3.10.55 \(\int 5 e^{-5+e^{1+x+e^5 x}-2 x} (1-2 x+e^{1+x+e^5 x} (x+e^5 x)) \, dx\)

Optimal. Leaf size=20 \[ 5 e^{-5+e^{1+x+e^5 x}-2 x} x \]

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Rubi [B]  time = 0.16, antiderivative size = 62, normalized size of antiderivative = 3.10, number of steps used = 2, number of rules used = 2, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {12, 2288} \begin {gather*} \frac {5 e^{-2 x+e^{e^5 x+x+1}-5} \left (2 x-\left (1+e^5\right ) e^{e^5 x+x+1} x\right )}{2-\left (1+e^5\right ) e^{e^5 x+x+1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 2288

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 {gather*} \begin {aligned} \text {integral} &=5 \int e^{-5+e^{1+x+e^5 x}-2 x} \left (1-2 x+e^{1+x+e^5 x} \left (x+e^5 x\right )\right ) \, dx\\ &=\frac {5 e^{-5+e^{1+x+e^5 x}-2 x} \left (2 x-e^{1+x+e^5 x} \left (1+e^5\right ) x\right )}{2-e^{1+x+e^5 x} \left (1+e^5\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.13, size = 20, normalized size = 1.00 \begin {gather*} 5 e^{-5+e^{1+x+e^5 x}-2 x} x \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

5*E^(-5 + E^(1 + x + E^5*x) - 2*x)*x

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fricas [A]  time = 0.93, size = 18, normalized size = 0.90 \begin {gather*} x e^{\left (-2 \, x + e^{\left (x e^{5} + x + 1\right )} + \log \relax (5) - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(5)+x)*exp(x*exp(5)+x+1)+1-2*x)*exp(exp(x*exp(5)+x+1)+log(5)-2*x-5),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int {\left ({\left (x e^{5} + x\right )} e^{\left (x e^{5} + x + 1\right )} - 2 \, x + 1\right )} e^{\left (-2 \, x + e^{\left (x e^{5} + x + 1\right )} + \log \relax (5) - 5\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(5)+x)*exp(x*exp(5)+x+1)+1-2*x)*exp(exp(x*exp(5)+x+1)+log(5)-2*x-5),x, algorithm="giac")

[Out]

integrate(((x*e^5 + x)*e^(x*e^5 + x + 1) - 2*x + 1)*e^(-2*x + e^(x*e^5 + x + 1) + log(5) - 5), x)

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maple [A]  time = 0.10, size = 18, normalized size = 0.90

method	result	size
risch	\(5 x \,{\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{5}+x +1}-5-2 x}\)	\(18\)
norman	\(x \,{\mathrm e}^{{\mathrm e}^{x \,{\mathrm e}^{5}+x +1}+\ln \relax (5)-2 x -5}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x*exp(5)+x)*exp(x*exp(5)+x+1)+1-2*x)*exp(exp(x*exp(5)+x+1)+ln(5)-2*x-5),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.81, size = 17, normalized size = 0.85 \begin {gather*} 5 \, x e^{\left (-2 \, x + e^{\left (x e^{5} + x + 1\right )} - 5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(5)+x)*exp(x*exp(5)+x+1)+1-2*x)*exp(exp(x*exp(5)+x+1)+log(5)-2*x-5),x, algorithm="maxima")

[Out]

5*x*e^(-2*x + e^(x*e^5 + x + 1) - 5)

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mupad [B]  time = 0.76, size = 20, normalized size = 1.00 \begin {gather*} 5\,x\,{\mathrm {e}}^{-2\,x}\,{\mathrm {e}}^{-5}\,{\mathrm {e}}^{\mathrm {e}\,{\mathrm {e}}^{x\,{\mathrm {e}}^5}\,{\mathrm {e}}^x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(log(5) - 2*x + exp(x + x*exp(5) + 1) - 5)*(exp(x + x*exp(5) + 1)*(x + x*exp(5)) - 2*x + 1),x)

[Out]

5*x*exp(-2*x)*exp(-5)*exp(exp(1)*exp(x*exp(5))*exp(x))

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sympy [A]  time = 0.22, size = 19, normalized size = 0.95 \begin {gather*} 5 x e^{- 2 x + e^{x + x e^{5} + 1} - 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x*exp(5)+x)*exp(x*exp(5)+x+1)+1-2*x)*exp(exp(x*exp(5)+x+1)+ln(5)-2*x-5),x)

[Out]

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

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