^4-4 x\right )-64 x-8 x^2+e^4 (32+8 x)\right ) \log \left (4-e^4+x\right )+\left (-16+4 e^4-4 x\right ) \log ^2\left (4-e^4+x\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (-82 \left (1-\frac {9 e^4}{41}\right )-e^{4+x} (-6+x)-16 x-e^x \left (24+2 x-x^2\right )-\left (8-2 e^4+2 x\right ) \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {-82 \left (1-\frac {9 e^4}{41}\right )-e^{4+x} (-6+x)-16 x-e^x \left (24+2 x-x^2\right )-\left (8-2 e^4+2 x\right ) \log \left (4-e^4+x\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \left (\frac {-6+x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )}+\frac {2 (5-x) \left (11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {-6+x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \frac {(5-x) \left (11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \left (-\frac {6}{8+e^x+2 x+2 \log \left (4-e^4+x\right )}+\frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )}\right ) \, dx+4 \int \frac {(5-x) \left (11+7 x+x^2-e^4 (3+x)+\left (4-e^4+x\right ) \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \left (\frac {-11 \left (1-\frac {3 e^4}{11}\right )-7 \left (1-\frac {e^4}{7}\right ) x-x^2-4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )-x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {\left (9-e^4\right ) \left (11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \frac {-11 \left (1-\frac {3 e^4}{11}\right )-7 \left (1-\frac {e^4}{7}\right ) x-x^2-4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )-x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {11 \left (1-\frac {3 e^4}{11}\right )+7 \left (1-\frac {e^4}{7}\right ) x+x^2+4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )+x \log \left (4-e^4+x\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \frac {-11-7 x-x^2+e^4 (3+x)+\left (-4+e^4-x\right ) \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {11+7 x+x^2-e^4 (3+x)-\left (-4+e^4-x\right ) \log \left (4-e^4+x\right )}{\left (4-e^4+x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+4 \int \left (-\frac {11 \left (1-\frac {3 e^4}{11}\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {7 \left (1-\frac {e^4}{7}\right ) x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx+\left (4 \left (9-e^4\right )\right ) \int \left (-\frac {11 \left (1-\frac {3 e^4}{11}\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {7 \left (1-\frac {e^4}{7}\right ) x}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x^2}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {4 \left (1-\frac {e^4}{4}\right ) \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-4 \int \frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-4 \int \frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-\left (4 \left (11-3 e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (9-e^4\right )\right ) \int \frac {x^2}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (9-e^4\right )\right ) \int \frac {x \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (11-3 e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right ) \left (9-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \frac {x}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-4 \int \frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-4 \int \frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-\left (4 \left (11-3 e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (9-e^4\right )\right ) \int \left (\frac {4 \left (1-\frac {e^4}{4}\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {\left (-4+e^4\right )^2}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}-\frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-\left (4 \left (9-e^4\right )\right ) \int \left (-\frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {\left (-4+e^4\right ) \log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx-\left (4 \left (11-3 e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right ) \left (9-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \left (-\frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}+\frac {-4+e^4}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2}\right ) \, dx\\ &=2 \int \frac {x}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-4 \int \frac {x^2}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-4 \int \frac {x \log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-12 \int \frac {1}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \, dx-\left (4 \left (11-3 e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (7-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {x}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (9-e^4\right )\right ) \int \frac {\log \left (4-e^4+x\right )}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (11-3 e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx-\left (4 \left (4-e^4\right )^2 \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx+\left (4 \left (4-e^4\right ) \left (7-e^4\right ) \left (9-e^4\right )\right ) \int \frac {1}{\left (-4+e^4-x\right ) \left (8+e^x+2 x+2 \log \left (4-e^4+x\right )\right )^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.07, size = 28, normalized size = 1.04 \begin {gather*} \frac {2 (5-x)}{8+e^x+2 x+2 \log \left (4-e^4+x\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.86, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.32, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 25, normalized size = 0.93
method | result | size |
risch | \(-\frac {2 \left (x -5\right )}{{\mathrm e}^{x}+2 \ln \left (4-{\mathrm e}^{4}+x \right )+2 x +8}\) | \(25\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 24, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (x - 5\right )}}{2 \, x + e^{x} + 2 \, \log \left (x - e^{4} + 4\right ) + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int -\frac {32\,x-36\,{\mathrm {e}}^4+\ln \left (x-{\mathrm {e}}^4+4\right )\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+{\mathrm {e}}^x\,\left (4\,x-2\,x^2+{\mathrm {e}}^4\,\left (2\,x-12\right )+48\right )+164}{192\,x+\ln \left (x-{\mathrm {e}}^4+4\right )\,\left (64\,x+{\mathrm {e}}^x\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+8\,x^2-{\mathrm {e}}^4\,\left (8\,x+32\right )+128\right )-{\mathrm {e}}^4\,\left (4\,x^2+32\,x+64\right )+{\mathrm {e}}^{2\,x}\,\left (x-{\mathrm {e}}^4+4\right )+{\mathrm {e}}^x\,\left (32\,x+4\,x^2-{\mathrm {e}}^4\,\left (4\,x+16\right )+64\right )+{\ln \left (x-{\mathrm {e}}^4+4\right )}^2\,\left (4\,x-4\,{\mathrm {e}}^4+16\right )+48\,x^2+4\,x^3+256} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.41, size = 22, normalized size = 0.81 \begin {gather*} \frac {10 - 2 x}{2 x + e^{x} + 2 \log {\left (x - e^{4} + 4 \right )} + 8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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