3.5.31 \(\int \frac {e^{e^{\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}}+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}} (48+361 x+152 x \log (x)+16 x \log ^2(x))}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx\)

Optimal. Leaf size=21 \[ e^{e^{-5+x+\frac {12}{6+\log (x)-5 (5+\log (x))}}} \]

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Rubi [F]  time = 14.92, 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 (\exp \left (\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right ) \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{361 x+152 x \log (x)+16 x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(E^((-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x])) + (-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Lo
g[x]))*(48 + 361*x + 152*x*Log[x] + 16*x*Log[x]^2))/(361*x + 152*x*Log[x] + 16*x*Log[x]^2),x]

[Out]

Defer[Int][E^(E^((-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x])) + (-107 + 19*x + (-20 + 4*x)*Log[x])/(19
+ 4*Log[x])), x] + 48*Defer[Int][E^(E^((-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x])) + (-107 + 19*x + (-
20 + 4*x)*Log[x])/(19 + 4*Log[x]))/(x*(19 + 4*Log[x])^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {\exp \left (\exp \left (\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right ) \left (48+361 x+152 x \log (x)+16 x \log ^2(x)\right )}{x (19+4 \log (x))^2} \, dx\\ &=\int \left (\exp \left (\exp \left (\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {48 \exp \left (\exp \left (\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )}{x (19+4 \log (x))^2}\right ) \, dx\\ &=48 \int \frac {\exp \left (\exp \left (\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )}{x (19+4 \log (x))^2} \, dx+\int \exp \left (\exp \left (\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right )+\frac {-107+19 x+(-20+4 x) \log (x)}{19+4 \log (x)}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 2.36, size = 17, normalized size = 0.81 \begin {gather*} e^{e^{-5+x-\frac {12}{19+4 \log (x)}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(E^((-107 + 19*x + (-20 + 4*x)*Log[x])/(19 + 4*Log[x])) + (-107 + 19*x + (-20 + 4*x)*Log[x])/(19
+ 4*Log[x]))*(48 + 361*x + 152*x*Log[x] + 16*x*Log[x]^2))/(361*x + 152*x*Log[x] + 16*x*Log[x]^2),x]

[Out]

E^E^(-5 + x - 12/(19 + 4*Log[x]))

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fricas [B]  time = 0.76, size = 74, normalized size = 3.52 \begin {gather*} e^{\left (\frac {{\left (4 \, \log \relax (x) + 19\right )} e^{\left (\frac {4 \, {\left (x - 5\right )} \log \relax (x) + 19 \, x - 107}{4 \, \log \relax (x) + 19}\right )} + 4 \, {\left (x - 5\right )} \log \relax (x) + 19 \, x - 107}{4 \, \log \relax (x) + 19} - \frac {4 \, {\left (x - 5\right )} \log \relax (x) + 19 \, x - 107}{4 \, \log \relax (x) + 19}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19))*exp(exp(((4*x-20
)*log(x)+19*x-107)/(4*log(x)+19)))/(16*x*log(x)^2+152*x*log(x)+361*x),x, algorithm="fricas")

[Out]

e^(((4*log(x) + 19)*e^((4*(x - 5)*log(x) + 19*x - 107)/(4*log(x) + 19)) + 4*(x - 5)*log(x) + 19*x - 107)/(4*lo
g(x) + 19) - (4*(x - 5)*log(x) + 19*x - 107)/(4*log(x) + 19))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (16 \, x \log \relax (x)^{2} + 152 \, x \log \relax (x) + 361 \, x + 48\right )} e^{\left (\frac {4 \, {\left (x - 5\right )} \log \relax (x) + 19 \, x - 107}{4 \, \log \relax (x) + 19} + e^{\left (\frac {4 \, {\left (x - 5\right )} \log \relax (x) + 19 \, x - 107}{4 \, \log \relax (x) + 19}\right )}\right )}}{16 \, x \log \relax (x)^{2} + 152 \, x \log \relax (x) + 361 \, x}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19))*exp(exp(((4*x-20
)*log(x)+19*x-107)/(4*log(x)+19)))/(16*x*log(x)^2+152*x*log(x)+361*x),x, algorithm="giac")

