Optimal. Leaf size=32 \[ \frac {1}{5} \left (-2+\frac {\left (3-e^5\right ) x^2}{5 \log (7) \left (1+\log \left (x^2\right )\right )}\right ) \]
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Rubi [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 0.20, antiderivative size = 134, normalized size of antiderivative = 4.19, number of steps
used = 10, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {6, 12, 6820,
2343, 2347, 2209, 2413, 6617} \begin {gather*} \frac {\left (3-e^5\right ) \log \left (x^2\right ) \text {ExpIntegralEi}\left (\log \left (x^2\right )+1\right )}{25 e \log (7)}-\frac {\left (3-e^5\right ) \left (\log \left (x^2\right )+1\right ) \text {ExpIntegralEi}\left (\log \left (x^2\right )+1\right )}{25 e \log (7)}+\frac {\left (3-e^5\right ) \text {ExpIntegralEi}\left (\log \left (x^2\right )+1\right )}{25 e \log (7)}-\frac {\left (3-e^5\right ) x^2 \log \left (x^2\right )}{25 \log (7) \left (\log \left (x^2\right )+1\right )}+\frac {\left (3-e^5\right ) x^2}{25 \log (7)} \end {gather*}
Antiderivative was successfully verified.
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Rule 6
Rule 12
Rule 2209
Rule 2343
Rule 2347
Rule 2413
Rule 6617
Rule 6820
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\left (6-2 e^5\right ) x \log \left (x^2\right )}{25 \log (7)+50 \log (7) \log \left (x^2\right )+25 \log (7) \log ^2\left (x^2\right )} \, dx\\ &=\left (2 \left (3-e^5\right )\right ) \int \frac {x \log \left (x^2\right )}{25 \log (7)+50 \log (7) \log \left (x^2\right )+25 \log (7) \log ^2\left (x^2\right )} \, dx\\ &=\left (2 \left (3-e^5\right )\right ) \int \frac {x \log \left (x^2\right )}{25 \log (7) \left (1+\log \left (x^2\right )\right )^2} \, dx\\ &=\frac {\left (2 \left (3-e^5\right )\right ) \int \frac {x \log \left (x^2\right )}{\left (1+\log \left (x^2\right )\right )^2} \, dx}{25 \log (7)}\\ &=\frac {\left (3-e^5\right ) \text {Ei}\left (1+\log \left (x^2\right )\right ) \log \left (x^2\right )}{25 e \log (7)}-\frac {\left (3-e^5\right ) x^2 \log \left (x^2\right )}{25 \log (7) \left (1+\log \left (x^2\right )\right )}-\frac {\left (4 \left (3-e^5\right )\right ) \int \left (\frac {\text {Ei}\left (1+\log \left (x^2\right )\right )}{2 e x}-\frac {x}{2 \left (1+\log \left (x^2\right )\right )}\right ) \, dx}{25 \log (7)}\\ &=\frac {\left (3-e^5\right ) \text {Ei}\left (1+\log \left (x^2\right )\right ) \log \left (x^2\right )}{25 e \log (7)}-\frac {\left (3-e^5\right ) x^2 \log \left (x^2\right )}{25 \log (7) \left (1+\log \left (x^2\right )\right )}+\frac {\left (2 \left (3-e^5\right )\right ) \int \frac {x}{1+\log \left (x^2\right )} \, dx}{25 \log (7)}-\frac {\left (2 \left (3-e^5\right )\right ) \int \frac {\text {Ei}\left (1+\log \left (x^2\right )\right )}{x} \, dx}{25 e \log (7)}\\ &=\frac {\left (3-e^5\right ) \text {Ei}\left (1+\log \left (x^2\right )\right ) \log \left (x^2\right )}{25 e \log (7)}-\frac {\left (3-e^5\right ) x^2 \log \left (x^2\right )}{25 \log (7) \left (1+\log \left (x^2\right )\right )}+\frac {\left (3-e^5\right ) \text {Subst}\left (\int \frac {e^x}{1+x} \, dx,x,\log \left (x^2\right )\right )}{25 \log (7)}-\frac {\left (3-e^5\right ) \text {Subst}\left (\int \text {Ei}(1+x) \, dx,x,\log \left (x^2\right )\right )}{25 e \log (7)}\\ &=\frac {\left (3-e^5\right ) x^2}{25 \log (7)}+\frac {\left (3-e^5\right ) \text {Ei}\left (1+\log \left (x^2\right )\right )}{25 e \log (7)}+\frac {\left (3-e^5\right ) \text {Ei}\left (1+\log \left (x^2\right )\right ) \log \left (x^2\right )}{25 e \log (7)}-\frac {\left (3-e^5\right ) x^2 \log \left (x^2\right )}{25 \log (7) \left (1+\log \left (x^2\right )\right )}-\frac {\left (3-e^5\right ) \text {Ei}\left (1+\log \left (x^2\right )\right ) \left (1+\log \left (x^2\right )\right )}{25 e \log (7)}\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.02, size = 24, normalized size = 0.75 \begin {gather*} -\frac {\left (-3+e^5\right ) x^2}{25 \log (7) \left (1+\log \left (x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.42, size = 22, normalized size = 0.69
method | result | size |
default | \(-\frac {\left ({\mathrm e}^{5}-3\right ) x^{2}}{25 \ln \left (7\right ) \left (\ln \left (x^{2}\right )+1\right )}\) | \(22\) |
norman | \(-\frac {\left ({\mathrm e}^{5}-3\right ) x^{2}}{25 \ln \left (7\right ) \left (\ln \left (x^{2}\right )+1\right )}\) | \(22\) |
risch | \(-\frac {\left ({\mathrm e}^{5}-3\right ) x^{2}}{25 \ln \left (7\right ) \left (\ln \left (x^{2}\right )+1\right )}\) | \(22\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.50, size = 20, normalized size = 0.62 \begin {gather*} -\frac {x^{2} {\left (e^{5} - 3\right )}}{25 \, {\left (2 \, \log \left (7\right ) \log \left (x\right ) + \log \left (7\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.41, size = 26, normalized size = 0.81 \begin {gather*} -\frac {x^{2} e^{5} - 3 \, x^{2}}{25 \, {\left (\log \left (7\right ) \log \left (x^{2}\right ) + \log \left (7\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.04, size = 26, normalized size = 0.81 \begin {gather*} \frac {- x^{2} e^{5} + 3 x^{2}}{25 \log {\left (7 \right )} \log {\left (x^{2} \right )} + 25 \log {\left (7 \right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 37, normalized size = 1.16 \begin {gather*} -\frac {x^{2} e^{5}}{25 \, {\left (\log \left (7\right ) \log \left (x^{2}\right ) + \log \left (7\right )\right )}} + \frac {3 \, x^{2}}{25 \, {\left (\log \left (7\right ) \log \left (x^{2}\right ) + \log \left (7\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.45, size = 21, normalized size = 0.66 \begin {gather*} -\frac {x^2\,\left ({\mathrm {e}}^5-3\right )}{25\,\ln \left (7\right )\,\left (\ln \left (x^2\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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