\(\int e^{-e^{1+x^2}+e (5+2 x)} (7+2 x+e (14 x+2 x^2)+e^{1+x^2} (-14 x^2-2 x^3)) \, dx\) [7666]

Optimal result

Integrand size = 55, antiderivative size = 21 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx=e^{e \left (5-e^{x^2}+2 x\right )} x (7+x) \]

[Out]

x*exp((2*x+5-exp(x^2))*exp(1))*(x+7)

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(62\) vs. \(2(21)=42\).

Time = 0.10 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.95, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.018, Rules used = {2326} \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx=\frac {e^{e (2 x+5)-e^{x^2+1}} \left (e \left (x^2+7 x\right )-e^{x^2+1} \left (x^3+7 x^2\right )\right )}{e-e^{x^2+1} x} \]

[In]

Int[E^(-E^(1 + x^2) + E*(5 + 2*x))*(7 + 2*x + E*(14*x + 2*x^2) + E^(1 + x^2)*(-14*x^2 - 2*x^3)),x]

[Out]

(E^(-E^(1 + x^2) + E*(5 + 2*x))*(E*(7*x + x^2) - E^(1 + x^2)*(7*x^2 + x^3)))/(E - E^(1 + x^2)*x)

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps \begin{align*} \text {integral}& = \frac {e^{-e^{1+x^2}+e (5+2 x)} \left (e \left (7 x+x^2\right )-e^{1+x^2} \left (7 x^2+x^3\right )\right )}{e-e^{1+x^2} x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.92 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx=e^{-e \left (-5+e^{x^2}-2 x\right )} x (7+x) \]

[In]

Integrate[E^(-E^(1 + x^2) + E*(5 + 2*x))*(7 + 2*x + E*(14*x + 2*x^2) + E^(1 + x^2)*(-14*x^2 - 2*x^3)),x]

[Out]

(x*(7 + x))/E^(E*(-5 + E^x^2 - 2*x))

Maple [A] (verified)

Time = 0.27 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19

method	result	size
risch	\(\left (x +7\right ) x \,{\mathrm e}^{2 x \,{\mathrm e}+5 \,{\mathrm e}-{\mathrm e}^{x^{2}+1}}\)	\(25\)
parallelrisch	\({\mathrm e}^{-{\mathrm e} \left ({\mathrm e}^{x^{2}}-2 x -5\right )} x^{2}+7 \,{\mathrm e}^{-{\mathrm e} \left ({\mathrm e}^{x^{2}}-2 x -5\right )} x\)	\(37\)
norman	\(x^{2} {\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{x^{2}}+\left (5+2 x \right ) {\mathrm e}}+7 x \,{\mathrm e}^{-{\mathrm e} \,{\mathrm e}^{x^{2}}+\left (5+2 x \right ) {\mathrm e}}\)	\(45\)

[In]

int(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,method=
_RETURNVERBOSE)

[Out]

(x+7)*x*exp(2*x*exp(1)+5*exp(1)-exp(x^2+1))

Fricas [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.24 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx={\left (x^{2} + 7 \, x\right )} e^{\left ({\left (2 \, x + 5\right )} e - e^{\left (x^{2} + 1\right )}\right )} \]

[In]

integrate(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,
algorithm="fricas")

[Out]

(x^2 + 7*x)*e^((2*x + 5)*e - e^(x^2 + 1))

Sympy [A] (verification not implemented)

Time = 0.53 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx=\left (x^{2} + 7 x\right ) e^{e \left (2 x + 5\right ) - e e^{x^{2}}} \]

[In]

integrate(((-2*x**3-14*x**2)*exp(1)*exp(x**2)+(2*x**2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x**2)+(5+2*x)*exp(1)
),x)

[Out]

(x**2 + 7*x)*exp(E*(2*x + 5) - E*exp(x**2))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx={\left (x^{2} e^{\left (5 \, e\right )} + 7 \, x e^{\left (5 \, e\right )}\right )} e^{\left (2 \, x e - e^{\left (x^{2} + 1\right )}\right )} \]

[In]

integrate(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,
algorithm="maxima")

[Out]

(x^2*e^(5*e) + 7*x*e^(5*e))*e^(2*x*e - e^(x^2 + 1))

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (20) = 40\).

Time = 0.31 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.19 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx=x^{2} e^{\left (2 \, x e + 5 \, e - e^{\left (x^{2} + 1\right )}\right )} + 7 \, x e^{\left (2 \, x e + 5 \, e - e^{\left (x^{2} + 1\right )}\right )} \]

[In]

integrate(((-2*x^3-14*x^2)*exp(1)*exp(x^2)+(2*x^2+14*x)*exp(1)+7+2*x)*exp(-exp(1)*exp(x^2)+(5+2*x)*exp(1)),x,
algorithm="giac")

[Out]

x^2*e^(2*x*e + 5*e - e^(x^2 + 1)) + 7*x*e^(2*x*e + 5*e - e^(x^2 + 1))

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int e^{-e^{1+x^2}+e (5+2 x)} \left (7+2 x+e \left (14 x+2 x^2\right )+e^{1+x^2} \left (-14 x^2-2 x^3\right )\right ) \, dx=x\,{\mathrm {e}}^{5\,\mathrm {e}}\,{\mathrm {e}}^{-{\mathrm {e}}^{x^2}\,\mathrm {e}}\,{\mathrm {e}}^{2\,x\,\mathrm {e}}\,\left (x+7\right ) \]

[In]

int(exp(exp(1)*(2*x + 5) - exp(x^2)*exp(1))*(2*x + exp(1)*(14*x + 2*x^2) - exp(x^2)*exp(1)*(14*x^2 + 2*x^3) +
7),x)

[Out]

x*exp(5*exp(1))*exp(-exp(x^2)*exp(1))*exp(2*x*exp(1))*(x + 7)