Optimal. Leaf size=20 \[ e^{e^x}+\frac {x^2}{400 \left (e^x+x\right )^2} \]
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Rubi [A] time = 0.46, antiderivative size = 21, normalized size of antiderivative = 1.05, number of steps used = 9, number of rules used = 8, integrand size = 84, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6741, 12, 6688, 6742, 2282, 2194, 6712, 32} \begin {gather*} e^{e^x}+\frac {1}{400 \left (\frac {e^x}{x}+1\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 2194
Rule 2282
Rule 6688
Rule 6712
Rule 6741
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^x \left (x-x^2\right )+e^{e^x} \left (200 e^{4 x}+600 e^{3 x} x+600 e^{2 x} x^2+200 e^x x^3\right )}{200 \left (e^x+x\right )^3} \, dx\\ &=\frac {1}{200} \int \frac {e^x \left (x-x^2\right )+e^{e^x} \left (200 e^{4 x}+600 e^{3 x} x+600 e^{2 x} x^2+200 e^x x^3\right )}{\left (e^x+x\right )^3} \, dx\\ &=\frac {1}{200} \int e^x \left (200 e^{e^x}-\frac {(-1+x) x}{\left (e^x+x\right )^3}\right ) \, dx\\ &=\frac {1}{200} \int \left (200 e^{e^x+x}-\frac {e^x (-1+x) x}{\left (e^x+x\right )^3}\right ) \, dx\\ &=-\left (\frac {1}{200} \int \frac {e^x (-1+x) x}{\left (e^x+x\right )^3} \, dx\right )+\int e^{e^x+x} \, dx\\ &=-\left (\frac {1}{200} \operatorname {Subst}\left (\int \frac {1}{(1+x)^3} \, dx,x,\frac {e^x}{x}\right )\right )+\operatorname {Subst}\left (\int e^x \, dx,x,e^x\right )\\ &=e^{e^x}+\frac {1}{400 \left (1+\frac {e^x}{x}\right )^2}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 1.10, size = 26, normalized size = 1.30 \begin {gather*} \frac {1}{200} \left (200 e^{e^x}+\frac {x^2}{2 \left (e^x+x\right )^2}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.61, size = 39, normalized size = 1.95 \begin {gather*} \frac {x^{2} + 400 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (e^{x}\right )}}{400 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 58, normalized size = 2.90 \begin {gather*} \frac {400 \, x^{2} e^{\left (x + e^{x}\right )} + x^{2} e^{x} + 800 \, x e^{\left (2 \, x + e^{x}\right )} + 400 \, e^{\left (3 \, x + e^{x}\right )}}{400 \, {\left (x^{2} e^{x} + 2 \, x e^{\left (2 \, x\right )} + e^{\left (3 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 16, normalized size = 0.80
method | result | size |
risch | \(\frac {x^{2}}{400 \left ({\mathrm e}^{x}+x \right )^{2}}+{\mathrm e}^{{\mathrm e}^{x}}\) | \(16\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.40, size = 39, normalized size = 1.95 \begin {gather*} \frac {x^{2} + 400 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )} e^{\left (e^{x}\right )}}{400 \, {\left (x^{2} + 2 \, x e^{x} + e^{\left (2 \, x\right )}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 27, normalized size = 1.35 \begin {gather*} {\mathrm {e}}^{{\mathrm {e}}^x}+\frac {x^2}{400\,\left ({\mathrm {e}}^{2\,x}+2\,x\,{\mathrm {e}}^x+x^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.22, size = 26, normalized size = 1.30 \begin {gather*} \frac {x^{2}}{400 x^{2} + 800 x e^{x} + 400 e^{2 x}} + e^{e^{x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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