Optimal. Leaf size=26 \[ \left (\frac {3}{9+\frac {e^x}{4 \left (-3+e^{25}\right )}}-x\right ) x \]
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Rubi [B] time = 0.55, antiderivative size = 119, normalized size of antiderivative = 4.58, number of steps used = 17, number of rules used = 12, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {6741, 6742, 2184, 2190, 2279, 2391, 2185, 2191, 2282, 36, 29, 31} \begin {gather*} -\frac {5 x^2}{6}-\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {x}{3}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \log \left (e^x-36 \left (3-e^{25}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 31
Rule 36
Rule 2184
Rule 2185
Rule 2190
Rule 2191
Rule 2279
Rule 2282
Rule 2391
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3888+\left (1-\frac {6}{e^{25}}\right ) e^{50} (432-2592 x)-23328 x-2 e^{2 x} x+e^x \left (-36+e^{25} (12-156 x)+468 x\right )}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2} \, dx\\ &=\int \left (\frac {12 \left (3-e^{25}\right ) (-1+x)}{e^x-108 \left (1-\frac {e^{25}}{3}\right )}-2 x+\frac {432 \left (3-e^{25}\right )^2 x}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2}\right ) \, dx\\ &=-x^2+\left (12 \left (3-e^{25}\right )\right ) \int \frac {-1+x}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx+\left (432 \left (3-e^{25}\right )^2\right ) \int \frac {x}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2} \, dx\\ &=-\frac {1}{6} (1-x)^2-x^2+\frac {1}{3} \int \frac {e^x (-1+x)}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx+\left (12 \left (3-e^{25}\right )\right ) \int \frac {e^x x}{\left (e^x-108 \left (1-\frac {e^{25}}{3}\right )\right )^2} \, dx-\left (12 \left (3-e^{25}\right )\right ) \int \frac {x}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx\\ &=-\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} \int \frac {e^x x}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx-\frac {1}{3} \int \log \left (1-\frac {e^x}{108 \left (1-\frac {e^{25}}{3}\right )}\right ) \, dx+\left (12 \left (3-e^{25}\right )\right ) \int \frac {1}{e^x-108 \left (1-\frac {e^{25}}{3}\right )} \, dx\\ &=-\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \int \log \left (1-\frac {e^x}{108 \left (1-\frac {e^{25}}{3}\right )}\right ) \, dx-\frac {1}{3} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{108 \left (1-\frac {e^{25}}{3}\right )}\right )}{x} \, dx,x,e^x\right )+\left (12 \left (3-e^{25}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{x \left (-108+36 e^{25}+x\right )} \, dx,x,e^x\right )\\ &=-\frac {1}{6} (1-x)^2-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \text {Li}_2\left (\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x} \, dx,x,e^x\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-108+36 e^{25}+x} \, dx,x,e^x\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {\log \left (1-\frac {x}{108 \left (1-\frac {e^{25}}{3}\right )}\right )}{x} \, dx,x,e^x\right )\\ &=-\frac {1}{6} (1-x)^2-\frac {x}{3}-\frac {12 \left (3-e^{25}\right ) x}{e^x-36 \left (3-e^{25}\right )}-\frac {5 x^2}{6}-\frac {1}{3} (1-x) \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )-\frac {1}{3} x \log \left (1-\frac {e^x}{36 \left (3-e^{25}\right )}\right )+\frac {1}{3} \log \left (e^x-36 \left (3-e^{25}\right )\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.15, size = 24, normalized size = 0.92 \begin {gather*} -x \left (-\frac {12 \left (-3+e^{25}\right )}{-108+36 e^{25}+e^x}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 40, normalized size = 1.54 \begin {gather*} -\frac {x^{2} e^{x} - 108 \, x^{2} + 12 \, {\left (3 \, x^{2} - x\right )} e^{25} + 36 \, x}{36 \, e^{25} + e^{x} - 108} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.87, size = 159, normalized size = 6.12 \begin {gather*} -\frac {108 \, x^{2} e^{50} - 648 \, x^{2} e^{25} + 3 \, x^{2} e^{\left (x + 25\right )} - 9 \, x^{2} e^{x} + 972 \, x^{2} - 36 \, x e^{50} + 216 \, x e^{25} + 468 \, e^{50} \log \left (36 \, e^{25} + e^{x} - 108\right ) - 1404 \, e^{25} \log \left (36 \, e^{25} + e^{x} - 108\right ) + 13 \, e^{\left (x + 25\right )} \log \left (36 \, e^{25} + e^{x} - 108\right ) - 468 \, e^{50} \log \left (-36 \, e^{25} - e^{x} + 108\right ) + 1404 \, e^{25} \log \left (-36 \, e^{25} - e^{x} + 108\right ) - 13 \, e^{\left (x + 25\right )} \log \left (-36 \, e^{25} - e^{x} + 108\right ) - 324 \, x}{3 \, {\left (36 \, e^{50} - 216 \, e^{25} + e^{\left (x + 25\right )} - 3 \, e^{x} + 324\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.39, size = 24, normalized size = 0.92
method | result | size |
risch | \(-x^{2}+\frac {12 \left ({\mathrm e}^{25}-3\right ) x}{{\mathrm e}^{x}+36 \,{\mathrm e}^{25}-108}\) | \(24\) |
norman | \(\frac {\left (-36 \,{\mathrm e}^{25}+108\right ) x^{2}+\left (12 \,{\mathrm e}^{25}-36\right ) x -{\mathrm e}^{x} x^{2}}{{\mathrm e}^{x}+36 \,{\mathrm e}^{25}-108}\) | \(38\) |
default | \(\text {Expression too large to display}\) | \(2970\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.58, size = 222, normalized size = 8.54 \begin {gather*} \frac {1}{3} \, {\left (\frac {x}{e^{50} - 6 \, e^{25} + 9} - \frac {\log \left (36 \, e^{25} + e^{x} - 108\right )}{e^{50} - 6 \, e^{25} + 9} + \frac {36}{{\left (e^{25} - 3\right )} e^{x} + 36 \, e^{50} - 216 \, e^{25} + 324}\right )} e^{50} - 2 \, {\left (\frac {x}{e^{50} - 6 \, e^{25} + 9} - \frac {\log \left (36 \, e^{25} + e^{x} - 108\right )}{e^{50} - 6 \, e^{25} + 9} + \frac {36}{{\left (e^{25} - 3\right )} e^{x} + 36 \, e^{50} - 216 \, e^{25} + 324}\right )} e^{25} + \frac {3 \, x}{e^{50} - 6 \, e^{25} + 9} - \frac {108 \, x^{2} {\left (e^{25} - 3\right )} + {\left (3 \, x^{2} + x\right )} e^{x} + 36 \, e^{25} - 108}{3 \, {\left (36 \, e^{25} + e^{x} - 108\right )}} - \frac {3 \, \log \left (36 \, e^{25} + e^{x} - 108\right )}{e^{50} - 6 \, e^{25} + 9} + \frac {108}{{\left (e^{25} - 3\right )} e^{x} + 36 \, e^{50} - 216 \, e^{25} + 324} + \frac {1}{3} \, \log \left (36 \, e^{25} + e^{x} - 108\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.19, size = 23, normalized size = 0.88 \begin {gather*} \frac {12\,x\,\left ({\mathrm {e}}^{25}-3\right )}{36\,{\mathrm {e}}^{25}+{\mathrm {e}}^x-108}-x^2 \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.14, size = 22, normalized size = 0.85 \begin {gather*} - x^{2} + \frac {- 36 x + 12 x e^{25}}{e^{x} - 108 + 36 e^{25}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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