\(\int \frac {e^{\frac {-x-3 x^4+(-x+x^4) \log (4+4 x)}{-3+\log (4+4 x)}} (3+7 x+36 x^3+36 x^4+(2+2 x-24 x^3-24 x^4) \log (4+4 x)+(-1-x+4 x^3+4 x^4) \log ^2(4+4 x))}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx\) [8811]

Optimal result

Integrand size = 129, antiderivative size = 22 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{-x+x^4-\frac {4 x}{-3+\log (4+4 x)}} \]

[Out]

exp(x^4-4/(ln(4+4*x)-3)*x-x)

Rubi [F]

\[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx \]

[In]

Int[(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(3 + 7*x + 36*x^3 + 36*x^4 + (2 + 2*x - 24
*x^3 - 24*x^4)*Log[4 + 4*x] + (-1 - x + 4*x^3 + 4*x^4)*Log[4 + 4*x]^2))/(9 + 9*x + (-6 - 6*x)*Log[4 + 4*x] + (
1 + x)*Log[4 + 4*x]^2),x]

[Out]

-29*Defer[Int][E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))/(-3 + Log[4*(1 + x)])^2, x] - 4*
Defer[Int][E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))/((1 + x)*(-3 + Log[4*(1 + x)])^2), x
] + 108*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(1 + x))/(-3 + Log[4*(1 + x
)])^2, x] - 108*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(1 + x)^2)/(-3 + Lo
g[4*(1 + x)])^2, x] + 36*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(1 + x)^3)
/(-3 + Log[4*(1 + x)])^2, x] + 2*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*Lo
g[4 + 4*x])/(3 - Log[4*(1 + x)])^2, x] - 24*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4
+ 4*x]))*x^3*Log[4 + 4*x])/(3 - Log[4*(1 + x)])^2, x] - Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/
(-3 + Log[4 + 4*x]))*Log[4 + 4*x]^2)/(3 - Log[4*(1 + x)])^2, x] + 4*Defer[Int][(E^((-x - 3*x^4 + (-x + x^4)*Lo
g[4 + 4*x])/(-3 + Log[4 + 4*x]))*x^3*Log[4 + 4*x]^2)/(3 - Log[4*(1 + x)])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{(1+x) (3-\log (4 (1+x)))^2} \, dx \\ & = \int \left (\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {7 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {36 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {36 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^4}{(1+x) (-3+\log (4 (1+x)))^2}+\frac {2 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (1-12 x^3\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2}+\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (-1+4 x^3\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (1-12 x^3\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^4}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \left (-1+4 x^3\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ & = 2 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \log (4+4 x)}{(3-\log (4 (1+x)))^2}-\frac {12 e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx+3 \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+7 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+36 \int \left (\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^2}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+36 \int \left (-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x}{(-3+\log (4 (1+x)))^2}-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^2}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3}{(-3+\log (4 (1+x)))^2}+\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}}}{(1+x) (-3+\log (4 (1+x)))^2}\right ) \, dx+\int \left (-\frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}+\frac {4 e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2}\right ) \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3}{(-3+\log (4 (1+x)))^2} \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+36 \int \left (-\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2}+\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)}{(-3+\log (4 (1+x)))^2}-\frac {3 \exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^2}{(-3+\log (4 (1+x)))^2}+\frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^3}{(-3+\log (4 (1+x)))^2}\right ) \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ & = 2 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+3 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx+4 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx+7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx-7 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(1+x) (-3+\log (4 (1+x)))^2} \, dx-24 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) x^3 \log (4+4 x)}{(3-\log (4 (1+x)))^2} \, dx-36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right )}{(-3+\log (4 (1+x)))^2} \, dx+36 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^3}{(-3+\log (4 (1+x)))^2} \, dx+108 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)}{(-3+\log (4 (1+x)))^2} \, dx-108 \int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) (1+x)^2}{(-3+\log (4 (1+x)))^2} \, dx-\int \frac {\exp \left (\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}\right ) \log ^2(4+4 x)}{(3-\log (4 (1+x)))^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.25 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.50 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\frac {x \left (-1-3 x^3+\left (-1+x^3\right ) \log (4 (1+x))\right )}{-3+\log (4 (1+x))}} \]

[In]

Integrate[(E^((-x - 3*x^4 + (-x + x^4)*Log[4 + 4*x])/(-3 + Log[4 + 4*x]))*(3 + 7*x + 36*x^3 + 36*x^4 + (2 + 2*
x - 24*x^3 - 24*x^4)*Log[4 + 4*x] + (-1 - x + 4*x^3 + 4*x^4)*Log[4 + 4*x]^2))/(9 + 9*x + (-6 - 6*x)*Log[4 + 4*
x] + (1 + x)*Log[4 + 4*x]^2),x]

[Out]

E^((x*(-1 - 3*x^3 + (-1 + x^3)*Log[4*(1 + x)]))/(-3 + Log[4*(1 + x)]))

Maple [A] (verified)

Time = 0.81 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.64

method	result	size
parallelrisch	\({\mathrm e}^{\frac {\left (x^{4}-x \right ) \ln \left (4+4 x \right )-3 x^{4}-x}{\ln \left (4+4 x \right )-3}}\)	\(36\)
risch	\({\mathrm e}^{\frac {x \left (\ln \left (4+4 x \right ) x^{3}-3 x^{3}-\ln \left (4+4 x \right )-1\right )}{\ln \left (4+4 x \right )-3}}\)	\(39\)

