Optimal. Leaf size=32 \[ \frac {\sqrt [6]{2} e^{\frac {x}{2+x}} \log (3)}{\sqrt [6]{\frac {x}{-5+4 x^2}}} \]
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Rubi [F]
time = 9.04, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {\exp \left (\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}\right ) \left (20-40 x+21 x^2+64 x^3+4 x^4\right ) \log (3)}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\log (3) \int \frac {\exp \left (\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}\right ) \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx\\ &=\log (3) \int \frac {e^{\frac {x}{2+x}} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{3 x^2 (2+x)^2} \, dx\\ &=\frac {1}{3} \log (3) \int \frac {e^{\frac {x}{2+x}} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{x^2 (2+x)^2} \, dx\\ &=\frac {\left (\left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \int \frac {e^{\frac {x}{2+x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{x^{7/6} (2+x)^2 \left (-10+8 x^2\right )^{5/6}} \, dx}{3 x^{5/6}}\\ &=\frac {\left (2 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} \left (20-40 x^6+21 x^{12}+64 x^{18}+4 x^{24}\right )}{x^2 \left (2+x^6\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (2 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \left (\frac {5 e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}}+\frac {48 e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}}+\frac {4 e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}}+\frac {132 e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right )^2 \left (-10+8 x^{12}\right )^{5/6}}-\frac {192 e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right ) \left (-10+8 x^{12}\right )^{5/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (8 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (10 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (96 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (264 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (384 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (2+x^6\right ) \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (8 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (10 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (96 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (264 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \left (-\frac {i e^{\frac {x^6}{2+x^6}} x}{4 \sqrt {2} \left (i \sqrt {2}-x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}}+\frac {i e^{\frac {x^6}{2+x^6}} x}{4 \sqrt {2} \left (i \sqrt {2}+x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (384 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \left (\frac {e^{\frac {x^6}{2+x^6}} x}{2 \left (-i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}}+\frac {e^{\frac {x^6}{2+x^6}} x}{2 \left (i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}}\right ) \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\frac {\left (8 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^{10}}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (10 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}}}{x^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (96 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x^4}{\left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (192 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (-i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (192 \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (i \sqrt {2}+x^3\right ) \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}-\frac {\left (33 i \sqrt {2} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (i \sqrt {2}-x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}+\frac {\left (33 i \sqrt {2} \left (\frac {x}{-10+8 x^2}\right )^{5/6} \left (-10+8 x^2\right )^{5/6} \log (3)\right ) \text {Subst}\left (\int \frac {e^{\frac {x^6}{2+x^6}} x}{\left (i \sqrt {2}+x^3\right )^2 \left (-10+8 x^{12}\right )^{5/6}} \, dx,x,\sqrt [6]{x}\right )}{x^{5/6}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}
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Mathematica [F] Contains unresolved integral.
time = 1.28, size = 84, normalized size = 2.62 \begin {gather*} \left (\int \frac {e^{\frac {6 x+(-2-x) \log \left (\frac {x}{-10+8 x^2}\right )}{12+6 x}} \left (20-40 x+21 x^2+64 x^3+4 x^4\right )}{-120 x-120 x^2+66 x^3+96 x^4+24 x^5} \, dx\right ) \log (3) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.30, size = 44, normalized size = 1.38
method | result | size |
risch | \(\ln \left (3\right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{8 x^{2}-10}\right ) x +2 \ln \left (\frac {x}{8 x^{2}-10}\right )-6 x}{6 \left (2+x \right )}}\) | \(44\) |
gosper | \(\ln \left (3\right ) {\mathrm e}^{-\frac {\ln \left (\frac {x}{8 x^{2}-10}\right ) x +2 \ln \left (\frac {x}{8 x^{2}-10}\right )-6 x}{6 \left (2+x \right )}}\) | \(46\) |
norman | \(\frac {x \ln \left (3\right ) {\mathrm e}^{\frac {\left (-x -2\right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}+2 \ln \left (3\right ) {\mathrm e}^{\frac {\left (-x -2\right ) \ln \left (\frac {x}{8 x^{2}-10}\right )+6 x}{6 x +12}}}{2+x}\) | \(78\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 32, normalized size = 1.00 \begin {gather*} e^{\left (-\frac {{\left (x + 2\right )} \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right ) - 6 \, x}{6 \, {\left (x + 2\right )}}\right )} \log \left (3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.40, size = 27, normalized size = 0.84 \begin {gather*} e^{\frac {6 x + \left (- x - 2\right ) \log {\left (\frac {x}{8 x^{2} - 10} \right )}}{6 x + 12}} \log {\left (3 \right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 53, normalized size = 1.66 \begin {gather*} e^{\left (-\frac {x \log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right )}{6 \, {\left (x + 2\right )}} + \frac {x}{x + 2} - \frac {\log \left (\frac {x}{2 \, {\left (4 \, x^{2} - 5\right )}}\right )}{3 \, {\left (x + 2\right )}}\right )} \log \left (3\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {{\mathrm {e}}^{\frac {6\,x-\ln \left (\frac {x}{8\,x^2-10}\right )\,\left (x+2\right )}{6\,x+12}}\,\ln \left (3\right )\,\left (4\,x^4+64\,x^3+21\,x^2-40\,x+20\right )}{24\,x^5+96\,x^4+66\,x^3-120\,x^2-120\,x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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