Optimal. Leaf size=27 \[ 1+\frac {e^x}{\log \left (2 (4-x) x \left (-\frac {1}{2}+25 e^x+x\right )\right )} \]
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Rubi [A] time = 1.31, antiderivative size = 25, normalized size of antiderivative = 0.93, number of steps used = 2, number of rules used = 2, integrand size = 154, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {6688, 2288} \begin {gather*} \frac {e^x}{\log \left (-\left (\left (-2 x-50 e^x+1\right ) (4-x) x\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 2288
Rule 6688
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (-4+18 x-6 x^2-50 e^x \left (-4-2 x+x^2\right )+(-4+x) x \left (-1+50 e^x+2 x\right ) \log \left (-\left ((-4+x) x \left (-1+50 e^x+2 x\right )\right )\right )\right )}{\left (1-50 e^x-2 x\right ) (4-x) x \log ^2\left (-\left ((-4+x) x \left (-1+50 e^x+2 x\right )\right )\right )} \, dx\\ &=\frac {e^x}{\log \left (-\left (\left (1-50 e^x-2 x\right ) (4-x) x\right )\right )}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.08, size = 23, normalized size = 0.85 \begin {gather*} \frac {e^x}{\log \left (-\left ((-4+x) x \left (-1+50 e^x+2 x\right )\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.02, size = 31, normalized size = 1.15 \begin {gather*} \frac {e^{x}}{\log \left (-2 \, x^{3} + 9 \, x^{2} - 50 \, {\left (x^{2} - 4 \, x\right )} e^{x} - 4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.37, size = 32, normalized size = 1.19 \begin {gather*} \frac {e^{x}}{\log \left (-2 \, x^{3} - 50 \, x^{2} e^{x} + 9 \, x^{2} + 200 \, x e^{x} - 4 \, x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.14, size = 222, normalized size = 8.22
method | result | size |
risch | \(\frac {2 i {\mathrm e}^{x}}{\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right ) \mathrm {csgn}\left (i x \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right )-\pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right )^{2}+2 \pi \mathrm {csgn}\left (i x \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right )^{2}-\pi \,\mathrm {csgn}\left (i \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right ) \mathrm {csgn}\left (i x \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right )^{2}-\pi \mathrm {csgn}\left (i x \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )\right )^{3}-2 \pi +2 i \ln \relax (x )+2 i \ln \left (x^{2}+\left (25 \,{\mathrm e}^{x}-\frac {9}{2}\right ) x -100 \,{\mathrm e}^{x}+2\right )}\) | \(222\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 22, normalized size = 0.81 \begin {gather*} \frac {e^{x}}{\log \left (x - 4\right ) + \log \relax (x) + \log \left (-2 \, x - 50 \, e^{x} + 1\right )} \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 {\ln \left ({\mathrm {e}}^x\,\left (200\,x-50\,x^2\right )-4\,x+9\,x^2-2\,x^3\right )\,\left ({\mathrm {e}}^{2\,x}\,\left (200\,x-50\,x^2\right )-{\mathrm {e}}^x\,\left (2\,x^3-9\,x^2+4\,x\right )\right )-{\mathrm {e}}^{2\,x}\,\left (-50\,x^2+100\,x+200\right )+{\mathrm {e}}^x\,\left (6\,x^2-18\,x+4\right )}{{\ln \left ({\mathrm {e}}^x\,\left (200\,x-50\,x^2\right )-4\,x+9\,x^2-2\,x^3\right )}^2\,\left (4\,x-{\mathrm {e}}^x\,\left (200\,x-50\,x^2\right )-9\,x^2+2\,x^3\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.37, size = 29, normalized size = 1.07 \begin {gather*} \frac {e^{x}}{\log {\left (- 2 x^{3} + 9 x^{2} - 4 x + \left (- 50 x^{2} + 200 x\right ) e^{x} \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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