3.42.2 \(\int \frac {4+e^{e^{18-2 e^x-2 x}+4 x-x \log (3)} (16+e^{18-2 e^x-2 x} (-8-8 e^x)-4 \log (3))}{25+e^{2 e^{18-2 e^x-2 x}+8 x-2 x \log (3)}-10 x+x^2+e^{e^{18-2 e^x-2 x}+4 x-x \log (3)} (-10+2 x)} \, dx\)

Optimal. Leaf size=31 \[ 5-\frac {4}{-5+e^{e^{18-2 e^x-2 x}-x (-4+\log (3))}+x} \]

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Rubi [F]  time = 7.44, 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 {4+e^{e^{18-2 e^x-2 x}+4 x-x \log (3)} \left (16+e^{18-2 e^x-2 x} \left (-8-8 e^x\right )-4 \log (3)\right )}{25+e^{2 e^{18-2 e^x-2 x}+8 x-2 x \log (3)}-10 x+x^2+e^{e^{18-2 e^x-2 x}+4 x-x \log (3)} (-10+2 x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(4 + E^(E^(18 - 2*E^x - 2*x) + 4*x - x*Log[3])*(16 + E^(18 - 2*E^x - 2*x)*(-8 - 8*E^x) - 4*Log[3]))/(25 +
E^(2*E^(18 - 2*E^x - 2*x) + 8*x - 2*x*Log[3]) - 10*x + x^2 + E^(E^(18 - 2*E^x - 2*x) + 4*x - x*Log[3])*(-10 +
2*x)),x]

[Out]

4*(4 - Log[3])*Defer[Int][(3^x*E^(E^(-2*(-9 + E^x + x)) + 4*x))/(E^(E^(-2*(-9 + E^x + x)) + 4*x) + 3^x*(-5 + x
))^2, x] + 4*Defer[Int][3^(2*x)/(-5*3^x + E^(E^(18 - 2*E^x - 2*x) + 4*x) + 3^x*x)^2, x] - 8*Defer[Int][(3^x*E^
(18 + E^(18 - 2*E^x - 2*x) - 2*E^x + 2*x))/(-5*3^x + E^(E^(18 - 2*E^x - 2*x) + 4*x) + 3^x*x)^2, x] - 8*Defer[I
nt][(3^x*E^(18 + E^(18 - 2*E^x - 2*x) - 2*E^x + 3*x))/(-5*3^x + E^(E^(18 - 2*E^x - 2*x) + 4*x) + 3^x*x)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {3^{2 x} \left (4+e^{e^{18-2 e^x-2 x}+4 x-x \log (3)} \left (16+e^{18-2 e^x-2 x} \left (-8-8 e^x\right )-4 \log (3)\right )\right )}{\left (5\ 3^x-e^{e^{18-2 e^x-2 x}+4 x}-3^x x\right )^2} \, dx\\ &=\int \left (\frac {4\ 3^{2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2}+\frac {4\ 3^x e^{e^{18-2 e^x-2 x}-2 e^x+2 x} \left (-2 e^{18}-2 e^{18+x}+4 e^{2 e^x+2 x} \left (1-\frac {\log (3)}{4}\right )\right )}{\left (5\ 3^x-e^{e^{18-2 e^x-2 x}+4 x}-3^x x\right )^2}\right ) \, dx\\ &=4 \int \frac {3^{2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx+4 \int \frac {3^x e^{e^{18-2 e^x-2 x}-2 e^x+2 x} \left (-2 e^{18}-2 e^{18+x}+4 e^{2 e^x+2 x} \left (1-\frac {\log (3)}{4}\right )\right )}{\left (5\ 3^x-e^{e^{18-2 e^x-2 x}+4 x}-3^x x\right )^2} \, dx\\ &=4 \int \frac {3^{2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx+4 \int \left (-\frac {2\ 3^x e^{18+e^{18-2 e^x-2 x}-2 e^x+2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2}-\frac {2\ 3^x e^{18+e^{18-2 e^x-2 x}-2 e^x+3 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2}+\frac {3^x \exp \left (e^{18-2 e^x-2 x}-2 e^x+2 x+2 \left (e^x+x\right )\right ) (4-\log (3))}{\left (5\ 3^x-e^{e^{18-2 e^x-2 x}+4 x}-3^x x\right )^2}\right ) \, dx\\ &=4 \int \frac {3^{2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx-8 \int \frac {3^x e^{18+e^{18-2 e^x-2 x}-2 e^x+2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx-8 \int \frac {3^x e^{18+e^{18-2 e^x-2 x}-2 e^x+3 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx+(4 (4-\log (3))) \int \frac {3^x \exp \left (e^{18-2 e^x-2 x}-2 e^x+2 x+2 \left (e^x+x\right )\right )}{\left (5\ 3^x-e^{e^{18-2 e^x-2 x}+4 x}-3^x x\right )^2} \, dx\\ &=4 \int \frac {3^{2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx-8 \int \frac {3^x e^{18+e^{18-2 e^x-2 x}-2 e^x+2 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx-8 \int \frac {3^x e^{18+e^{18-2 e^x-2 x}-2 e^x+3 x}}{\left (-5 3^x+e^{e^{18-2 e^x-2 x}+4 x}+3^x x\right )^2} \, dx+(4 (4-\log (3))) \int \frac {3^x e^{e^{-2 \left (-9+e^x+x\right )}+4 x}}{\left (e^{e^{-2 \left (-9+e^x+x\right )}+4 x}+3^x (-5+x)\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 31, normalized size = 1.00 \begin {gather*} -\frac {4\ 3^x}{e^{e^{-2 \left (-9+e^x+x\right )}+4 x}+3^x (-5+x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(4 + E^(E^(18 - 2*E^x - 2*x) + 4*x - x*Log[3])*(16 + E^(18 - 2*E^x - 2*x)*(-8 - 8*E^x) - 4*Log[3]))/
(25 + E^(2*E^(18 - 2*E^x - 2*x) + 8*x - 2*x*Log[3]) - 10*x + x^2 + E^(E^(18 - 2*E^x - 2*x) + 4*x - x*Log[3])*(
-10 + 2*x)),x]

