Optimal. Leaf size=30 \[ \frac {70}{9}-x \left (4+e^{e^x}+\frac {x}{2 \left (-e^2+x\right )}\right )^2 \]
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Rubi [F]
time = 4.45, 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 {-64 e^6+224 e^4 x-243 e^2 x^2+81 x^3+e^{2 e^x} \left (-4 e^6+12 e^4 x-12 e^2 x^2+4 x^3+e^x \left (-8 e^6 x+24 e^4 x^2-24 e^2 x^3+8 x^4\right )\right )+e^{e^x} \left (-32 e^6+104 e^4 x-108 e^2 x^2+36 x^3+e^x \left (-32 e^6 x+100 e^4 x^2-104 e^2 x^3+36 x^4\right )\right )}{4 e^6-12 e^4 x+12 e^2 x^2-4 x^3} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right ) \left (-8 e^4-2 e^{4+e^x}+19 e^2 x+4 e^{2+e^x} x-4 e^{4+e^x+x} x-9 x^2-2 e^{e^x} x^2+8 e^{2+e^x+x} x^2-4 e^{e^x+x} x^3\right )}{4 \left (e^2-x\right )^3} \, dx\\ &=\frac {1}{4} \int \frac {\left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right ) \left (-8 e^4-2 e^{4+e^x}+19 e^2 x+4 e^{2+e^x} x-4 e^{4+e^x+x} x-9 x^2-2 e^{e^x} x^2+8 e^{2+e^x+x} x^2-4 e^{e^x+x} x^3\right )}{\left (e^2-x\right )^3} \, dx\\ &=\frac {1}{4} \int \left (-\frac {8 e^4 \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3}-\frac {2 e^{4+e^x} \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3}+\frac {19 e^2 x \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3}+\frac {4 e^{2+e^x} x \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3}-\frac {4 e^{e^x+x} x \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{e^2-x}-\frac {9 x^2 \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3}-\frac {2 e^{e^x} x^2 \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{4+e^x} \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3} \, dx\right )-\frac {1}{2} \int \frac {e^{e^x} x^2 \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3} \, dx-\frac {9}{4} \int \frac {x^2 \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3} \, dx+\frac {1}{4} \left (19 e^2\right ) \int \frac {x \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3} \, dx-\left (2 e^4\right ) \int \frac {8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x}{\left (e^2-x\right )^3} \, dx+\int \frac {e^{2+e^x} x \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{\left (e^2-x\right )^3} \, dx-\int \frac {e^{e^x+x} x \left (8 e^2+2 e^{2+e^x}-9 x-2 e^{e^x} x\right )}{e^2-x} \, dx\\ &=-\left (\frac {1}{2} \int \left (\frac {e^{4+e^x} \left (8 e^2-9 x\right )}{\left (e^2-x\right )^3}+\frac {2 e^{4+2 e^x}}{\left (e^2-x\right )^2}\right ) \, dx\right )-\frac {1}{2} \int \left (\frac {e^{e^x} \left (8 e^2-9 x\right ) x^2}{\left (e^2-x\right )^3}+\frac {2 e^{2 e^x} x^2}{\left (e^2-x\right )^2}\right ) \, dx-\frac {9}{4} \int \left (\frac {\left (8 e^2-9 x\right ) x^2}{\left (e^2-x\right )^3}+\frac {2 e^{e^x} x^2}{\left (e^2-x\right )^2}\right ) \, dx+\frac {1}{4} \left (19 e^2\right ) \int \left (\frac {\left (8 e^2-9 x\right ) x}{\left (e^2-x\right )^3}+\frac {2 e^{e^x} x}{\left (e^2-x\right )^2}\right ) \, dx-\left (2 e^4\right ) \int \left (\frac {8 e^2-9 x}{\left (e^2-x\right )^3}+\frac {2 e^{e^x}}{\left (e^2-x\right )^2}\right ) \, dx+\int \left (\frac {e^{2+e^x} \left (8 e^2-9 x\right ) x}{\left (e^2-x\right )^3}+\frac {2 e^{2+2 e^x} x}{\left (e^2-x\right )^2}\right ) \, dx-\int \left (2 e^{2 e^x+x} x+\frac {e^{e^x+x} \left (8 e^2-9 x\right ) x}{e^2-x}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{4+e^x} \left (8 e^2-9 x\right )}{\left (e^2-x\right )^3} \, dx\right )-\frac {1}{2} \int \frac {e^{e^x} \left (8 e^2-9 x\right ) x^2}{\left (e^2-x\right )^3} \, dx-2 \int e^{2 e^x+x} x \, dx+2 \int \frac {e^{2+2 e^x} x}{\left (e^2-x\right )^2} \, dx-\frac {9}{4} \int \frac {\left (8 e^2-9 x\right ) x^2}{\left (e^2-x\right )^3} \, dx-\frac {9}{2} \int \frac {e^{e^x} x^2}{\left (e^2-x\right )^2} \, dx+\frac {1}{4} \left (19 e^2\right ) \int \frac {\left (8 e^2-9 x\right ) x}{\left (e^2-x\right )^3} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{e^x} x}{\left (e^2-x\right )^2} \, dx-\left (2 e^4\right ) \int \frac {8 e^2-9 x}{\left (e^2-x\right )^3} \, dx-\left (4 e^4\right ) \int \frac {e^{e^x}}{\left (e^2-x\right )^2} \, dx-\int \frac {e^{4+2 e^x}}{\left (e^2-x\right )^2} \, dx+\int \frac {e^{2+e^x} \left (8 e^2-9 x\right ) x}{\left (e^2-x\right )^3} \, dx-\int \frac {e^{e^x+x} \left (8 e^2-9 x\right ) x}{e^2-x} \, dx-\int \frac {e^{2 e^x} x^2}{\left (e^2-x\right )^2} \, dx\\ &=\frac {e^2 \left (8 e^2-9 x\right )^2}{\left (e^2-x\right )^2}-\frac {1}{2} \int \left (-\frac {e^{6+e^x}}{\left (e^2-x\right )^3}+\frac {9 e^{4+e^x}}{\left (e^2-x\right )^2}\right ) \, dx-\frac {1}{2} \int \left (9 e^{e^x}-\frac {e^{6+e^x}}{\left (e^2-x\right )^3}+\frac {11 e^{4+e^x}}{\left (e^2-x\right )^2}-\frac {19 e^{2+e^x}}{e^2-x}\right ) \, dx-2 \int e^{2 e^x+x} x \, dx+2 \int \frac {e^{2 \left (1+e^x\right )} x}{\left (e^2-x\right )^2} \, dx-\frac {9}{4} \int \left (9-\frac {e^6}{\left (e^2-x\right )^3}+\frac {11 e^4}{\left (e^2-x\right )^2}-\frac {19 e^2}{e^2-x}\right ) \, dx-\frac {9}{2} \int \left (e^{e^x}+\frac {e^{4+e^x}}{\left (e^2-x\right )^2}-\frac {2 e^{2+e^x}}{e^2-x}\right ) \, dx+\frac {1}{4} \left (19 e^2\right ) \int \left (-\frac {e^4}{\left (e^2-x\right )^3}+\frac {10 e^2}{\left (e^2-x\right )^2}-\frac {9}{e^2-x}\right ) \, dx+\frac {1}{2} \left (19 e^2\right ) \int \left (\frac {e^{2+e^x}}{\left (e^2-x\right )^2}+\frac {e^{e^x}}{-e^2+x}\right ) \, dx-\left (4 e^4\right ) \int \frac {e^{e^x}}{\left (e^2-x\right )^2} \, dx+\int \left (-\frac {e^{6+e^x}}{\left (e^2-x\right )^3}+\frac {10 e^{4+e^x}}{\left (e^2-x\right )^2}-\frac {9 e^{2+e^x}}{e^2-x}\right ) \, dx-\int \left (e^{2 e^x}+\frac {e^{4+2 e^x}}{\left (e^2-x\right )^2}-\frac {2 e^{2+2 e^x}}{e^2-x}\right ) \, dx-\int \frac {e^{2 \left (2+e^x\right )}}{\left (e^2-x\right )^2} \, dx-\int \left (e^{2+e^x+x}-\frac {e^{4+e^x+x}}{e^2-x}+9 e^{e^x+x} x\right ) \, dx\\ &=-\frac {5 e^6}{4 \left (e^2-x\right )^2}+\frac {e^2 \left (8 e^2-9 x\right )^2}{\left (e^2-x\right )^2}+\frac {91 e^4}{4 \left (e^2-x\right )}-\frac {81 x}{4}+2 \left (\frac {1}{2} \int \frac {e^{6+e^x}}{\left (e^2-x\right )^3} \, dx\right )+2 \int \frac {e^{2+2 e^x}}{e^2-x} \, dx-2 \int e^{2 e^x+x} x \, dx+2 \int \left (\frac {e^{2+2 \left (1+e^x\right )}}{\left (e^2-x\right )^2}+\frac {e^{2 \left (1+e^x\right )}}{-e^2+x}\right ) \, dx-2 \left (\frac {9}{2} \int e^{e^x} \, dx\right )-2 \left (\frac {9}{2} \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx\right )-\frac {11}{2} \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx-9 \int e^{e^x+x} x \, dx+\frac {19}{2} \int \frac {e^{2+e^x}}{e^2-x} \, dx+10 \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{2+e^x}}{\left (e^2-x\right )^2} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{e^x}}{-e^2+x} \, dx-\left (4 e^4\right ) \int \frac {e^{e^x}}{\left (e^2-x\right )^2} \, dx-\int e^{2 e^x} \, dx-\int e^{2+e^x+x} \, dx-\int \frac {e^{6+e^x}}{\left (e^2-x\right )^3} \, dx-\int \frac {e^{2 \left (2+e^x\right )}}{\left (e^2-x\right )^2} \, dx-\int \frac {e^{4+2 e^x}}{\left (e^2-x\right )^2} \, dx+\int \frac {e^{4+e^x+x}}{e^2-x} \, dx\\ &=-\frac {5 e^6}{4 \left (e^2-x\right )^2}+\frac {e^2 \left (8 e^2-9 x\right )^2}{\left (e^2-x\right )^2}+\frac {91 e^4}{4 \left (e^2-x\right )}-\frac {81 x}{4}+2 \left (\frac {1}{2} \int \frac {e^{6+e^x}}{\left (e^2-x\right )^3} \, dx\right )+2 \int \frac {e^{2+2 \left (1+e^x\right )}}{\left (e^2-x\right )^2} \, dx+2 \int \frac {e^{2 \left (1+e^x\right )}}{e^2-x} \, dx-2 \int e^{2 e^x+x} x \, dx+2 \int \frac {e^{2 \left (1+e^x\right )}}{-e^2+x} \, dx-2 \left (\frac {9}{2} \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx\right )-2 \left (\frac {9}{2} \text {Subst}\left (\int \frac {e^x}{x} \, dx,x,e^x\right )\right )-\frac {11}{2} \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx-9 \int e^{e^x+x} x \, dx+\frac {19}{2} \int \frac {e^{2+e^x}}{e^2-x} \, dx+10 \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{2+e^x}}{\left (e^2-x\right )^2} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{e^x}}{-e^2+x} \, dx-\left (4 e^4\right ) \int \frac {e^{e^x}}{\left (e^2-x\right )^2} \, dx-\int \frac {e^{6+e^x}}{\left (e^2-x\right )^3} \, dx-2 \int \frac {e^{2 \left (2+e^x\right )}}{\left (e^2-x\right )^2} \, dx+\int \frac {e^{4+e^x+x}}{e^2-x} \, dx-\text {Subst}\left (\int e^{2+x} \, dx,x,e^x\right )-\text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,e^x\right )\\ &=-e^{2+e^x}-\frac {5 e^6}{4 \left (e^2-x\right )^2}+\frac {e^2 \left (8 e^2-9 x\right )^2}{\left (e^2-x\right )^2}+\frac {91 e^4}{4 \left (e^2-x\right )}-\frac {81 x}{4}-9 \text {Ei}\left (e^x\right )-\text {Ei}\left (2 e^x\right )+2 \left (\frac {1}{2} \int \frac {e^{6+e^x}}{\left (e^2-x\right )^3} \, dx\right )+2 \int \frac {e^{2 \left (2+e^x\right )}}{\left (e^2-x\right )^2} \, dx+2 \int \frac {e^{2 \left (1+e^x\right )}}{e^2-x} \, dx-2 \int e^{2 e^x+x} x \, dx+2 \int \frac {e^{2 \left (1+e^x\right )}}{-e^2+x} \, dx-2 \left (\frac {9}{2} \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx\right )-\frac {11}{2} \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx-9 \int e^{e^x+x} x \, dx+\frac {19}{2} \int \frac {e^{2+e^x}}{e^2-x} \, dx+10 \int \frac {e^{4+e^x}}{\left (e^2-x\right )^2} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{2+e^x}}{\left (e^2-x\right )^2} \, dx+\frac {1}{2} \left (19 e^2\right ) \int \frac {e^{e^x}}{-e^2+x} \, dx-\left (4 e^4\right ) \int \frac {e^{e^x}}{\left (e^2-x\right )^2} \, dx-\int \frac {e^{6+e^x}}{\left (e^2-x\right )^3} \, dx-2 \int \frac {e^{2 \left (2+e^x\right )}}{\left (e^2-x\right )^2} \, dx+\int \frac {e^{4+e^x+x}}{e^2-x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(72\) vs. \(2(30)=60\).
time = 0.13, size = 72, normalized size = 2.40 \begin {gather*} \frac {1}{4} \left (-\frac {e^6}{\left (e^2-x\right )^2}-81 x-4 e^{2 e^x} x-\frac {4 e^{e^x} \left (8 e^2-9 x\right ) x}{e^2-x}-\frac {19 e^4}{-e^2+x}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(58\) vs.
