Optimal. Leaf size=28 \[ 9 x^2 \left (\frac {5}{4}-\frac {\log (4)+\log \left (x \left (x-e^4 x\right )\right )}{x}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 33, normalized size of antiderivative = 1.18, number of steps
used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {12, 2503, 2332}
\begin {gather*} \frac {45 x^2}{4}-9 x \log \left (\left (1-e^4\right ) x^2\right )+18 x-9 x (2+\log (4)) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 2332
Rule 2503
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \left (-36+45 x-18 \log (4)-18 \log \left (x^2-e^4 x^2\right )\right ) \, dx\\ &=\frac {45 x^2}{4}-9 x (2+\log (4))-9 \int \log \left (x^2-e^4 x^2\right ) \, dx\\ &=\frac {45 x^2}{4}-9 x (2+\log (4))-9 \int \log \left (\left (1-e^4\right ) x^2\right ) \, dx\\ &=18 x+\frac {45 x^2}{4}-9 x (2+\log (4))-9 x \log \left (\left (1-e^4\right ) x^2\right )\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.01, size = 28, normalized size = 1.00 \begin {gather*} \frac {45 x^2}{4}-9 x \log (4)-9 x \log \left (\left (1-e^4\right ) x^2\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.64, size = 27, normalized size = 0.96
method | result | size |
default | \(\frac {45 x^{2}}{4}-18 x \ln \left (2\right )-9 x \ln \left (-x^{2} {\mathrm e}^{4}+x^{2}\right )\) | \(27\) |
norman | \(\frac {45 x^{2}}{4}-18 x \ln \left (2\right )-9 x \ln \left (-x^{2} {\mathrm e}^{4}+x^{2}\right )\) | \(27\) |
risch | \(\frac {45 x^{2}}{4}-18 x \ln \left (2\right )-9 x \ln \left (-x^{2} {\mathrm e}^{4}+x^{2}\right )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.25, size = 26, normalized size = 0.93 \begin {gather*} \frac {45}{4} \, x^{2} - 18 \, x \log \left (2\right ) - 9 \, x \log \left (-x^{2} e^{4} + x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 26, normalized size = 0.93 \begin {gather*} \frac {45}{4} \, x^{2} - 18 \, x \log \left (2\right ) - 9 \, x \log \left (-x^{2} e^{4} + x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 27, normalized size = 0.96 \begin {gather*} \frac {45 x^{2}}{4} - 9 x \log {\left (- x^{2} e^{4} + x^{2} \right )} - 18 x \log {\left (2 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 26, normalized size = 0.93 \begin {gather*} \frac {45}{4} \, x^{2} - 18 \, x \log \left (2\right ) - 9 \, x \log \left (-x^{2} e^{4} + x^{2}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.01, size = 23, normalized size = 0.82 \begin {gather*} -\frac {9\,x\,\left (4\,\ln \left (x^2-x^2\,{\mathrm {e}}^4\right )-5\,x+\ln \left (256\right )\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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