\(\int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} (-6+8 x-6 x^2)+(2 x-2 e^{20} x-x^2) \log (5))}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx\) [7648]

Optimal result

Integrand size = 121, antiderivative size = 32 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=e^{x-\frac {x^2 (1-2 x-\log (5))}{3 \left (1-e^{20}-x\right )}} \]

[Out]

exp(x-1/3*x^2*(1-ln(5)-2*x)/(1-x-exp(20)))

Rubi [F]

\[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=\int \frac {\exp \left (\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}\right ) \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx \]

[In]

Int[(E^((-3*x + 3*E^20*x + 4*x^2 - 2*x^3 - x^2*Log[5])/(-3 + 3*E^20 + 3*x))*(3 + 3*E^40 - 8*x + 10*x^2 - 4*x^3
+ E^20*(-6 + 8*x - 6*x^2) + (2*x - 2*E^20*x - x^2)*Log[5]))/(3 + 3*E^40 - 6*x + 3*x^2 + E^20*(-6 + 6*x)),x]

[Out]

((2 + 2*E^20 - Log[5])*Defer[Int][E^((x*(-3*(1 - E^20) - 2*x^2 + x*(4 - Log[5])))/(-3 + 3*E^20 + 3*x)), x])/3
- (4*Defer[Int][E^((x*(-3*(1 - E^20) - 2*x^2 + x*(4 - Log[5])))/(-3 + 3*E^20 + 3*x))*x, x])/3 + ((1 - E^20)^2*
(1 - 2*E^20 + Log[5])*Defer[Int][E^((x*(-3*(1 - E^20) - 2*x^2 + x*(4 - Log[5])))/(-3 + 3*E^20 + 3*x))/(-1 + E^
20 + x)^2, x])/3

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) \left (3 \left (1-e^{20}\right )^2-4 x^3-2 \left (1-e^{20}\right ) x (4-\log (5))+x^2 \left (10-6 e^{20}-\log (5)\right )\right )}{3 \left (1-e^{20}\right )^2-6 \left (1-e^{20}\right ) x+3 x^2} \, dx \\ & = \int \frac {\exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) \left (3 \left (1-e^{20}\right )^2-4 x^3-2 \left (1-e^{20}\right ) x (4-\log (5))+x^2 \left (10-6 e^{20}-\log (5)\right )\right )}{3 \left (-1+e^{20}+x\right )^2} \, dx \\ & = \frac {1}{3} \int \frac {\exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) \left (3 \left (1-e^{20}\right )^2-4 x^3-2 \left (1-e^{20}\right ) x (4-\log (5))+x^2 \left (10-6 e^{20}-\log (5)\right )\right )}{\left (-1+e^{20}+x\right )^2} \, dx \\ & = \frac {1}{3} \int \left (-4 \exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) x-\frac {\exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) \left (-1+e^{20}\right )^2 \left (-1+2 e^{20}-\log (5)\right )}{\left (-1+e^{20}+x\right )^2}+2 \exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) \left (1+e^{20}-\frac {\log (5)}{2}\right )\right ) \, dx \\ & = -\left (\frac {4}{3} \int \exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) x \, dx\right )+\frac {1}{3} \left (2+2 e^{20}-\log (5)\right ) \int \exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right ) \, dx+\frac {1}{3} \left (\left (1-e^{20}\right )^2 \left (1-2 e^{20}+\log (5)\right )\right ) \int \frac {\exp \left (\frac {x \left (-3 \left (1-e^{20}\right )-2 x^2+x (4-\log (5))\right )}{-3+3 e^{20}+3 x}\right )}{\left (-1+e^{20}+x\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=5^{-\frac {x^2}{3 \left (-1+e^{20}+x\right )}} e^{-\frac {x \left (3-3 e^{20}-4 x+2 x^2\right )}{3 \left (-1+e^{20}+x\right )}} \]

[In]

Integrate[(E^((-3*x + 3*E^20*x + 4*x^2 - 2*x^3 - x^2*Log[5])/(-3 + 3*E^20 + 3*x))*(3 + 3*E^40 - 8*x + 10*x^2 -
4*x^3 + E^20*(-6 + 8*x - 6*x^2) + (2*x - 2*E^20*x - x^2)*Log[5]))/(3 + 3*E^40 - 6*x + 3*x^2 + E^20*(-6 + 6*x)
),x]

[Out]

1/(5^(x^2/(3*(-1 + E^20 + x)))*E^((x*(3 - 3*E^20 - 4*x + 2*x^2))/(3*(-1 + E^20 + x))))

Maple [A] (verified)