[Out]

(-4*3^x)/(E^(E^(-2*(-9 + E^x + x)) + 4*x) + 3^x*(-5 + x))

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fricas [A]  time = 0.64, size = 27, normalized size = 0.87 \begin {gather*} -\frac {4}{x + e^{\left (-x \log \relax (3) + 4 \, x + e^{\left (-2 \, x - 2 \, e^{x} + 18\right )}\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*exp(x)-8)*exp(-exp(x)+9-x)^2-4*log(3)+16)*exp(exp(-exp(x)+9-x)^2-x*log(3)+4*x)+4)/(exp(exp(-ex
p(x)+9-x)^2-x*log(3)+4*x)^2+(2*x-10)*exp(exp(-exp(x)+9-x)^2-x*log(3)+4*x)+x^2-10*x+25),x, algorithm="fricas")

[Out]

-4/(x + e^(-x*log(3) + 4*x + e^(-2*x - 2*e^x + 18)) - 5)

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giac [A]  time = 0.78, size = 27, normalized size = 0.87 \begin {gather*} -\frac {4}{x + e^{\left (-x \log \relax (3) + 4 \, x + e^{\left (-2 \, x - 2 \, e^{x} + 18\right )}\right )} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*exp(x)-8)*exp(-exp(x)+9-x)^2-4*log(3)+16)*exp(exp(-exp(x)+9-x)^2-x*log(3)+4*x)+4)/(exp(exp(-ex
p(x)+9-x)^2-x*log(3)+4*x)^2+(2*x-10)*exp(exp(-exp(x)+9-x)^2-x*log(3)+4*x)+x^2-10*x+25),x, algorithm="giac")

[Out]

-4/(x + e^(-x*log(3) + 4*x + e^(-2*x - 2*e^x + 18)) - 5)

