Optimal. Leaf size=32 \[ 2 e^{\frac {x^2}{\left (e^5+\frac {8 e^3}{x}\right ) \left (e^x+x^2\right )}} x \]
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Rubi [F] time = 177.90, 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 {\exp \left (\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}\right ) \left (128 e^6 x^4+16 e^3 x^5+32 e^8 x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\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 {\exp \left (\frac {x^3}{8 e^3 x^2+e^5 x^3+e^x \left (8 e^3+e^5 x\right )}\right ) \left (128 e^6 x^4+\left (16 e^3+32 e^8\right ) x^5+2 e^{10} x^6+e^{2 x} \left (128 e^6+32 e^8 x+2 e^{10} x^2\right )+e^x \left (256 e^6 x^2+4 e^{10} x^4+e^3 \left (48 x^3-16 x^4\right )+e^5 \left (64 e^3 x^3+4 x^4-2 x^5\right )\right )\right )}{64 e^6 x^4+16 e^8 x^5+e^{10} x^6+e^{2 x} \left (64 e^6+16 e^8 x+e^{10} x^2\right )+e^x \left (128 e^6 x^2+32 e^8 x^3+2 e^{10} x^4\right )} \, dx\\ &=\int \frac {2 e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (64 e^{3+2 x}+16 e^{5+2 x} x+128 e^{3+x} x^2+e^{7+2 x} x^2+32 e^{5+x} x^3-8 e^x (-3+x) x^3+64 e^3 x^4+2 e^{7+x} x^4-e^{2+x} (-2+x) x^4+8 \left (1+2 e^5\right ) x^5+e^7 x^6\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )^2} \, dx\\ &=2 \int \frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (64 e^{3+2 x}+16 e^{5+2 x} x+128 e^{3+x} x^2+e^{7+2 x} x^2+32 e^{5+x} x^3-8 e^x (-3+x) x^3+64 e^3 x^4+2 e^{7+x} x^4-e^{2+x} (-2+x) x^4+8 \left (1+2 e^5\right ) x^5+e^7 x^6\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )^2} \, dx\\ &=2 \int \left (e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}+\frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} (-2+x) x^5}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2}+\frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3 \left (24-2 \left (4-e^2\right ) x-e^2 x^2\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )}\right ) \, dx\\ &=2 \int e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \, dx+2 \int \frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} (-2+x) x^5}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2} \, dx+2 \int \frac {e^{-3+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3 \left (24-2 \left (4-e^2\right ) x-e^2 x^2\right )}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )} \, dx\\ &=2 \int e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \, dx+2 \int \left (-\frac {8192 e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right )}{\left (e^x+x^2\right )^2}+\frac {1024 e^{-13+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x}{\left (e^x+x^2\right )^2}-\frac {128 e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^2}{\left (e^x+x^2\right )^2}+\frac {16 e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^3}{\left (e^x+x^2\right )^2}-\frac {2 e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^4}{\left (e^x+x^2\right )^2}+\frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^5}{\left (e^x+x^2\right )^2}+\frac {65536 e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right )}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2}\right ) \, dx+2 \int \left (\frac {512 e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{e^x+x^2}-\frac {8 e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (8+e^2\right ) x}{e^x+x^2}+\frac {2 e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (4+e^2\right ) x^2}{e^x+x^2}-\frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3}{e^x+x^2}-\frac {4096 e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )}+\frac {512 e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \left (-8+e^2\right )}{\left (8+e^2 x\right ) \left (e^x+x^2\right )}\right ) \, dx\\ &=2 \int e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} \, dx+2 \int \frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^5}{\left (e^x+x^2\right )^2} \, dx-2 \int \frac {e^{-5+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3}{e^x+x^2} \, dx+1024 \int \frac {e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{e^x+x^2} \, dx-8192 \int \frac {e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right )^2 \left (e^x+x^2\right )} \, dx-\left (1024 \left (8-e^2\right )\right ) \int \frac {e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right ) \left (e^x+x^2\right )} \, dx-\left (4 \left (4+e^2\right )\right ) \int \frac {e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^4}{\left (e^x+x^2\right )^2} \, dx+\left (4 \left (4+e^2\right )\right ) \int \frac {e^{-7+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^2}{e^x+x^2} \, dx+\left (32 \left (4+e^2\right )\right ) \int \frac {e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^3}{\left (e^x+x^2\right )^2} \, dx-\left (256 \left (4+e^2\right )\right ) \int \frac {e^{-11+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x^2}{\left (e^x+x^2\right )^2} \, dx+\left (2048 \left (4+e^2\right )\right ) \int \frac {e^{-13+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x}{\left (e^x+x^2\right )^2} \, dx-\left (16384 \left (4+e^2\right )\right ) \int \frac {e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (e^x+x^2\right )^2} \, dx+\left (131072 \left (4+e^2\right )\right ) \int \frac {e^{-15+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}}}{\left (8+e^2 x\right ) \left (e^x+x^2\right )^2} \, dx-\left (16 \left (8+e^2\right )\right ) \int \frac {e^{-9+\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x}{e^x+x^2} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.21, size = 30, normalized size = 0.94 \begin {gather*} 2 e^{\frac {x^3}{e^3 \left (8+e^2 x\right ) \left (e^x+x^2\right )}} x \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 36, normalized size = 1.12 \begin {gather*} 2 \, x e^{\left (\frac {x^{3}}{x^{3} e^{5} + 8 \, x^{2} e^{3} + {\left (x e^{5} + 8 \, e^{3}\right )} e^{x}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \mathit {sage}_{0} x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 28, normalized size = 0.88
method | result | size |
risch | \(2 x \,{\mathrm e}^{\frac {x^{3}}{\left (x^{2}+{\mathrm e}^{x}\right ) \left (x \,{\mathrm e}^{5}+8 \,{\mathrm e}^{3}\right )}}\) | \(28\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 2 \, \int \frac {{\left (x^{6} e^{10} + 16 \, x^{5} e^{8} + 8 \, x^{5} e^{3} + 64 \, x^{4} e^{6} + {\left (x^{2} e^{10} + 16 \, x e^{8} + 64 \, e^{6}\right )} e^{\left (2 \, x\right )} + {\left (2 \, x^{4} e^{10} + 128 \, x^{2} e^{6} - {\left (x^{5} - 2 \, x^{4} - 32 \, x^{3} e^{3}\right )} e^{5} - 8 \, {\left (x^{4} - 3 \, x^{3}\right )} e^{3}\right )} e^{x}\right )} e^{\left (\frac {x^{3}}{x^{3} e^{5} + 8 \, x^{2} e^{3} + {\left (x e^{5} + 8 \, e^{3}\right )} e^{x}}\right )}}{x^{6} e^{10} + 16 \, x^{5} e^{8} + 64 \, x^{4} e^{6} + {\left (x^{2} e^{10} + 16 \, x e^{8} + 64 \, e^{6}\right )} e^{\left (2 \, x\right )} + 2 \, {\left (x^{4} e^{10} + 16 \, x^{3} e^{8} + 64 \, x^{2} e^{6}\right )} e^{x}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 36, normalized size = 1.12 \begin {gather*} 2\,x\,{\mathrm {e}}^{\frac {x^3}{8\,x^2\,{\mathrm {e}}^3+x^3\,{\mathrm {e}}^5+8\,{\mathrm {e}}^3\,{\mathrm {e}}^x+x\,{\mathrm {e}}^5\,{\mathrm {e}}^x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.46, size = 36, normalized size = 1.12 \begin {gather*} 2 x e^{\frac {x^{3}}{x^{3} e^{5} + 8 x^{2} e^{3} + \left (x e^{5} + 8 e^{3}\right ) e^{x}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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