\(2(23)=46\).
time = 1.46, size = 59, normalized size = 1.97
method | result | size |
risch | \(-\frac {81 x}{4}+\frac {\frac {9 \,{\mathrm e}^{6}}{2}-\frac {19 x \,{\mathrm e}^{4}}{4}}{{\mathrm e}^{4}-2 \,{\mathrm e}^{2} x +x^{2}}-{\mathrm e}^{2 \,{\mathrm e}^{x}} x -\frac {\left (8 \,{\mathrm e}^{2}-9 x \right ) x \,{\mathrm e}^{{\mathrm e}^{x}}}{{\mathrm e}^{2}-x}\) | \(59\) |
norman | \(\frac {56 x \,{\mathrm e}^{4}-\frac {81 x^{3}}{4}-9 x^{3} {\mathrm e}^{{\mathrm e}^{x}}-{\mathrm e}^{2 \,{\mathrm e}^{x}} x^{3}+17 \,{\mathrm e}^{2} {\mathrm e}^{{\mathrm e}^{x}} x^{2}+2 \,{\mathrm e}^{2} {\mathrm e}^{2 \,{\mathrm e}^{x}} x^{2}-8 \,{\mathrm e}^{4} {\mathrm e}^{{\mathrm e}^{x}} x -{\mathrm e}^{4} {\mathrm e}^{2 \,{\mathrm e}^{x}} x -36 \,{\mathrm e}^{6}}{\left ({\mathrm e}^{2}-x \right )^{2}}\) | \(91\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 161 vs.
\(2 (24) = 48\).
time = 0.33, size = 161, normalized size = 5.37 \begin {gather*} -\frac {243}{8} \, {\left (\frac {4 \, x e^{2} - 3 \, e^{4}}{x^{2} - 2 \, x e^{2} + e^{4}} - 2 \, \log \left (x - e^{2}\right )\right )} e^{2} - \frac {243}{4} \, e^{2} \log \left (x - e^{2}\right ) - \frac {81}{4} \, x + \frac {28 \, {\left (2 \, x - e^{2}\right )} e^{4}}{x^{2} - 2 \, x e^{2} + e^{4}} + \frac {81 \, {\left (6 \, x e^{4} - 5 \, e^{6}\right )}}{8 \, {\left (x^{2} - 2 \, x e^{2} + e^{4}\right )}} - \frac {{\left (x^{2} - x e^{2}\right )} e^{\left (2 \, e^{x}\right )} + {\left (9 \, x^{2} - 8 \, x e^{2}\right )} e^{\left (e^{x}\right )}}{x - e^{2}} - \frac {8 \, e^{6}}{x^{2} - 2 \, x e^{2} + e^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 82 vs.
\(2 (24) = 48\).
time = 0.41, size = 82, normalized size = 2.73 \begin {gather*} -\frac {81 \, x^{3} - 162 \, x^{2} e^{2} + 100 \, x e^{4} + 4 \, {\left (x^{3} - 2 \, x^{2} e^{2} + x e^{4}\right )} e^{\left (2 \, e^{x}\right )} + 4 \, {\left (9 \, x^{3} - 17 \, x^{2} e^{2} + 8 \, x e^{4}\right )} e^{\left (e^{x}\right )} - 18 \, e^{6}}{4 \, {\left (x^{2} - 2 \, x e^{2} + e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs.
\(2 (22) = 44\).
time = 0.23, size = 70, normalized size = 2.33 \begin {gather*} - \frac {81 x}{4} - \frac {19 x e^{4} - 18 e^{6}}{4 x^{2} - 8 x e^{2} + 4 e^{4}} + \frac {\left (- 9 x^{2} + 8 x e^{2}\right ) e^{e^{x}} + \left (- x^{2} + x e^{2}\right ) e^{2 e^{x}}}{x - e^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs.
\(2 (24) = 48\).
time = 0.42, size = 95, normalized size = 3.17 \begin {gather*} -\frac {4 \, x^{3} e^{\left (2 \, e^{x}\right )} + 36 \, x^{3} e^{\left (e^{x}\right )} + 81 \, x^{3} - 162 \, x^{2} e^{2} - 8 \, x^{2} e^{\left (2 \, e^{x} + 2\right )} - 68 \, x^{2} e^{\left (e^{x} + 2\right )} + 100 \, x e^{4} + 4 \, x e^{\left (2 \, e^{x} + 4\right )} + 32 \, x e^{\left (e^{x} + 4\right )} - 18 \, e^{6}}{4 \, {\left (x^{2} - 2 \, x e^{2} + e^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.59, size = 63, normalized size = 2.10 \begin {gather*} \frac {18\,{\mathrm {e}}^6-19\,x\,{\mathrm {e}}^4}{4\,x^2-8\,{\mathrm {e}}^2\,x+4\,{\mathrm {e}}^4}-x\,{\mathrm {e}}^{2\,{\mathrm {e}}^x}-\frac {81\,x}{4}+\frac {{\mathrm {e}}^{{\mathrm {e}}^x}\,\left (8\,x\,{\mathrm {e}}^2-9\,x^2\right )}{x-{\mathrm {e}}^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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