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maple [A]  time = 0.08, size = 27, normalized size = 0.87

method	result	size
risch	\(-\frac {4}{-5+\left (\frac {1}{3}\right )^{x} {\mathrm e}^{{\mathrm e}^{-2 \,{\mathrm e}^{x}+18-2 x}+4 x}+x}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-8*exp(x)-8)*exp(-exp(x)+9-x)^2-4*ln(3)+16)*exp(exp(-exp(x)+9-x)^2-x*ln(3)+4*x)+4)/(exp(exp(-exp(x)+9-x
)^2-x*ln(3)+4*x)^2+(2*x-10)*exp(exp(-exp(x)+9-x)^2-x*ln(3)+4*x)+x^2-10*x+25),x,method=_RETURNVERBOSE)

[Out]

-4/(-5+(1/3)^x*exp(exp(-2*exp(x)+18-2*x)+4*x)+x)

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maxima [A]  time = 0.55, size = 30, normalized size = 0.97 \begin {gather*} -\frac {4 \cdot 3^{x}}{3^{x} {\left (x - 5\right )} + e^{\left (4 \, x + e^{\left (-2 \, x - 2 \, e^{x} + 18\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*exp(x)-8)*exp(-exp(x)+9-x)^2-4*log(3)+16)*exp(exp(-exp(x)+9-x)^2-x*log(3)+4*x)+4)/(exp(exp(-ex
p(x)+9-x)^2-x*log(3)+4*x)^2+(2*x-10)*exp(exp(-exp(x)+9-x)^2-x*log(3)+4*x)+x^2-10*x+25),x, algorithm="maxima")

[Out]

-4*3^x/(3^x*(x - 5) + e^(4*x + e^(-2*x - 2*e^x + 18)))

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mupad [F]  time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int -\frac {{\mathrm {e}}^{4\,x+{\mathrm {e}}^{18-2\,{\mathrm {e}}^x-2\,x}-x\,\ln \relax (3)}\,\left (4\,\ln \relax (3)+{\mathrm {e}}^{18-2\,{\mathrm {e}}^x-2\,x}\,\left (8\,{\mathrm {e}}^x+8\right )-16\right )-4}{{\mathrm {e}}^{8\,x+2\,{\mathrm {e}}^{18-2\,{\mathrm {e}}^x-2\,x}-2\,x\,\ln \relax (3)}-10\,x+{\mathrm {e}}^{4\,x+{\mathrm {e}}^{18-2\,{\mathrm {e}}^x-2\,x}-x\,\ln \relax (3)}\,\left (2\,x-10\right )+x^2+25} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(4*x + exp(18 - 2*exp(x) - 2*x) - x*log(3))*(4*log(3) + exp(18 - 2*exp(x) - 2*x)*(8*exp(x) + 8) - 16)
- 4)/(exp(8*x + 2*exp(18 - 2*exp(x) - 2*x) - 2*x*log(3)) - 10*x + exp(4*x + exp(18 - 2*exp(x) - 2*x) - x*log(
3))*(2*x - 10) + x^2 + 25),x)

[Out]

int(-(exp(4*x + exp(18 - 2*exp(x) - 2*x) - x*log(3))*(4*log(3) + exp(18 - 2*exp(x) - 2*x)*(8*exp(x) + 8) - 16)
- 4)/(exp(8*x + 2*exp(18 - 2*exp(x) - 2*x) - 2*x*log(3)) - 10*x + exp(4*x + exp(18 - 2*exp(x) - 2*x) - x*log(
3))*(2*x - 10) + x^2 + 25), x)

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sympy [A]  time = 0.35, size = 27, normalized size = 0.87 \begin {gather*} - \frac {4}{x + e^{- x \log {\relax (3 )} + 4 x + e^{- 2 x - 2 e^{x} + 18}} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-8*exp(x)-8)*exp(-exp(x)+9-x)**2-4*ln(3)+16)*exp(exp(-exp(x)+9-x)**2-x*ln(3)+4*x)+4)/(exp(exp(-ex
p(x)+9-x)**2-x*ln(3)+4*x)**2+(2*x-10)*exp(exp(-exp(x)+9-x)**2-x*ln(3)+4*x)+x**2-10*x+25),x)

[Out]

-4/(x + exp(-x*log(3) + 4*x + exp(-2*x - 2*exp(x) + 18)) - 5)

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