Optimal. Leaf size=22 \[ 5+x-x^2+\frac {x \left (5+x^2\right )}{e^7 \log (x)} \]
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Rubi [A]
time = 0.17, antiderivative size = 33, normalized size of antiderivative = 1.50, number of steps
used = 15, number of rules used = 8, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {12, 6874,
2367, 2334, 2335, 2343, 2346, 2209} \begin {gather*} \frac {x^3}{e^7 \log (x)}-\frac {1}{4} (1-2 x)^2+\frac {5 x}{e^7 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2209
Rule 2334
Rule 2335
Rule 2343
Rule 2346
Rule 2367
Rule 6874
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {-5-x^2+\left (5+3 x^2\right ) \log (x)+e^7 (1-2 x) \log ^2(x)}{\log ^2(x)} \, dx}{e^7}\\ &=\frac {\int \left (-e^7 (-1+2 x)+\frac {-5-x^2}{\log ^2(x)}+\frac {5+3 x^2}{\log (x)}\right ) \, dx}{e^7}\\ &=-\frac {1}{4} (1-2 x)^2+\frac {\int \frac {-5-x^2}{\log ^2(x)} \, dx}{e^7}+\frac {\int \frac {5+3 x^2}{\log (x)} \, dx}{e^7}\\ &=-\frac {1}{4} (1-2 x)^2+\frac {\int \left (-\frac {5}{\log ^2(x)}-\frac {x^2}{\log ^2(x)}\right ) \, dx}{e^7}+\frac {\int \left (\frac {5}{\log (x)}+\frac {3 x^2}{\log (x)}\right ) \, dx}{e^7}\\ &=-\frac {1}{4} (1-2 x)^2-\frac {\int \frac {x^2}{\log ^2(x)} \, dx}{e^7}+\frac {3 \int \frac {x^2}{\log (x)} \, dx}{e^7}-\frac {5 \int \frac {1}{\log ^2(x)} \, dx}{e^7}+\frac {5 \int \frac {1}{\log (x)} \, dx}{e^7}\\ &=-\frac {1}{4} (1-2 x)^2+\frac {5 x}{e^7 \log (x)}+\frac {x^3}{e^7 \log (x)}+\frac {5 \text {li}(x)}{e^7}-\frac {3 \int \frac {x^2}{\log (x)} \, dx}{e^7}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )}{e^7}-\frac {5 \int \frac {1}{\log (x)} \, dx}{e^7}\\ &=-\frac {1}{4} (1-2 x)^2+\frac {3 \text {Ei}(3 \log (x))}{e^7}+\frac {5 x}{e^7 \log (x)}+\frac {x^3}{e^7 \log (x)}-\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )}{e^7}\\ &=-\frac {1}{4} (1-2 x)^2+\frac {5 x}{e^7 \log (x)}+\frac {x^3}{e^7 \log (x)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.05, size = 24, normalized size = 1.09 \begin {gather*} \frac {x \left (5+x^2-e^7 (-1+x) \log (x)\right )}{e^7 \log (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 33, normalized size = 1.50
method | result | size |
risch | \(-x \left (x -1\right )+\frac {\left (x^{2}+5\right ) x \,{\mathrm e}^{-7}}{\ln \left (x \right )}\) | \(21\) |
default | \({\mathrm e}^{-7} \left (-x^{2} {\mathrm e}^{7}+x \,{\mathrm e}^{7}+\frac {x^{3}}{\ln \left (x \right )}+\frac {5 x}{\ln \left (x \right )}\right )\) | \(33\) |
norman | \(\frac {x \ln \left (x \right )+{\mathrm e}^{-7} x^{3}-x^{2} \ln \left (x \right )+5 \,{\mathrm e}^{-7} x}{\ln \left (x \right )}\) | \(33\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order
3.
time = 0.29, size = 44, normalized size = 2.00 \begin {gather*} -{\left (x^{2} e^{7} - x e^{7} - 3 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - 5 \, {\rm Ei}\left (\log \left (x\right )\right ) + 5 \, \Gamma \left (-1, -\log \left (x\right )\right ) + 3 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right )\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 27, normalized size = 1.23 \begin {gather*} \frac {{\left (x^{3} - {\left (x^{2} - x\right )} e^{7} \log \left (x\right ) + 5 \, x\right )} e^{\left (-7\right )}}{\log \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.03, size = 17, normalized size = 0.77 \begin {gather*} - x^{2} + x + \frac {x^{3} + 5 x}{e^{7} \log {\left (x \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 32, normalized size = 1.45 \begin {gather*} -{\left (x^{2} e^{7} - x e^{7} - \frac {x^{3}}{\log \left (x\right )} - \frac {5 \, x}{\log \left (x\right )}\right )} e^{\left (-7\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.56, size = 26, normalized size = 1.18 \begin {gather*} x\,{\mathrm {e}}^{-7}\,\left ({\mathrm {e}}^7-x\,{\mathrm {e}}^7\right )+\frac {x\,{\mathrm {e}}^{-7}\,\left (x^2+5\right )}{\ln \left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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