\(\int (6-2 e^x+e^{x-4 e^{25} x} (-2+8 e^{25})) \, dx\) [7503]

Optimal result

Integrand size = 25, antiderivative size = 27 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=2 x+2 \left (-e^x-e^{x-4 e^{25} x}+2 x\right ) \]

[Out]

6*x-2*exp(x)-2*exp(-4*x*exp(25)+x)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2225, 2259} \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=6 x-2 e^x-2 e^{\left (1-4 e^{25}\right ) x} \]

[In]

Int[6 - 2*E^x + E^(x - 4*E^25*x)*(-2 + 8*E^25),x]

[Out]

-2*E^x - 2*E^((1 - 4*E^25)*x) + 6*x

Rule 2225

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2259

Int[(u_.)*(F_)^((a_.) + (b_.)*(v_)), x_Symbol] :> Int[u*F^(a + b*NormalizePowerOfLinear[v, x]), x] /; FreeQ[{F
, a, b}, x] && PolynomialQ[u, x] && PowerOfLinearQ[v, x] &&  !PowerOfLinearMatchQ[v, x]

Rubi steps \begin{align*} \text {integral}& = 6 x-2 \int e^x \, dx-\left (2 \left (1-4 e^{25}\right )\right ) \int e^{x-4 e^{25} x} \, dx \\ & = -2 e^x+6 x-\left (2 \left (1-4 e^{25}\right )\right ) \int e^{\left (1-4 e^{25}\right ) x} \, dx \\ & = -2 e^x-2 e^{\left (1-4 e^{25}\right ) x}+6 x \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=-2 \left (e^x+e^{x-4 e^{25} x}-3 x\right ) \]

[In]

Integrate[6 - 2*E^x + E^(x - 4*E^25*x)*(-2 + 8*E^25),x]

[Out]

-2*(E^x + E^(x - 4*E^25*x) - 3*x)

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.70

method	result	size
norman	\(6 x -2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{-4 x \,{\mathrm e}^{25}+x}\)	\(19\)
parallelrisch	\(6 x -2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{-4 x \,{\mathrm e}^{25}+x}\)	\(19\)
risch	\(6 x -2 \,{\mathrm e}^{x}-2 \,{\mathrm e}^{-x \left (4 \,{\mathrm e}^{25}-1\right )}\)	\(21\)
default	\(6 x +\frac {\left (8 \,{\mathrm e}^{25}-2\right ) {\mathrm e}^{-4 x \,{\mathrm e}^{25}+x}}{-4 \,{\mathrm e}^{25}+1}-2 \,{\mathrm e}^{x}\)	\(32\)
parts	\(6 x +\frac {\left (8 \,{\mathrm e}^{25}-2\right ) {\mathrm e}^{-4 x \,{\mathrm e}^{25}+x}}{-4 \,{\mathrm e}^{25}+1}-2 \,{\mathrm e}^{x}\)	\(32\)

[In]

int(-2*exp(x)+(8*exp(25)-2)*exp(-4*x*exp(25)+x)+6,x,method=_RETURNVERBOSE)

[Out]

6*x-2*exp(x)-2*exp(-4*x*exp(25)+x)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=6 \, x - 2 \, e^{\left (-4 \, x e^{25} + x\right )} - 2 \, e^{x} \]

[In]

integrate(-2*exp(x)+(8*exp(25)-2)*exp(-4*x*exp(25)+x)+6,x, algorithm="fricas")

[Out]

6*x - 2*e^(-4*x*e^25 + x) - 2*e^x

Sympy [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=6 x - 2 e^{x} - \frac {\left (-2 + 8 e^{25}\right ) e^{- 4 x e^{25} + x}}{-1 + 4 e^{25}} \]

[In]

integrate(-2*exp(x)+(8*exp(25)-2)*exp(-4*x*exp(25)+x)+6,x)

[Out]

6*x - 2*exp(x) - (-2 + 8*exp(25))*exp(-4*x*exp(25) + x)/(-1 + 4*exp(25))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=6 \, x - 2 \, e^{\left (-4 \, x e^{25} + x\right )} - 2 \, e^{x} \]

[In]

integrate(-2*exp(x)+(8*exp(25)-2)*exp(-4*x*exp(25)+x)+6,x, algorithm="maxima")

[Out]

6*x - 2*e^(-4*x*e^25 + x) - 2*e^x

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=6 \, x - 2 \, e^{\left (-4 \, x e^{25} + x\right )} - 2 \, e^{x} \]

[In]

integrate(-2*exp(x)+(8*exp(25)-2)*exp(-4*x*exp(25)+x)+6,x, algorithm="giac")

[Out]

6*x - 2*e^(-4*x*e^25 + x) - 2*e^x

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.67 \[ \int \left (6-2 e^x+e^{x-4 e^{25} x} \left (-2+8 e^{25}\right )\right ) \, dx=6\,x-2\,{\mathrm {e}}^x-2\,{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{25}}\,{\mathrm {e}}^x \]

[In]

int(exp(x - 4*x*exp(25))*(8*exp(25) - 2) - 2*exp(x) + 6,x)

[Out]

6*x - 2*exp(x) - 2*exp(-4*x*exp(25))*exp(x)