[In]

int(((4*x^4+4*x^3-x-1)*ln(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*ln(4+4*x)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*ln(4+4*x
)-3*x^4-x)/(ln(4+4*x)-3))/((1+x)*ln(4+4*x)^2+(-6*x-6)*ln(4+4*x)+9*x+9),x,method=_RETURNVERBOSE)

[Out]

exp(((x^4-x)*ln(4+4*x)-3*x^4-x)/(ln(4+4*x)-3))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.59 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (-\frac {3 \, x^{4} - {\left (x^{4} - x\right )} \log \left (4 \, x + 4\right ) + x}{\log \left (4 \, x + 4\right ) - 3}\right )} \]

[In]

integrate(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*
log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/((1+x)*log(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.23 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\frac {- 3 x^{4} - x + \left (x^{4} - x\right ) \log {\left (4 x + 4 \right )}}{\log {\left (4 x + 4 \right )} - 3}} \]

[In]

integrate(((4*x**4+4*x**3-x-1)*ln(4+4*x)**2+(-24*x**4-24*x**3+2*x+2)*ln(4+4*x)+36*x**4+36*x**3+7*x+3)*exp(((x*
*4-x)*ln(4+4*x)-3*x**4-x)/(ln(4+4*x)-3))/((1+x)*ln(4+4*x)**2+(-6*x-6)*ln(4+4*x)+9*x+9),x)

[Out]

exp((-3*x**4 - x + (x**4 - x)*log(4*x + 4))/(log(4*x + 4) - 3))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (21) = 42\).

Time = 0.43 (sec) , antiderivative size = 109, normalized size of antiderivative = 4.95 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (\frac {2 \, x^{4} \log \left (2\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} + \frac {x^{4} \log \left (x + 1\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {3 \, x^{4}}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {2 \, x \log \left (2\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {x \log \left (x + 1\right )}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3} - \frac {x}{2 \, \log \left (2\right ) + \log \left (x + 1\right ) - 3}\right )} \]

[In]

integrate(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*
log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/((1+x)*log(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x, algorithm="maxima")

[Out]

e^(2*x^4*log(2)/(2*log(2) + log(x + 1) - 3) + x^4*log(x + 1)/(2*log(2) + log(x + 1) - 3) - 3*x^4/(2*log(2) + l
og(x + 1) - 3) - 2*x*log(2)/(2*log(2) + log(x + 1) - 3) - x*log(x + 1)/(2*log(2) + log(x + 1) - 3) - x/(2*log(
2) + log(x + 1) - 3))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (21) = 42\).

Time = 0.83 (sec) , antiderivative size = 69, normalized size of antiderivative = 3.14 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=e^{\left (\frac {x^{4} \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {3 \, x^{4}}{\log \left (4 \, x + 4\right ) - 3} - \frac {x \log \left (4 \, x + 4\right )}{\log \left (4 \, x + 4\right ) - 3} - \frac {x}{\log \left (4 \, x + 4\right ) - 3}\right )} \]

[In]

integrate(((4*x^4+4*x^3-x-1)*log(4+4*x)^2+(-24*x^4-24*x^3+2*x+2)*log(4+4*x)+36*x^4+36*x^3+7*x+3)*exp(((x^4-x)*
log(4+4*x)-3*x^4-x)/(log(4+4*x)-3))/((1+x)*log(4+4*x)^2+(-6*x-6)*log(4+4*x)+9*x+9),x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 15.84 (sec) , antiderivative size = 57, normalized size of antiderivative = 2.59 \[ \int \frac {e^{\frac {-x-3 x^4+\left (-x+x^4\right ) \log (4+4 x)}{-3+\log (4+4 x)}} \left (3+7 x+36 x^3+36 x^4+\left (2+2 x-24 x^3-24 x^4\right ) \log (4+4 x)+\left (-1-x+4 x^3+4 x^4\right ) \log ^2(4+4 x)\right )}{9+9 x+(-6-6 x) \log (4+4 x)+(1+x) \log ^2(4+4 x)} \, dx=\frac {{\mathrm {e}}^{-\frac {x}{\ln \left (4\,x+4\right )-3}}\,{\mathrm {e}}^{-\frac {3\,x^4}{\ln \left (4\,x+4\right )-3}}}{{\left (4\,x+4\right )}^{\frac {x-x^4}{\ln \left (4\,x+4\right )-3}}} \]

[In]

int((exp(-(x + log(4*x + 4)*(x - x^4) + 3*x^4)/(log(4*x + 4) - 3))*(7*x + log(4*x + 4)*(2*x - 24*x^3 - 24*x^4
+ 2) - log(4*x + 4)^2*(x - 4*x^3 - 4*x^4 + 1) + 36*x^3 + 36*x^4 + 3))/(9*x - log(4*x + 4)*(6*x + 6) + log(4*x
+ 4)^2*(x + 1) + 9),x)

[Out]

(exp(-x/(log(4*x + 4) - 3))*exp(-(3*x^4)/(log(4*x + 4) - 3)))/(4*x + 4)^((x - x^4)/(log(4*x + 4) - 3))