Optimal. Leaf size=19 \[ \left (1+e^{-\frac {1}{2} x \left (\frac {625}{x^8}+\log (x)\right )}\right ) x \]
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Rubi [B] time = 1.45, antiderivative size = 67, normalized size of antiderivative = 3.53, number of steps used = 6, number of rules used = 5, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 6742, 6688, 8, 2288} \begin {gather*} x-\frac {e^{-\frac {625}{2 x^7}} x^{-\frac {x}{2}-7} \left (-x^8+x^8 (-\log (x))+4375\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (x^8 \log (x)+625\right )}{x^8}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 2288
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7+2 e^{\frac {625+x^8 \log (x)}{2 x^7}} x^7-x^8-x^8 \log (x)\right )}{x^7} \, dx\\ &=\frac {1}{2} \int \left (2 e^{\frac {625}{2 x^7}-\frac {625+x^8 \log (x)}{2 x^7}} x^{x/2}+\frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7-x^8-x^8 \log (x)\right )}{x^7}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{-\frac {625+x^8 \log (x)}{2 x^7}} \left (4375+2 x^7-x^8-x^8 \log (x)\right )}{x^7} \, dx+\int e^{\frac {625}{2 x^7}-\frac {625+x^8 \log (x)}{2 x^7}} x^{x/2} \, dx\\ &=-\frac {e^{-\frac {625}{2 x^7}} x^{-7-\frac {x}{2}} \left (4375-x^8-x^8 \log (x)\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (625+x^8 \log (x)\right )}{x^8}}+\int 1 \, dx\\ &=x-\frac {e^{-\frac {625}{2 x^7}} x^{-7-\frac {x}{2}} \left (4375-x^8-x^8 \log (x)\right )}{\frac {x^7+8 x^7 \log (x)}{x^7}-\frac {7 \left (625+x^8 \log (x)\right )}{x^8}}\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.35, size = 28, normalized size = 1.47 \begin {gather*} \frac {1}{2} \left (2 x+2 e^{-\frac {625}{2 x^7}} x^{1-\frac {x}{2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.56, size = 33, normalized size = 1.74 \begin {gather*} {\left (x e^{\left (\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )} + x\right )} e^{\left (-\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\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*} \int -\frac {{\left (x^{8} \log \relax (x) + x^{8} - 2 \, x^{7} e^{\left (\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )} - 2 \, x^{7} - 4375\right )} e^{\left (-\frac {x^{8} \log \relax (x) + 625}{2 \, x^{7}}\right )}}{2 \, x^{7}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 18, normalized size = 0.95
method | result | size |
risch | \(x +x \,x^{-\frac {x}{2}} {\mathrm e}^{-\frac {625}{2 x^{7}}}\) | \(18\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 16, normalized size = 0.84 \begin {gather*} x e^{\left (-\frac {1}{2} \, x \log \relax (x) - \frac {625}{2 \, x^{7}}\right )} + x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.34, size = 16, normalized size = 0.84 \begin {gather*} x\,\left ({\mathrm {e}}^{-\frac {x\,\ln \relax (x)}{2}-\frac {625}{2\,x^7}}+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.40, size = 19, normalized size = 1.00 \begin {gather*} x + x e^{- \frac {\frac {x^{8} \log {\relax (x )}}{2} + \frac {625}{2}}{x^{7}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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