Optimal. Leaf size=23 \[ 5+x^2 \log (x) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \]
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Rubi [B] time = 3.50, antiderivative size = 62, normalized size of antiderivative = 2.70, number of steps used = 33, number of rules used = 12, integrand size = 95, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.126, Rules used = {6688, 14, 6742, 2554, 43, 2334, 12, 2303, 2551, 2304, 30, 2557} \begin {gather*} -\frac {x^3}{3}+\frac {1}{3} x^3 \log (x)+\frac {x^2}{2}-x^2 \log (x) \log \left (x^4-e^x\right )-\frac {1}{6} \left (3 x^2-2 x^3\right ) (2 \log (x)+1) \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 14
Rule 30
Rule 43
Rule 2303
Rule 2304
Rule 2334
Rule 2551
Rule 2554
Rule 2557
Rule 6688
Rule 6742
Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int x \left (-1+x-\log \left (-e^x+x^4\right )+\frac {\log (x) \left (2 e^x (-1+x)-3 (-2+x) x^4-2 \left (e^x-x^4\right ) \log \left (-e^x+x^4\right )\right )}{e^x-x^4}\right ) \, dx\\ &=\int \left (\frac {(-4+x) x^5 \log (x)}{-e^x+x^4}+x (1+2 \log (x)) \left (-1+x-\log \left (-e^x+x^4\right )\right )\right ) \, dx\\ &=\int \frac {(-4+x) x^5 \log (x)}{-e^x+x^4} \, dx+\int x (1+2 \log (x)) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \, dx\\ &=\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx+\int \left ((-1+x) x (1+2 \log (x))-x (1+2 \log (x)) \log \left (-e^x+x^4\right )\right ) \, dx-\int \frac {-4 \int \frac {x^5}{-e^x+x^4} \, dx+\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx+\int (-1+x) x (1+2 \log (x)) \, dx-\int x (1+2 \log (x)) \log \left (-e^x+x^4\right ) \, dx-\int \left (-\frac {4 \int \frac {x^5}{-e^x+x^4} \, dx}{x}+\frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x}\right ) \, dx\\ &=-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-2 \int \frac {1}{6} x (-3+2 x) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx-\int \left (x \log \left (-e^x+x^4\right )+2 x \log (x) \log \left (-e^x+x^4\right )\right ) \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-\frac {1}{3} \int x (-3+2 x) \, dx-2 \int x \log (x) \log \left (-e^x+x^4\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx-\int x \log \left (-e^x+x^4\right ) \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-\frac {1}{2} x^2 \log \left (-e^x+x^4\right )-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \left (-3 x+2 x^2\right ) \, dx+\frac {1}{2} \int \frac {x^2 \left (-e^x+4 x^3\right )}{-e^x+x^4} \, dx+2 \int \frac {x^2 \left (-e^x+4 x^3\right ) \log (x)}{2 \left (-e^x+x^4\right )} \, dx+2 \int \frac {1}{2} x \log \left (-e^x+x^4\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-\frac {1}{2} x^2 \log \left (-e^x+x^4\right )-x^2 \log (x) \log \left (-e^x+x^4\right )+\frac {1}{2} \int \left (x^2-\frac {(-4+x) x^5}{-e^x+x^4}\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx+\log (x) \int \frac {x^6}{-e^x+x^4} \, dx-(4 \log (x)) \int \frac {x^5}{-e^x+x^4} \, dx+\int \frac {x^2 \left (-e^x+4 x^3\right ) \log (x)}{-e^x+x^4} \, dx+\int x \log \left (-e^x+x^4\right ) \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {x^3}{18}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{2} \int \frac {(-4+x) x^5}{-e^x+x^4} \, dx-\frac {1}{2} \int \frac {x^2 \left (-e^x+4 x^3\right )}{-e^x+x^4} \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\int \frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx-3 \int \frac {x^6}{-e^x+x^4} \, dx}{3 x} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {x^3}{18}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx-3 \int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx-\frac {1}{2} \int \left (x^2-\frac {(-4+x) x^5}{-e^x+x^4}\right ) \, dx-\frac {1}{2} \int \left (-\frac {4 x^5}{-e^x+x^4}+\frac {x^6}{-e^x+x^4}\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \left (\frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx}{x}-\frac {3 \int \frac {x^6}{-e^x+x^4} \, dx}{x}\right ) \, dx+\frac {1}{2} \int \frac {(-4+x) x^5}{-e^x+x^4} \, dx-\frac {1}{2} \int \frac {x^6}{-e^x+x^4} \, dx+2 \int \frac {x^5}{-e^x+x^4} \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\int \frac {\int \frac {x^6}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \frac {x^3+12 \int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx-\frac {1}{2} \int \frac {x^6}{-e^x+x^4} \, dx+\frac {1}{2} \int \left (-\frac {4 x^5}{-e^x+x^4}+\frac {x^6}{-e^x+x^4}\right ) \, dx+2 \int \frac {x^5}{-e^x+x^4} \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {2 x^3}{9}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )-\frac {1}{3} \int \left (x^2+\frac {12 \int \frac {x^5}{-e^x+x^4} \, dx}{x}\right ) \, dx+4 \int \frac {\int \frac {x^5}{-e^x+x^4} \, dx}{x} \, dx\\ &=\frac {x^2}{2}-\frac {x^3}{3}+\frac {1}{3} x^3 \log (x)-\frac {1}{6} \left (3 x^2-2 x^3\right ) (1+2 \log (x))-x^2 \log (x) \log \left (-e^x+x^4\right )\\ \end {aligned} \end {gather*}
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Mathematica [A] time = 0.36, size = 21, normalized size = 0.91 \begin {gather*} x^2 \log (x) \left (-1+x-\log \left (-e^x+x^4\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 29, normalized size = 1.26 \begin {gather*} -x^{2} \log \left (x^{4} - e^{x}\right ) \log \relax (x) + {\left (x^{3} - x^{2}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.45, size = 30, normalized size = 1.30 \begin {gather*} x^{3} \log \relax (x) - x^{2} \log \left (x^{4} - e^{x}\right ) \log \relax (x) - x^{2} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 27, normalized size = 1.17
method | result | size |
risch | \(-x^{2} \ln \relax (x ) \ln \left (-{\mathrm e}^{x}+x^{4}\right )+x^{2} \left (x -1\right ) \ln \relax (x )\) | \(27\) |
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 29, normalized size = 1.26 \begin {gather*} -x^{2} \log \left (x^{4} - e^{x}\right ) \log \relax (x) + {\left (x^{3} - x^{2}\right )} \log \relax (x) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.63, size = 21, normalized size = 0.91 \begin {gather*} -x^2\,\ln \relax (x)\,\left (\ln \left (x^4-{\mathrm {e}}^x\right )-x+1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.58, size = 24, normalized size = 1.04 \begin {gather*} - x^{2} \log {\relax (x )} \log {\left (x^{4} - e^{x} \right )} + \left (x^{3} - x^{2}\right ) \log {\relax (x )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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