3.61.31 \(\int \frac {e^{\frac {4+(1+x) \log (\frac {x^2}{7})+\log (\frac {x^2}{7}) \log (\log (4+x))}{\log (\frac {x^2}{7})}} ((-32-8 x) \log (4+x)+\log ^2(\frac {x^2}{7}) (x+(4 x+x^2) \log (4+x)))}{(4 x+x^2) \log ^2(\frac {x^2}{7}) \log (4+x)} \, dx\)

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 = 3.74, 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.

[In]

Int[(E^((4 + (1 + x)*Log[x^2/7] + Log[x^2/7]*Log[Log[4 + x]])/Log[x^2/7])*((-32 - 8*x)*Log[4 + x] + Log[x^2/7]
^2*(x + (4*x + x^2)*Log[4 + x])))/((4*x + x^2)*Log[x^2/7]^2*Log[4 + x]),x]

[Out]

Defer[Int][E^(1 + x + 4/Log[x^2/7])/(4 + x), x] + Defer[Int][E^(1 + x + 4/Log[x^2/7])*Log[4 + x], x] - 8*Defer
[Int][(E^(1 + x + 4/Log[x^2/7])*Log[4 + x])/(x*Log[x^2/7]^2), x]

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 = 0.34, 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.

[In]

Integrate[(E^((4 + (1 + x)*Log[x^2/7] + Log[x^2/7]*Log[Log[4 + x]])/Log[x^2/7])*((-32 - 8*x)*Log[4 + x] + Log[
x^2/7]^2*(x + (4*x + x^2)*Log[4 + x])))/((4*x + x^2)*Log[x^2/7]^2*Log[4 + x]),x]

[Out]

E^(1 + x + 4/Log[x^2/7])*Log[4 + x]

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fricas [A]  time = 0.74, 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.

[In]

integrate((((x^2+4*x)*log(4+x)+x)*log(1/7*x^2)^2+(-8*x-32)*log(4+x))*exp((log(1/7*x^2)*log(log(4+x))+(x+1)*log
(1/7*x^2)+4)/log(1/7*x^2))/(x^2+4*x)/log(4+x)/log(1/7*x^2)^2,x, algorithm="fricas")

[Out]

e^(((x + 1)*log(1/7*x^2) + log(1/7*x^2)*log(log(x + 4)) + 4)/log(1/7*x^2))

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giac [A]  time = 4.15, 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.

[In]

integrate((((x^2+4*x)*log(4+x)+x)*log(1/7*x^2)^2+(-8*x-32)*log(4+x))*exp((log(1/7*x^2)*log(log(4+x))+(x+1)*log
(1/7*x^2)+4)/log(1/7*x^2))/(x^2+4*x)/log(4+x)/log(1/7*x^2)^2,x, algorithm="giac")

[Out]

e^(x + 4/log(1/7*x^2) + log(log(x + 4)) + 1)

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maple [C]  time = 1.61, size = 266, normalized size = 12.09

method	result	size
risch	\({\mathrm e}^{\frac {i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}+i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+i \ln \left (\ln \left (4+x \right )\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \ln \left (\ln \left (4+x \right )\right ) \pi \,\mathrm {csgn}\left (i x^{2}\right ) \mathrm {csgn}\left (i x \right )^{2}+i x \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \ln \left (\ln \left (4+x \right )\right ) \pi \mathrm {csgn}\left (i x^{2}\right )^{2} \mathrm {csgn}\left (i x \right )-2 i x \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i x \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 \ln \left (\ln \left (4+x \right )\right ) \ln \relax (x )-4 x \ln \relax (x )+2 \ln \left (\ln \left (4+x \right )\right ) \ln \relax (7)+2 x \ln \relax (7)-4 \ln \relax (x )+2 \ln \relax (7)-8}{i \pi \mathrm {csgn}\left (i x^{2}\right )^{3}-2 i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+i \pi \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-4 \ln \relax (x )+2 \ln \relax (7)}}\)	\(266\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2+4*x)*ln(4+x)+x)*ln(1/7*x^2)^2+(-8*x-32)*ln(4+x))*exp((ln(1/7*x^2)*ln(ln(4+x))+(x+1)*ln(1/7*x^2)+4)/
ln(1/7*x^2))/(x^2+4*x)/ln(4+x)/ln(1/7*x^2)^2,x,method=_RETURNVERBOSE)

