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

Optimal. Leaf size=27 \[ 2 x+2 \left (-e^x-e^{x-4 e^{25} x}+2 x\right ) \]

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Rubi [A]  time = 0.02, antiderivative size = 22, normalized size of antiderivative = 0.81, number of steps used = 4, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2194, 2227} \begin {gather*} 6 x-2 e^x-2 e^{\left (1-4 e^{25}\right ) x} \end {gather*}

Antiderivative was successfully verified.

[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 2194

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 2227

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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.70 \begin {gather*} -2 \left (e^x+e^{x-4 e^{25} x}-3 x\right ) \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.58, size = 18, normalized size = 0.67 \begin {gather*} 6 \, x - 2 \, e^{\left (-4 \, x e^{25} + x\right )} - 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.13, size = 18, normalized size = 0.67 \begin {gather*} 6 \, x - 2 \, e^{\left (-4 \, x e^{25} + x\right )} - 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.03, size = 19, normalized size = 0.70

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [A]  time = 0.37, size = 18, normalized size = 0.67 \begin {gather*} 6 \, x - 2 \, e^{\left (-4 \, x e^{25} + x\right )} - 2 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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mupad [B]  time = 5.25, size = 18, normalized size = 0.67 \begin {gather*} 6\,x-2\,{\mathrm {e}}^x-2\,{\mathrm {e}}^{-4\,x\,{\mathrm {e}}^{25}}\,{\mathrm {e}}^x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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sympy [A]  time = 0.12, size = 31, normalized size = 1.15 \begin {gather*} 6 x - 2 e^{x} - \frac {\left (-2 + 8 e^{25}\right ) e^{- 4 x e^{25} + x}}{-1 + 4 e^{25}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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