Time = 7.22 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.94

method	result	size
gosper	\({\mathrm e}^{-\frac {x \left (x \ln \left (5\right )+2 x^{2}-3 \,{\mathrm e}^{20}-4 x +3\right )}{3 \left ({\mathrm e}^{20}+x -1\right )}}\)	\(30\)
risch	\({\mathrm e}^{\frac {\left (-x \ln \left (5\right )-2 x^{2}+3 \,{\mathrm e}^{20}+4 x -3\right ) x}{3 \,{\mathrm e}^{20}+3 x -3}}\)	\(31\)
parallelrisch	\({\mathrm e}^{\frac {\left (-x \ln \left (5\right )-2 x^{2}+3 \,{\mathrm e}^{20}+4 x -3\right ) x}{3 \,{\mathrm e}^{20}+3 x -3}}\)	\(31\)
norman	\(\frac {x \,{\mathrm e}^{\frac {-x^{2} \ln \left (5\right )+3 x \,{\mathrm e}^{20}-2 x^{3}+4 x^{2}-3 x}{3 \,{\mathrm e}^{20}+3 x -3}}+\left ({\mathrm e}^{20}-1\right ) {\mathrm e}^{\frac {-x^{2} \ln \left (5\right )+3 x \,{\mathrm e}^{20}-2 x^{3}+4 x^{2}-3 x}{3 \,{\mathrm e}^{20}+3 x -3}}}{{\mathrm e}^{20}+x -1}\)	\(95\)

[In]

int(((-2*x*exp(20)-x^2+2*x)*ln(5)+3*exp(20)^2+(-6*x^2+8*x-6)*exp(20)-4*x^3+10*x^2-8*x+3)*exp((-x^2*ln(5)+3*x*e
xp(20)-2*x^3+4*x^2-3*x)/(3*exp(20)+3*x-3))/(3*exp(20)^2+(6*x-6)*exp(20)+3*x^2-6*x+3),x,method=_RETURNVERBOSE)

[Out]

exp(-1/3*x*(x*ln(5)+2*x^2-3*exp(20)-4*x+3)/(exp(20)+x-1))

Fricas [A] (verification not implemented)

none

Time = 0.35 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=e^{\left (-\frac {2 \, x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 3 \, x e^{20} + 3 \, x}{3 \, {\left (x + e^{20} - 1\right )}}\right )} \]

[In]

integrate(((-2*x*exp(20)-x^2+2*x)*log(5)+3*exp(20)^2+(-6*x^2+8*x-6)*exp(20)-4*x^3+10*x^2-8*x+3)*exp((-x^2*log(
5)+3*x*exp(20)-2*x^3+4*x^2-3*x)/(3*exp(20)+3*x-3))/(3*exp(20)^2+(6*x-6)*exp(20)+3*x^2-6*x+3),x, algorithm="fri
cas")

[Out]

e^(-1/3*(2*x^3 + x^2*log(5) - 4*x^2 - 3*x*e^20 + 3*x)/(x + e^20 - 1))

Sympy [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.16 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=e^{\frac {- 2 x^{3} - x^{2} \log {\left (5 \right )} + 4 x^{2} - 3 x + 3 x e^{20}}{3 x - 3 + 3 e^{20}}} \]

[In]

integrate(((-2*x*exp(20)-x**2+2*x)*ln(5)+3*exp(20)**2+(-6*x**2+8*x-6)*exp(20)-4*x**3+10*x**2-8*x+3)*exp((-x**2
*ln(5)+3*x*exp(20)-2*x**3+4*x**2-3*x)/(3*exp(20)+3*x-3))/(3*exp(20)**2+(6*x-6)*exp(20)+3*x**2-6*x+3),x)

[Out]

exp((-2*x**3 - x**2*log(5) + 4*x**2 - 3*x + 3*x*exp(20))/(3*x - 3 + 3*exp(20)))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 115 vs. \(2 (22) = 44\).

Time = 0.74 (sec) , antiderivative size = 115, normalized size of antiderivative = 3.59 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=5^{\frac {1}{3} \, e^{20} - \frac {1}{3}} e^{\left (-\frac {2}{3} \, x^{2} + \frac {2}{3} \, x e^{20} - \frac {1}{3} \, x \log \left (5\right ) + \frac {2}{3} \, x - \frac {e^{40} \log \left (5\right )}{3 \, {\left (x + e^{20} - 1\right )}} + \frac {2 \, e^{20} \log \left (5\right )}{3 \, {\left (x + e^{20} - 1\right )}} + \frac {2 \, e^{60}}{3 \, {\left (x + e^{20} - 1\right )}} - \frac {5 \, e^{40}}{3 \, {\left (x + e^{20} - 1\right )}} + \frac {4 \, e^{20}}{3 \, {\left (x + e^{20} - 1\right )}} - \frac {\log \left (5\right )}{3 \, {\left (x + e^{20} - 1\right )}} - \frac {1}{3 \, {\left (x + e^{20} - 1\right )}} - \frac {2}{3} \, e^{40} + e^{20} - \frac {1}{3}\right )} \]

