\(\int e^{(4+x)^{x^2} (-10 x+10 x^3)} (4+x)^{-1+x^2} (-40-10 x+120 x^2+20 x^3+10 x^5+(-80 x^2-20 x^3+80 x^4+20 x^5) \log (4+x)) \, dx\) [6789]

Optimal result

Integrand size = 75, antiderivative size = 21 \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=e^{5 (4+x)^{x^2} (-2+2 x) \left (x+x^2\right )} \]

[Out]

exp(exp(x^2*ln(4+x))*(10*x-10)*(x^2+x))

Rubi [F]

\[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=\int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx \]

[In]

Int[E^((4 + x)^x^2*(-10*x + 10*x^3))*(4 + x)^(-1 + x^2)*(-40 - 10*x + 120*x^2 + 20*x^3 + 10*x^5 + (-80*x^2 - 2
0*x^3 + 80*x^4 + 20*x^5)*Log[4 + x]),x]

[Out]

-20*Log[4 + x]*Defer[Int][E^(10*x*(4 + x)^x^2*(-1 + x^2))*x^2*(4 + x)^x^2, x] + 20*Log[4 + x]*Defer[Int][E^(10
*x*(4 + x)^x^2*(-1 + x^2))*x^4*(4 + x)^x^2, x] - 40*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*(4 + x)^(-1 +
x^2), x] - 10*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*x*(4 + x)^(-1 + x^2), x] + 120*Defer[Int][E^(x*(4 +
x)^x^2*(-10 + 10*x^2))*x^2*(4 + x)^(-1 + x^2), x] + 20*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*x^3*(4 + x)
^(-1 + x^2), x] + 10*Defer[Int][E^(x*(4 + x)^x^2*(-10 + 10*x^2))*x^5*(4 + x)^(-1 + x^2), x] + 20*Defer[Int][De
fer[Int][E^(10*x*(4 + x)^x^2*(-1 + x^2))*x^2*(4 + x)^x^2, x]/(4 + x), x] - 20*Defer[Int][Defer[Int][E^(10*x*(4
+ x)^x^2*(-1 + x^2))*x^4*(4 + x)^x^2, x]/(4 + x), x]

Rubi steps \begin{align*} \text {integral}& = \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx \\ & = \int \left (-40 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2}-10 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2}+120 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2}+20 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2}+10 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2}+20 e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \left (-4-x+4 x^2+x^3\right ) \log (4+x)\right ) \, dx \\ & = -\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \left (-4-x+4 x^2+x^3\right ) \log (4+x) \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx \\ & = -\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx-20 \int \frac {-\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx}{4+x} \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx-(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx \\ & = -\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx-20 \int \left (-\frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx}{4+x}+\frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx}{4+x}\right ) \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx-(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx \\ & = -\left (10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x (4+x)^{-1+x^2} \, dx\right )+10 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^5 (4+x)^{-1+x^2} \, dx+20 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^3 (4+x)^{-1+x^2} \, dx+20 \int \frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx}{4+x} \, dx-20 \int \frac {\int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx}{4+x} \, dx-40 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} (4+x)^{-1+x^2} \, dx+120 \int e^{x (4+x)^{x^2} \left (-10+10 x^2\right )} x^2 (4+x)^{-1+x^2} \, dx-(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^2 (4+x)^{x^2} \, dx+(20 \log (4+x)) \int e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} x^4 (4+x)^{x^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 5.05 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=e^{10 x (4+x)^{x^2} \left (-1+x^2\right )} \]

[In]

Integrate[E^((4 + x)^x^2*(-10*x + 10*x^3))*(4 + x)^(-1 + x^2)*(-40 - 10*x + 120*x^2 + 20*x^3 + 10*x^5 + (-80*x
^2 - 20*x^3 + 80*x^4 + 20*x^5)*Log[4 + x]),x]

[Out]

