\(\int e^{-\frac {5}{6}+e^x} (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x (30-30 x+6 x^2)) \, dx\) [7451]

Optimal result

Integrand size = 43, antiderivative size = 24 \[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=6 e^{-\frac {5}{6}+e^x} \left (-e^x+x\right ) \left (-5+e^x+x\right ) \]

[Out]

6*(exp(x)+x-5)*(x-exp(x))/exp(5/6)*exp(exp(x))

Rubi [F]

\[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=\int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx \]

[In]

Int[E^(-5/6 + E^x)*(-30 + 18*E^(2*x) - 6*E^(3*x) + 12*x + E^x*(30 - 30*x + 6*x^2)),x]

[Out]

30*E^(-5/6 + E^x + x) - 6*E^(-5/6 + E^x + 2*x) - (30*ExpIntegralEi[E^x])/E^(5/6) + 12*Defer[Int][E^(-5/6 + E^x
)*x, x] - 30*Defer[Int][E^(-5/6 + E^x + x)*x, x] + 6*Defer[Int][E^(-5/6 + E^x + x)*x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-30 e^{-\frac {5}{6}+e^x}+18 e^{-\frac {5}{6}+e^x+2 x}-6 e^{-\frac {5}{6}+e^x+3 x}+12 e^{-\frac {5}{6}+e^x} x+6 e^{-\frac {5}{6}+e^x+x} \left (5-5 x+x^2\right )\right ) \, dx \\ & = -\left (6 \int e^{-\frac {5}{6}+e^x+3 x} \, dx\right )+6 \int e^{-\frac {5}{6}+e^x+x} \left (5-5 x+x^2\right ) \, dx+12 \int e^{-\frac {5}{6}+e^x} x \, dx+18 \int e^{-\frac {5}{6}+e^x+2 x} \, dx-30 \int e^{-\frac {5}{6}+e^x} \, dx \\ & = 6 \int \left (5 e^{-\frac {5}{6}+e^x+x}-5 e^{-\frac {5}{6}+e^x+x} x+e^{-\frac {5}{6}+e^x+x} x^2\right ) \, dx-6 \text {Subst}\left (\int e^{-\frac {5}{6}+x} x^2 \, dx,x,e^x\right )+12 \int e^{-\frac {5}{6}+e^x} x \, dx+18 \text {Subst}\left (\int e^{-\frac {5}{6}+x} x \, dx,x,e^x\right )-30 \text {Subst}\left (\int \frac {e^{-\frac {5}{6}+x}}{x} \, dx,x,e^x\right ) \\ & = 18 e^{-\frac {5}{6}+e^x+x}-6 e^{-\frac {5}{6}+e^x+2 x}-\frac {30 \operatorname {ExpIntegralEi}\left (e^x\right )}{e^{5/6}}+6 \int e^{-\frac {5}{6}+e^x+x} x^2 \, dx+12 \int e^{-\frac {5}{6}+e^x} x \, dx+12 \text {Subst}\left (\int e^{-\frac {5}{6}+x} x \, dx,x,e^x\right )-18 \text {Subst}\left (\int e^{-\frac {5}{6}+x} \, dx,x,e^x\right )+30 \int e^{-\frac {5}{6}+e^x+x} \, dx-30 \int e^{-\frac {5}{6}+e^x+x} x \, dx \\ & = -18 e^{-\frac {5}{6}+e^x}+30 e^{-\frac {5}{6}+e^x+x}-6 e^{-\frac {5}{6}+e^x+2 x}-\frac {30 \operatorname {ExpIntegralEi}\left (e^x\right )}{e^{5/6}}+6 \int e^{-\frac {5}{6}+e^x+x} x^2 \, dx+12 \int e^{-\frac {5}{6}+e^x} x \, dx-12 \text {Subst}\left (\int e^{-\frac {5}{6}+x} \, dx,x,e^x\right )-30 \int e^{-\frac {5}{6}+e^x+x} x \, dx+30 \text {Subst}\left (\int e^{-\frac {5}{6}+x} \, dx,x,e^x\right ) \\ & = 30 e^{-\frac {5}{6}+e^x+x}-6 e^{-\frac {5}{6}+e^x+2 x}-\frac {30 \operatorname {ExpIntegralEi}\left (e^x\right )}{e^{5/6}}+6 \int e^{-\frac {5}{6}+e^x+x} x^2 \, dx+12 \int e^{-\frac {5}{6}+e^x} x \, dx-30 \int e^{-\frac {5}{6}+e^x+x} x \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.71 \[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=-6 e^{-\frac {5}{3}+e^x} \left (-5 e^{\frac {5}{6}+x}+e^{\frac {5}{6}+2 x}-e^{5/6} (-5+x) x\right ) \]

[In]

Integrate[E^(-5/6 + E^x)*(-30 + 18*E^(2*x) - 6*E^(3*x) + 12*x + E^x*(30 - 30*x + 6*x^2)),x]

[Out]

-6*E^(-5/3 + E^x)*(-5*E^(5/6 + x) + E^(5/6 + 2*x) - E^(5/6)*(-5 + x)*x)

