Optimal. Leaf size=22 \[ e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \log (4+x) \]
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Rubi [F]
time = 2.71, 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 {4+(1+x) \log \left (\frac {x^2}{7}\right )+\log \left (\frac {x^2}{7}\right ) \log (\log (4+x))}{\log \left (\frac {x^2}{7}\right )}\right ) \left ((-32-8 x) \log (4+x)+\log ^2\left (\frac {x^2}{7}\right ) \left (x+\left (4 x+x^2\right ) \log (4+x)\right )\right )}{\left (4 x+x^2\right ) \log ^2\left (\frac {x^2}{7}\right ) \log (4+x)} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\frac {4+(1+x) \log \left (\frac {x^2}{7}\right )+\log \left (\frac {x^2}{7}\right ) \log (\log (4+x))}{\log \left (\frac {x^2}{7}\right )}\right ) \left ((-32-8 x) \log (4+x)+\log ^2\left (\frac {x^2}{7}\right ) \left (x+\left (4 x+x^2\right ) \log (4+x)\right )\right )}{x (4+x) \log ^2\left (\frac {x^2}{7}\right ) \log (4+x)} \, dx\\ &=\int \frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \left (-8 (4+x) \log (4+x)+\log ^2\left (\frac {x^2}{7}\right ) (x+x (4+x) \log (4+x))\right )}{x (4+x) \log ^2\left (\frac {x^2}{7}\right )} \, dx\\ &=\int \left (\frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}}}{4+x}+\frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \left (-8+x \log ^2\left (\frac {x^2}{7}\right )\right ) \log (4+x)}{x \log ^2\left (\frac {x^2}{7}\right )}\right ) \, dx\\ &=\int \frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}}}{4+x} \, dx+\int \frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \left (-8+x \log ^2\left (\frac {x^2}{7}\right )\right ) \log (4+x)}{x \log ^2\left (\frac {x^2}{7}\right )} \, dx\\ &=\int \frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}}}{4+x} \, dx+\int \left (e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \log (4+x)-\frac {8 e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \log (4+x)}{x \log ^2\left (\frac {x^2}{7}\right )}\right ) \, dx\\ &=-\left (8 \int \frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \log (4+x)}{x \log ^2\left (\frac {x^2}{7}\right )} \, dx\right )+\int \frac {e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}}}{4+x} \, dx+\int e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \log (4+x) \, dx\\ \end {aligned} \end {gather*}
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Mathematica [A]
time = 0.17, size = 22, normalized size = 1.00 \begin {gather*} e^{1+x+\frac {4}{\log \left (\frac {x^2}{7}\right )}} \log (4+x) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains higher order function than in optimal. Order 9 vs. order
3.
time = 3.76, size = 266, normalized size = 12.09
method | result | size |
risch | \({\mathrm e}^{\frac {i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-2 i \ln \left (\ln \left (4+x \right )\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )+i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+i \ln \left (\ln \left (4+x \right )\right ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \ln \left (\ln \left (4+x \right )\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-4 \ln \left (\ln \left (4+x \right )\right ) \ln \left (x \right )-4 x \ln \left (x \right )+2 \ln \left (\ln \left (4+x \right )\right ) \ln \left (7\right )+2 x \ln \left (7\right )-4 \ln \left (x \right )+2 \ln \left (7\right )-8}{i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 \ln \left (x \right )+2 \ln \left (7\right )}}\) | \(266\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.74, size = 20, normalized size = 0.91 \begin {gather*} e^{\left (x - \frac {4}{\log \left (7\right ) - 2 \, \log \left (x\right )} + 1\right )} \log \left (x + 4\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 34, normalized size = 1.55 \begin {gather*} e^{\left (\frac {{\left (x + 1\right )} \log \left (\frac {1}{7} \, x^{2}\right ) + \log \left (\frac {1}{7} \, x^{2}\right ) \log \left (\log \left (x + 4\right )\right ) + 4}{\log \left (\frac {1}{7} \, x^{2}\right )}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.93, size = 32, normalized size = 1.45 \begin {gather*} e^{\frac {\left (x + 1\right ) \log {\left (\frac {x^{2}}{7} \right )} + \log {\left (\frac {x^{2}}{7} \right )} \log {\left (\log {\left (x + 4 \right )} \right )} + 4}{\log {\left (\frac {x^{2}}{7} \right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.17, size = 19, normalized size = 0.86 \begin {gather*} e^{\left (x + \frac {4}{\log \left (\frac {1}{7} \, x^{2}\right )} + \log \left (\log \left (x + 4\right )\right ) + 1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.70, size = 121, normalized size = 5.50 \begin {gather*} \frac {{\ln \left (x+4\right )}^{\frac {\ln \left (x^2\right )}{\ln \left (x^2\right )-\ln \left (7\right )}}\,{\mathrm {e}}^{\frac {4}{\ln \left (x^2\right )-\ln \left (7\right )}}\,{\left (x^2\right )}^{\frac {1}{\ln \left (x^2\right )-\ln \left (7\right )}}\,{\left (x^2\right )}^{\frac {x}{\ln \left (x^2\right )-\ln \left (7\right )}}}{7^{\frac {x}{\ln \left (x^2\right )-\ln \left (7\right )}}\,7^{\frac {1}{\ln \left (x^2\right )-\ln \left (7\right )}}\,{\ln \left (x+4\right )}^{\frac {\ln \left (7\right )}{\ln \left (x^2\right )-\ln \left (7\right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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