\(\int e^{-2+e^x} ((216+432 x+288 x^2+64 x^3) \log (9)+e^x (81+216 x+216 x^2+96 x^3+16 x^4) \log (9)) \, dx\) [3848]

Optimal result

Integrand size = 53, antiderivative size = 17 \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=e^{-2+e^x} (3+2 x)^4 \log (9) \]

[Out]

2*exp(exp(x))*ln(3)*(3+2*x)^4/exp(2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.019, Rules used = {2326} \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=e^{e^x-2} \left (16 x^4+96 x^3+216 x^2+216 x+81\right ) \log (9) \]

[In]

Int[E^(-2 + E^x)*((216 + 432*x + 288*x^2 + 64*x^3)*Log[9] + E^x*(81 + 216*x + 216*x^2 + 96*x^3 + 16*x^4)*Log[9
]),x]

[Out]

E^(-2 + E^x)*(81 + 216*x + 216*x^2 + 96*x^3 + 16*x^4)*Log[9]

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}& = e^{-2+e^x} \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=e^{-2+e^x} (3+2 x)^4 \log (9) \]

[In]

Integrate[E^(-2 + E^x)*((216 + 432*x + 288*x^2 + 64*x^3)*Log[9] + E^x*(81 + 216*x + 216*x^2 + 96*x^3 + 16*x^4)
*Log[9]),x]

[Out]

E^(-2 + E^x)*(3 + 2*x)^4*Log[9]

Maple [A] (verified)

Time = 1.70 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.76

method	result	size
risch	\(2 \left (16 x^{4}+96 x^{3}+216 x^{2}+216 x +81\right ) \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{x}-2}\)	\(30\)
parallelrisch	\({\mathrm e}^{-2} \left (162 \,{\mathrm e}^{{\mathrm e}^{x}} \ln \left (3\right )+432 \,{\mathrm e}^{{\mathrm e}^{x}} x \ln \left (3\right )+432 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2} \ln \left (3\right )+192 \,{\mathrm e}^{{\mathrm e}^{x}} x^{3} \ln \left (3\right )+32 \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{x}} x^{4}\right )\)	\(52\)
norman	\(162 \,{\mathrm e}^{-2} \ln \left (3\right ) {\mathrm e}^{{\mathrm e}^{x}}+432 \,{\mathrm e}^{-2} \ln \left (3\right ) x \,{\mathrm e}^{{\mathrm e}^{x}}+432 \,{\mathrm e}^{-2} \ln \left (3\right ) x^{2} {\mathrm e}^{{\mathrm e}^{x}}+192 \,{\mathrm e}^{-2} \ln \left (3\right ) x^{3} {\mathrm e}^{{\mathrm e}^{x}}+32 \,{\mathrm e}^{-2} \ln \left (3\right ) x^{4} {\mathrm e}^{{\mathrm e}^{x}}\)	\(67\)

[In]

int((2*(16*x^4+96*x^3+216*x^2+216*x+81)*ln(3)*exp(x)+2*(64*x^3+288*x^2+432*x+216)*ln(3))*exp(exp(x))/exp(2),x,
method=_RETURNVERBOSE)

[Out]

2*(16*x^4+96*x^3+216*x^2+216*x+81)*ln(3)*exp(exp(x)-2)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.71 \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=2 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} e^{\left (e^{x} - 2\right )} \log \left (3\right ) \]

[In]

integrate((2*(16*x^4+96*x^3+216*x^2+216*x+81)*log(3)*exp(x)+2*(64*x^3+288*x^2+432*x+216)*log(3))*exp(exp(x))/e
xp(2),x, algorithm="fricas")

[Out]

2*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*e^(e^x - 2)*log(3)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 44 vs. \(2 (19) = 38\).

Time = 0.16 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.59 \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=\frac {\left (32 x^{4} \log {\left (3 \right )} + 192 x^{3} \log {\left (3 \right )} + 432 x^{2} \log {\left (3 \right )} + 432 x \log {\left (3 \right )} + 162 \log {\left (3 \right )}\right ) e^{e^{x}}}{e^{2}} \]

[In]

integrate((2*(16*x**4+96*x**3+216*x**2+216*x+81)*ln(3)*exp(x)+2*(64*x**3+288*x**2+432*x+216)*ln(3))*exp(exp(x)
)/exp(2),x)

[Out]

(32*x**4*log(3) + 192*x**3*log(3) + 432*x**2*log(3) + 432*x*log(3) + 162*log(3))*exp(-2)*exp(exp(x))

Maxima [F]

\[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=\int { 2 \, {\left ({\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} e^{x} \log \left (3\right ) + 8 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (3\right )\right )} e^{\left (e^{x} - 2\right )} \,d x } \]

[In]

integrate((2*(16*x^4+96*x^3+216*x^2+216*x+81)*log(3)*exp(x)+2*(64*x^3+288*x^2+432*x+216)*log(3))*exp(exp(x))/e
xp(2),x, algorithm="maxima")

[Out]

432*Ei(e^x)*e^(-2)*log(3) - 432*e^(-2)*integrate(e^(e^x), x)*log(3) + 2*(16*x^4*log(3) + 96*x^3*log(3) + 216*x
^2*log(3) + 216*x*log(3) + 81*log(3))*e^(e^x - 2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (16) = 32\).

Time = 0.26 (sec) , antiderivative size = 64, normalized size of antiderivative = 3.76 \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=2 \, {\left (16 \, x^{4} e^{\left (x + e^{x}\right )} \log \left (3\right ) + 96 \, x^{3} e^{\left (x + e^{x}\right )} \log \left (3\right ) + 216 \, x^{2} e^{\left (x + e^{x}\right )} \log \left (3\right ) + 216 \, x e^{\left (x + e^{x}\right )} \log \left (3\right ) + 81 \, e^{\left (x + e^{x}\right )} \log \left (3\right )\right )} e^{\left (-x - 2\right )} \]

[In]

integrate((2*(16*x^4+96*x^3+216*x^2+216*x+81)*log(3)*exp(x)+2*(64*x^3+288*x^2+432*x+216)*log(3))*exp(exp(x))/e
xp(2),x, algorithm="giac")

[Out]

2*(16*x^4*e^(x + e^x)*log(3) + 96*x^3*e^(x + e^x)*log(3) + 216*x^2*e^(x + e^x)*log(3) + 216*x*e^(x + e^x)*log(
3) + 81*e^(x + e^x)*log(3))*e^(-x - 2)

Mupad [B] (verification not implemented)

Time = 9.53 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.94 \[ \int e^{-2+e^x} \left (\left (216+432 x+288 x^2+64 x^3\right ) \log (9)+e^x \left (81+216 x+216 x^2+96 x^3+16 x^4\right ) \log (9)\right ) \, dx=2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-2}\,\ln \left (3\right )\,{\left (2\,x+3\right )}^4 \]

[In]

int(exp(exp(x))*exp(-2)*(2*log(3)*(432*x + 288*x^2 + 64*x^3 + 216) + 2*exp(x)*log(3)*(216*x + 216*x^2 + 96*x^3
+ 16*x^4 + 81)),x)

[Out]

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