[Out]

integrate((16*x*log(x)^2 + 152*x*log(x) + 361*x + 48)*e^((4*(x - 5)*log(x) + 19*x - 107)/(4*log(x) + 19) + e^(
(4*(x - 5)*log(x) + 19*x - 107)/(4*log(x) + 19)))/(16*x*log(x)^2 + 152*x*log(x) + 361*x), x)

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maple [A]  time = 0.04, size = 26, normalized size = 1.24

method	result	size
risch	\({\mathrm e}^{{\mathrm e}^{\frac {4 x \ln \relax (x )-20 \ln \relax (x )+19 x -107}{4 \ln \relax (x )+19}}}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x*ln(x)^2+152*x*ln(x)+361*x+48)*exp(((4*x-20)*ln(x)+19*x-107)/(4*ln(x)+19))*exp(exp(((4*x-20)*ln(x)+19
*x-107)/(4*ln(x)+19)))/(16*x*ln(x)^2+152*x*ln(x)+361*x),x,method=_RETURNVERBOSE)

[Out]

exp(exp((4*x*ln(x)-20*ln(x)+19*x-107)/(4*ln(x)+19)))

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maxima [B]  time = 1.17, size = 49, normalized size = 2.33 \begin {gather*} e^{\left (e^{\left (\frac {4 \, x \log \relax (x)}{4 \, \log \relax (x) + 19} + \frac {19 \, x}{4 \, \log \relax (x) + 19} - \frac {20 \, \log \relax (x)}{4 \, \log \relax (x) + 19} - \frac {107}{4 \, \log \relax (x) + 19}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*log(x)^2+152*x*log(x)+361*x+48)*exp(((4*x-20)*log(x)+19*x-107)/(4*log(x)+19))*exp(exp(((4*x-20
)*log(x)+19*x-107)/(4*log(x)+19)))/(16*x*log(x)^2+152*x*log(x)+361*x),x, algorithm="maxima")

[Out]

e^(e^(4*x*log(x)/(4*log(x) + 19) + 19*x/(4*log(x) + 19) - 20*log(x)/(4*log(x) + 19) - 107/(4*log(x) + 19)))

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mupad [B]  time = 0.72, size = 40, normalized size = 1.90 \begin {gather*} {\mathrm {e}}^{x^{\frac {4\,\left (x-5\right )}{4\,\ln \relax (x)+19}}\,{\mathrm {e}}^{\frac {19\,x}{4\,\ln \relax (x)+19}}\,{\mathrm {e}}^{-\frac {107}{4\,\ln \relax (x)+19}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((19*x + log(x)*(4*x - 20) - 107)/(4*log(x) + 19))*exp(exp((19*x + log(x)*(4*x - 20) - 107)/(4*log(x)
+ 19)))*(361*x + 16*x*log(x)^2 + 152*x*log(x) + 48))/(361*x + 16*x*log(x)^2 + 152*x*log(x)),x)

[Out]

exp(x^((4*(x - 5))/(4*log(x) + 19))*exp((19*x)/(4*log(x) + 19))*exp(-107/(4*log(x) + 19)))

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sympy [A]  time = 1.36, size = 22, normalized size = 1.05 \begin {gather*} e^{e^{\frac {19 x + \left (4 x - 20\right ) \log {\relax (x )} - 107}{4 \log {\relax (x )} + 19}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((16*x*ln(x)**2+152*x*ln(x)+361*x+48)*exp(((4*x-20)*ln(x)+19*x-107)/(4*ln(x)+19))*exp(exp(((4*x-20)*l
n(x)+19*x-107)/(4*ln(x)+19)))/(16*x*ln(x)**2+152*x*ln(x)+361*x),x)

[Out]

exp(exp((19*x + (4*x - 20)*log(x) - 107)/(4*log(x) + 19)))

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