Optimal. Leaf size=33 \[ \left (-\frac {5 x}{4}+\frac {e^{4 \left (-e^x+x+x^2\right )}}{\frac {2}{x}+x}\right ) \log (3) \]
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Rubi [F]
time = 6.76, 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 {\left (-20-20 x^2-5 x^4\right ) \log (3)+e^{-4 e^x+4 x+4 x^2} \left (e^x \left (-32 x-16 x^3\right ) \log (3)+\left (8+32 x+60 x^2+16 x^3+32 x^4\right ) \log (3)\right )}{16+16 x^2+4 x^4} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=4 \int \frac {\left (-20-20 x^2-5 x^4\right ) \log (3)+e^{-4 e^x+4 x+4 x^2} \left (e^x \left (-32 x-16 x^3\right ) \log (3)+\left (8+32 x+60 x^2+16 x^3+32 x^4\right ) \log (3)\right )}{\left (8+4 x^2\right )^2} \, dx\\ &=4 \int \left (-\frac {5 \log (3)}{16}+\frac {e^{-4 \left (e^x-x-x^2\right )} \left (2+8 x-8 e^x x+15 x^2+4 x^3-4 e^x x^3+8 x^4\right ) \log (3)}{4 \left (2+x^2\right )^2}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)+\log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )} \left (2+8 x-8 e^x x+15 x^2+4 x^3-4 e^x x^3+8 x^4\right )}{\left (2+x^2\right )^2} \, dx\\ &=-\frac {5}{4} x \log (3)+\log (3) \int \left (\frac {2 e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2}+\frac {8 e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2}+\frac {15 e^{-4 \left (e^x-x-x^2\right )} x^2}{\left (2+x^2\right )^2}+\frac {4 e^{-4 \left (e^x-x-x^2\right )} x^3}{\left (2+x^2\right )^2}+\frac {8 e^{-4 \left (e^x-x-x^2\right )} x^4}{\left (2+x^2\right )^2}-\frac {4 e^{x-4 \left (e^x-x-x^2\right )} x}{2+x^2}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2} \, dx+(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x^3}{\left (2+x^2\right )^2} \, dx-(4 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )} x}{2+x^2} \, dx+(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2} \, dx+(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x^4}{\left (2+x^2\right )^2} \, dx+(15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x^2}{\left (2+x^2\right )^2} \, dx\\ &=-\frac {5}{4} x \log (3)+(2 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{4 \left (-2-x^2\right )}\right ) \, dx-(4 \log (3)) \int \left (-\frac {e^{x-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}-x\right )}+\frac {e^{x-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}+x\right )}\right ) \, dx+(4 \log (3)) \int \left (-\frac {2 e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2}+\frac {e^{-4 \left (e^x-x-x^2\right )} x}{2+x^2}\right ) \, dx+(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x}{\left (2+x^2\right )^2} \, dx+(8 \log (3)) \int \left (e^{-4 \left (e^x-x-x^2\right )}+\frac {4 e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2}-\frac {4 e^{-4 \left (e^x-x-x^2\right )}}{2+x^2}\right ) \, dx+(15 \log (3)) \int \left (-\frac {2 e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2}+\frac {e^{-4 \left (e^x-x-x^2\right )}}{2+x^2}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {1}{2} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{-2-x^2} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )} x}{2+x^2} \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx+(15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{2+x^2} \, dx-(30 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2} \, dx+(32 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (2+x^2\right )^2} \, dx-(32 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{2+x^2} \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-\frac {1}{2} \log (3) \int \left (-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+(4 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}-x\right )}+\frac {e^{-4 \left (e^x-x-x^2\right )}}{2 \left (i \sqrt {2}+x\right )}\right ) \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx+(15 \log (3)) \int \left (\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}+\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx-(30 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{4 \left (-2-x^2\right )}\right ) \, dx-(32 \log (3)) \int \left (\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}+\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+(32 \log (3)) \int \left (-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}-x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{8 \left (i \sqrt {2}+x\right )^2}-\frac {e^{-4 \left (e^x-x-x^2\right )}}{4 \left (-2-x^2\right )}\right ) \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {1}{2} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{-2-x^2} \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx-(8 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{-2-x^2} \, dx+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2}}-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx+\frac {1}{2} (15 \log (3)) \int \left (-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx-(8 \log (3)) \int \left (-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}-x\right )}-\frac {i e^{-4 \left (e^x-x-x^2\right )}}{2 \sqrt {2} \left (i \sqrt {2}+x\right )}\right ) \, dx+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2}}-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx\\ &=-\frac {5}{4} x \log (3)-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-\frac {1}{4} \log (3) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+(2 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-(2 \log (3)) \int \frac {e^{x-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx+\frac {1}{4} (15 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}-x\right )^2} \, dx-(4 \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{\left (i \sqrt {2}+x\right )^2} \, dx+(8 \log (3)) \int e^{-4 \left (e^x-x-x^2\right )} \, dx+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}+\frac {(i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}-\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{4 \sqrt {2}}-\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{4 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx}{2 \sqrt {2}}+\frac {(15 i \log (3)) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx}{2 \sqrt {2}}+\left (2 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx+\left (2 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}-x} \, dx-\left (8 i \sqrt {2} \log (3)\right ) \int \frac {e^{-4 \left (e^x-x-x^2\right )}}{i \sqrt {2}+x} \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.23, size = 32, normalized size = 0.97 \begin {gather*} \frac {1}{4} x \left (-5+\frac {4 e^{-4 e^x+4 x (1+x)}}{2+x^2}\right ) \log (3) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.08, size = 32, normalized size = 0.97
method | result | size |
risch | \(-\frac {5 x \ln \left (3\right )}{4}+\frac {\ln \left (3\right ) x \,{\mathrm e}^{-4 \,{\mathrm e}^{x}+4 x^{2}+4 x}}{x^{2}+2}\) | \(32\) |
norman | \(\frac {x \ln \left (3\right ) {\mathrm e}^{-4 \,{\mathrm e}^{x}+4 x^{2}+4 x}-\frac {5 x \ln \left (3\right )}{2}-\frac {5 x^{3} \ln \left (3\right )}{4}}{x^{2}+2}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 108 vs.
\(2 (33) = 66\).
time = 0.53, size = 108, normalized size = 3.27 \begin {gather*} \frac {5}{8} \, {\left (3 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - 2 \, x - \frac {2 \, x}{x^{2} + 2}\right )} \log \left (3\right ) - \frac {5}{8} \, {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) + \frac {2 \, x}{x^{2} + 2}\right )} \log \left (3\right ) - \frac {5}{4} \, {\left (\sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} x\right ) - \frac {2 \, x}{x^{2} + 2}\right )} \log \left (3\right ) + \frac {x e^{\left (4 \, x^{2} + 4 \, x - 4 \, e^{x}\right )} \log \left (3\right )}{x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 40, normalized size = 1.21 \begin {gather*} \frac {4 \, x e^{\left (4 \, x^{2} + 4 \, x - 4 \, e^{x}\right )} \log \left (3\right ) - 5 \, {\left (x^{3} + 2 \, x\right )} \log \left (3\right )}{4 \, {\left (x^{2} + 2\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.11, size = 32, normalized size = 0.97 \begin {gather*} - \frac {5 x \log {\left (3 \right )}}{4} + \frac {x e^{4 x^{2} + 4 x - 4 e^{x}} \log {\left (3 \right )}}{x^{2} + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.52, size = 32, normalized size = 0.97 \begin {gather*} \frac {x\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{-4\,{\mathrm {e}}^x}\,\ln \left (3\right )}{x^2+2}-\frac {5\,x\,\ln \left (3\right )}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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