[In]

integrate(((-2*x*exp(20)-x^2+2*x)*log(5)+3*exp(20)^2+(-6*x^2+8*x-6)*exp(20)-4*x^3+10*x^2-8*x+3)*exp((-x^2*log(
5)+3*x*exp(20)-2*x^3+4*x^2-3*x)/(3*exp(20)+3*x-3))/(3*exp(20)^2+(6*x-6)*exp(20)+3*x^2-6*x+3),x, algorithm="max
ima")

[Out]

5^(1/3*e^20 - 1/3)*e^(-2/3*x^2 + 2/3*x*e^20 - 1/3*x*log(5) + 2/3*x - 1/3*e^40*log(5)/(x + e^20 - 1) + 2/3*e^20
*log(5)/(x + e^20 - 1) + 2/3*e^60/(x + e^20 - 1) - 5/3*e^40/(x + e^20 - 1) + 4/3*e^20/(x + e^20 - 1) - 1/3*log
(5)/(x + e^20 - 1) - 1/3/(x + e^20 - 1) - 2/3*e^40 + e^20 - 1/3)

Giac [A] (verification not implemented)

none

Time = 0.59 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.09 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=e^{\left (-\frac {2 \, x^{3} + x^{2} \log \left (5\right ) - 4 \, x^{2} - 3 \, x e^{20} + 3 \, x}{3 \, {\left (x + e^{20} - 1\right )}}\right )} \]

[In]

integrate(((-2*x*exp(20)-x^2+2*x)*log(5)+3*exp(20)^2+(-6*x^2+8*x-6)*exp(20)-4*x^3+10*x^2-8*x+3)*exp((-x^2*log(
5)+3*x*exp(20)-2*x^3+4*x^2-3*x)/(3*exp(20)+3*x-3))/(3*exp(20)^2+(6*x-6)*exp(20)+3*x^2-6*x+3),x, algorithm="gia
c")

[Out]

e^(-1/3*(2*x^3 + x^2*log(5) - 4*x^2 - 3*x*e^20 + 3*x)/(x + e^20 - 1))

Mupad [B] (verification not implemented)

Time = 14.73 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.69 \[ \int \frac {e^{\frac {-3 x+3 e^{20} x+4 x^2-2 x^3-x^2 \log (5)}{-3+3 e^{20}+3 x}} \left (3+3 e^{40}-8 x+10 x^2-4 x^3+e^{20} \left (-6+8 x-6 x^2\right )+\left (2 x-2 e^{20} x-x^2\right ) \log (5)\right )}{3+3 e^{40}-6 x+3 x^2+e^{20} (-6+6 x)} \, dx=\frac {{\mathrm {e}}^{-\frac {3\,x}{3\,x+3\,{\mathrm {e}}^{20}-3}}\,{\mathrm {e}}^{-\frac {2\,x^3}{3\,x+3\,{\mathrm {e}}^{20}-3}}\,{\mathrm {e}}^{\frac {4\,x^2}{3\,x+3\,{\mathrm {e}}^{20}-3}}\,{\mathrm {e}}^{\frac {3\,x\,{\mathrm {e}}^{20}}{3\,x+3\,{\mathrm {e}}^{20}-3}}}{5^{\frac {x^2}{3\,x+3\,{\mathrm {e}}^{20}-3}}} \]

[In]

int(-(exp(-(3*x - 3*x*exp(20) + x^2*log(5) - 4*x^2 + 2*x^3)/(3*x + 3*exp(20) - 3))*(8*x - 3*exp(40) + log(5)*(
2*x*exp(20) - 2*x + x^2) + exp(20)*(6*x^2 - 8*x + 6) - 10*x^2 + 4*x^3 - 3))/(3*exp(40) - 6*x + 3*x^2 + exp(20)
*(6*x - 6) + 3),x)

[Out]

(exp(-(3*x)/(3*x + 3*exp(20) - 3))*exp(-(2*x^3)/(3*x + 3*exp(20) - 3))*exp((4*x^2)/(3*x + 3*exp(20) - 3))*exp(
(3*x*exp(20))/(3*x + 3*exp(20) - 3)))/5^(x^2/(3*x + 3*exp(20) - 3))