3.103.53 \(\int \frac {1}{2} e^{\frac {1}{4} (-11+4 e^x)} (1+e^{\frac {1}{4} (11-4 e^x)} (20 e^x+4 e^{2 x})+e^x x) \, dx\)

Optimal. Leaf size=25 \[ \left (5+e^x\right )^2+\frac {1}{2} \left (-4+e^{-\frac {11}{4}+e^x} x\right ) \]

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Rubi [F]  time = 0.17, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{2} e^{\frac {1}{4} \left (-11+4 e^x\right )} \left (1+e^{\frac {1}{4} \left (11-4 e^x\right )} \left (20 e^x+4 e^{2 x}\right )+e^x x\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-11 + 4*E^x)/4)*(1 + E^((11 - 4*E^x)/4)*(20*E^x + 4*E^(2*x)) + E^x*x))/2,x]

[Out]

10*E^x + E^(2*x) + ExpIntegralEi[E^x]/(2*E^(11/4)) + Defer[Int][E^(-11/4 + E^x + x)*x, x]/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int e^{\frac {1}{4} \left (-11+4 e^x\right )} \left (1+e^{\frac {1}{4} \left (11-4 e^x\right )} \left (20 e^x+4 e^{2 x}\right )+e^x x\right ) \, dx\\ &=\frac {1}{2} \int \left (e^{-\frac {11}{4}+e^x}+20 e^x+4 e^{2 x}+e^{-\frac {11}{4}+e^x+x} x\right ) \, dx\\ &=\frac {1}{2} \int e^{-\frac {11}{4}+e^x} \, dx+\frac {1}{2} \int e^{-\frac {11}{4}+e^x+x} x \, dx+2 \int e^{2 x} \, dx+10 \int e^x \, dx\\ &=10 e^x+e^{2 x}+\frac {1}{2} \int e^{-\frac {11}{4}+e^x+x} x \, dx+\frac {1}{2} \operatorname {Subst}\left (\int \frac {e^{-\frac {11}{4}+x}}{x} \, dx,x,e^x\right )\\ &=10 e^x+e^{2 x}+\frac {\text {Ei}\left (e^x\right )}{2 e^{11/4}}+\frac {1}{2} \int e^{-\frac {11}{4}+e^x+x} x \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 28, normalized size = 1.12 \begin {gather*} \frac {1}{2} \left (20 e^x+2 e^{2 x}+e^{-\frac {11}{4}+e^x} x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-11 + 4*E^x)/4)*(1 + E^((11 - 4*E^x)/4)*(20*E^x + 4*E^(2*x)) + E^x*x))/2,x]

[Out]

(20*E^x + 2*E^(2*x) + E^(-11/4 + E^x)*x)/2

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fricas [A]  time = 0.76, size = 17, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, x e^{\left (e^{x} - \frac {11}{4}\right )} + e^{\left (2 \, x\right )} + 10 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((4*exp(x)^2+20*exp(x))*exp(-exp(x)+11/4)+exp(x)*x+1)/exp(-exp(x)+11/4),x, algorithm="fricas")

[Out]

1/2*x*e^(e^x - 11/4) + e^(2*x) + 10*e^x

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giac [A]  time = 0.14, size = 17, normalized size = 0.68 \begin {gather*} \frac {1}{2} \, x e^{\left (e^{x} - \frac {11}{4}\right )} + e^{\left (2 \, x\right )} + 10 \, e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((4*exp(x)^2+20*exp(x))*exp(-exp(x)+11/4)+exp(x)*x+1)/exp(-exp(x)+11/4),x, algorithm="giac")

[Out]

1/2*x*e^(e^x - 11/4) + e^(2*x) + 10*e^x

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maple [A]  time = 0.05, size = 18, normalized size = 0.72

method	result	size
risch	\({\mathrm e}^{2 x}+10 \,{\mathrm e}^{x}+\frac {x \,{\mathrm e}^{{\mathrm e}^{x}-\frac {11}{4}}}{2}\)	\(18\)
norman	\(\left ({\mathrm e}^{2 x} {\mathrm e}^{-{\mathrm e}^{x}+\frac {11}{4}}+\frac {x}{2}+10 \,{\mathrm e}^{x} {\mathrm e}^{-{\mathrm e}^{x}+\frac {11}{4}}\right ) {\mathrm e}^{{\mathrm e}^{x}-\frac {11}{4}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*((4*exp(x)^2+20*exp(x))*exp(-exp(x)+11/4)+exp(x)*x+1)/exp(-exp(x)+11/4),x,method=_RETURNVERBOSE)

[Out]

exp(2*x)+10*exp(x)+1/2*x*exp(exp(x)-11/4)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {1}{2} \, {\rm Ei}\left (e^{x}\right ) e^{\left (-\frac {11}{4}\right )} + \frac {1}{2} \, x e^{\left (e^{x} - \frac {11}{4}\right )} + e^{\left (2 \, x\right )} + 10 \, e^{x} - \frac {1}{2} \, \int e^{\left (e^{x} - \frac {11}{4}\right )}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((4*exp(x)^2+20*exp(x))*exp(-exp(x)+11/4)+exp(x)*x+1)/exp(-exp(x)+11/4),x, algorithm="maxima")

[Out]

1/2*Ei(e^x)*e^(-11/4) + 1/2*x*e^(e^x - 11/4) + e^(2*x) + 10*e^x - 1/2*integrate(e^(e^x - 11/4), x)

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mupad [B]  time = 0.07, size = 17, normalized size = 0.68 \begin {gather*} {\mathrm {e}}^{2\,x}+10\,{\mathrm {e}}^x+\frac {x\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{-\frac {11}{4}}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(exp(exp(x) - 11/4)*((exp(11/4 - exp(x))*(4*exp(2*x) + 20*exp(x)))/2 + (x*exp(x))/2 + 1/2),x)

[Out]

exp(2*x) + 10*exp(x) + (x*exp(exp(x))*exp(-11/4))/2

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sympy [A]  time = 0.21, size = 20, normalized size = 0.80 \begin {gather*} \frac {x e^{e^{x} - \frac {11}{4}}}{2} + e^{2 x} + 10 e^{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*((4*exp(x)**2+20*exp(x))*exp(-exp(x)+11/4)+exp(x)*x+1)/exp(-exp(x)+11/4),x)

[Out]

x*exp(exp(x) - 11/4)/2 + exp(2*x) + 10*exp(x)

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