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
risch	\(\left (6 x^{2}-6 \,{\mathrm e}^{2 x}-30 x +30 \,{\mathrm e}^{x}\right ) {\mathrm e}^{-\frac {5}{6}+{\mathrm e}^{x}}\)	\(26\)
parallelrisch	\({\mathrm e}^{-\frac {5}{6}} \left (6 \,{\mathrm e}^{{\mathrm e}^{x}} x^{2}-6 \,{\mathrm e}^{{\mathrm e}^{x}} {\mathrm e}^{2 x}-30 x \,{\mathrm e}^{{\mathrm e}^{x}}+30 \,{\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}\right )\)	\(37\)
norman	\(-30 \,{\mathrm e}^{-\frac {5}{6}} x \,{\mathrm e}^{{\mathrm e}^{x}}+6 \,{\mathrm e}^{-\frac {5}{6}} x^{2} {\mathrm e}^{{\mathrm e}^{x}}+30 \,{\mathrm e}^{-\frac {5}{6}} {\mathrm e}^{x} {\mathrm e}^{{\mathrm e}^{x}}-6 \,{\mathrm e}^{-\frac {5}{6}} {\mathrm e}^{2 x} {\mathrm e}^{{\mathrm e}^{x}}\)	\(48\)

[In]

int((-6*exp(x)^3+18*exp(x)^2+(6*x^2-30*x+30)*exp(x)+12*x-30)*exp(exp(x))/exp(5/6),x,method=_RETURNVERBOSE)

[Out]

(6*x^2-6*exp(2*x)-30*x+30*exp(x))*exp(-5/6+exp(x))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=6 \, {\left (x^{2} - 5 \, x - e^{\left (2 \, x\right )} + 5 \, e^{x}\right )} e^{\left (e^{x} - \frac {5}{6}\right )} \]

[In]

integrate((-6*exp(x)^3+18*exp(x)^2+(6*x^2-30*x+30)*exp(x)+12*x-30)*exp(exp(x))/exp(5/6),x, algorithm="fricas")

[Out]

6*(x^2 - 5*x - e^(2*x) + 5*e^x)*e^(e^x - 5/6)

Sympy [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.21 \[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=\frac {\left (6 x^{2} - 30 x - 6 e^{2 x} + 30 e^{x}\right ) e^{e^{x}}}{e^{\frac {5}{6}}} \]

[In]

integrate((-6*exp(x)**3+18*exp(x)**2+(6*x**2-30*x+30)*exp(x)+12*x-30)*exp(exp(x))/exp(5/6),x)

[Out]

(6*x**2 - 30*x - 6*exp(2*x) + 30*exp(x))*exp(-5/6)*exp(exp(x))

Maxima [F]

\[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=\int { 6 \, {\left ({\left (x^{2} - 5 \, x + 5\right )} e^{x} + 2 \, x - e^{\left (3 \, x\right )} + 3 \, e^{\left (2 \, x\right )} - 5\right )} e^{\left (e^{x} - \frac {5}{6}\right )} \,d x } \]

[In]

integrate((-6*exp(x)^3+18*exp(x)^2+(6*x^2-30*x+30)*exp(x)+12*x-30)*exp(exp(x))/exp(5/6),x, algorithm="maxima")

[Out]

-30*Ei(e^x)*e^(-5/6) + 6*(x^2*e^(1/6) - 5*x*e^(1/6) - e^(2*x + 1/6) + 5*e^(x + 1/6))*e^(e^x - 1) + 30*integrat
e(e^(e^x - 5/6), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 50 vs. \(2 (18) = 36\).

Time = 0.26 (sec) , antiderivative size = 50, normalized size of antiderivative = 2.08 \[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=6 \, {\left (x^{2} e^{\left (3 \, x + e^{x} - \frac {5}{6}\right )} - 5 \, x e^{\left (3 \, x + e^{x} - \frac {5}{6}\right )} - e^{\left (5 \, x + e^{x} - \frac {5}{6}\right )} + 5 \, e^{\left (4 \, x + e^{x} - \frac {5}{6}\right )}\right )} e^{\left (-3 \, x\right )} \]

[In]

integrate((-6*exp(x)^3+18*exp(x)^2+(6*x^2-30*x+30)*exp(x)+12*x-30)*exp(exp(x))/exp(5/6),x, algorithm="giac")

[Out]

6*(x^2*e^(3*x + e^x - 5/6) - 5*x*e^(3*x + e^x - 5/6) - e^(5*x + e^x - 5/6) + 5*e^(4*x + e^x - 5/6))*e^(-3*x)

Mupad [B] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int e^{-\frac {5}{6}+e^x} \left (-30+18 e^{2 x}-6 e^{3 x}+12 x+e^x \left (30-30 x+6 x^2\right )\right ) \, dx=-{\mathrm {e}}^{{\mathrm {e}}^x-\frac {5}{6}}\,\left (30\,x+6\,{\mathrm {e}}^{2\,x}-30\,{\mathrm {e}}^x-6\,x^2\right ) \]

[In]

int(exp(exp(x))*exp(-5/6)*(12*x + 18*exp(2*x) - 6*exp(3*x) + exp(x)*(6*x^2 - 30*x + 30) - 30),x)

[Out]

-exp(exp(x) - 5/6)*(30*x + 6*exp(2*x) - 30*exp(x) - 6*x^2)