[Out]

exp((I*Pi*csgn(I*x^2)^3+I*Pi*csgn(I*x)^2*csgn(I*x^2)+I*ln(ln(4+x))*Pi*csgn(I*x^2)^3-2*I*Pi*csgn(I*x)*csgn(I*x^
2)^2+I*ln(ln(4+x))*Pi*csgn(I*x^2)*csgn(I*x)^2+I*x*Pi*csgn(I*x^2)^3-2*I*ln(ln(4+x))*Pi*csgn(I*x^2)^2*csgn(I*x)-
2*I*x*Pi*csgn(I*x)*csgn(I*x^2)^2+I*x*Pi*csgn(I*x)^2*csgn(I*x^2)-4*ln(ln(4+x))*ln(x)-4*x*ln(x)+2*ln(ln(4+x))*ln
(7)+2*x*ln(7)-4*ln(x)+2*ln(7)-8)/(I*Pi*csgn(I*x^2)^3-2*I*Pi*csgn(I*x)*csgn(I*x^2)^2+I*Pi*csgn(I*x)^2*csgn(I*x^
2)-4*ln(x)+2*ln(7)))

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maxima [A]  time = 0.58, size = 20, normalized size = 0.91 \begin {gather*} e^{\left (x - \frac {4}{\log \relax (7) - 2 \, \log \relax (x)} + 1\right )} \log \left (x + 4\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+4*x)*log(4+x)+x)*log(1/7*x^2)^2+(-8*x-32)*log(4+x))*exp((log(1/7*x^2)*log(log(4+x))+(x+1)*log
(1/7*x^2)+4)/log(1/7*x^2))/(x^2+4*x)/log(4+x)/log(1/7*x^2)^2,x, algorithm="maxima")

[Out]

e^(x - 4/(log(7) - 2*log(x)) + 1)*log(x + 4)

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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 \relax (7)}}\,{\mathrm {e}}^{\frac {4}{\ln \left (x^2\right )-\ln \relax (7)}}\,{\left (x^2\right )}^{\frac {1}{\ln \left (x^2\right )-\ln \relax (7)}}\,{\left (x^2\right )}^{\frac {x}{\ln \left (x^2\right )-\ln \relax (7)}}}{7^{\frac {x}{\ln \left (x^2\right )-\ln \relax (7)}}\,7^{\frac {1}{\ln \left (x^2\right )-\ln \relax (7)}}\,{\ln \left (x+4\right )}^{\frac {\ln \relax (7)}{\ln \left (x^2\right )-\ln \relax (7)}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp((log(x^2/7)*log(log(x + 4)) + log(x^2/7)*(x + 1) + 4)/log(x^2/7))*(log(x + 4)*(8*x + 32) - log(x^2/7
)^2*(x + log(x + 4)*(4*x + x^2))))/(log(x + 4)*log(x^2/7)^2*(4*x + x^2)),x)

[Out]

(log(x + 4)^(log(x^2)/(log(x^2) - log(7)))*exp(4/(log(x^2) - log(7)))*(x^2)^(1/(log(x^2) - log(7)))*(x^2)^(x/(
log(x^2) - log(7))))/(7^(x/(log(x^2) - log(7)))*7^(1/(log(x^2) - log(7)))*log(x + 4)^(log(7)/(log(x^2) - log(7
))))

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sympy [A]  time = 1.77, 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.

[In]

integrate((((x**2+4*x)*ln(4+x)+x)*ln(1/7*x**2)**2+(-8*x-32)*ln(4+x))*exp((ln(1/7*x**2)*ln(ln(4+x))+(x+1)*ln(1/
7*x**2)+4)/ln(1/7*x**2))/(x**2+4*x)/ln(4+x)/ln(1/7*x**2)**2,x)

[Out]

exp(((x + 1)*log(x**2/7) + log(x**2/7)*log(log(x + 4)) + 4)/log(x**2/7))

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