E^(10*x*(4 + x)^x^2*(-1 + x^2))

Maple [A] (verified)

Time = 1.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.86

method	result	size
risch	\({\mathrm e}^{10 x \left (-1+x \right ) \left (1+x \right ) \left (4+x \right )^{x^{2}}}\)	\(18\)
parallelrisch	\({\mathrm e}^{\left (10 x^{3}-10 x \right ) {\mathrm e}^{x^{2} \ln \left (4+x \right )}}\)	\(21\)

[In]

int(((20*x^5+80*x^4-20*x^3-80*x^2)*ln(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*ln(4+x))*exp((10*x^3-10*x)*e
xp(x^2*ln(4+x)))/(4+x),x,method=_RETURNVERBOSE)

[Out]

exp(10*x*(-1+x)*(1+x)*(4+x)^(x^2))

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.81 \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=e^{\left (10 \, {\left (x^{3} - x\right )} {\left (x + 4\right )}^{\left (x^{2}\right )}\right )} \]

[In]

integrate(((20*x^5+80*x^4-20*x^3-80*x^2)*log(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*log(4+x))*exp((10*x^3
-10*x)*exp(x^2*log(4+x)))/(4+x),x, algorithm="fricas")

[Out]

e^(10*(x^3 - x)*(x + 4)^(x^2))

Sympy [A] (verification not implemented)

Time = 0.91 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=e^{\left (10 x^{3} - 10 x\right ) e^{x^{2} \log {\left (x + 4 \right )}}} \]

[In]

integrate(((20*x**5+80*x**4-20*x**3-80*x**2)*ln(4+x)+10*x**5+20*x**3+120*x**2-10*x-40)*exp(x**2*ln(4+x))*exp((
10*x**3-10*x)*exp(x**2*ln(4+x)))/(4+x),x)

[Out]

exp((10*x**3 - 10*x)*exp(x**2*log(x + 4)))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=e^{\left (10 \, {\left (x + 4\right )}^{\left (x^{2}\right )} x^{3} - 10 \, {\left (x + 4\right )}^{\left (x^{2}\right )} x\right )} \]

[In]

integrate(((20*x^5+80*x^4-20*x^3-80*x^2)*log(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*log(4+x))*exp((10*x^3
-10*x)*exp(x^2*log(4+x)))/(4+x),x, algorithm="maxima")

[Out]

e^(10*(x + 4)^(x^2)*x^3 - 10*(x + 4)^(x^2)*x)

Giac [F(-1)]

Timed out. \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx=\text {Timed out} \]

[In]

integrate(((20*x^5+80*x^4-20*x^3-80*x^2)*log(4+x)+10*x^5+20*x^3+120*x^2-10*x-40)*exp(x^2*log(4+x))*exp((10*x^3
-10*x)*exp(x^2*log(4+x)))/(4+x),x, algorithm="giac")

[Out]

Timed out

Mupad [B] (verification not implemented)

Time = 11.82 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int e^{(4+x)^{x^2} \left (-10 x+10 x^3\right )} (4+x)^{-1+x^2} \left (-40-10 x+120 x^2+20 x^3+10 x^5+\left (-80 x^2-20 x^3+80 x^4+20 x^5\right ) \log (4+x)\right ) \, dx={\mathrm {e}}^{-10\,x\,{\left (x+4\right )}^{x^2}}\,{\mathrm {e}}^{10\,x^3\,{\left (x+4\right )}^{x^2}} \]

[In]

int(-(exp(-exp(x^2*log(x + 4))*(10*x - 10*x^3))*exp(x^2*log(x + 4))*(10*x - 120*x^2 - 20*x^3 - 10*x^5 + log(x
+ 4)*(80*x^2 + 20*x^3 - 80*x^4 - 20*x^5) + 40))/(x + 4),x)

[Out]

exp(-10*x*(x + 4)^(x^2))*exp(10*x^3*(x + 4)^(x^2))