3.95.60 \(\int \frac {e^{2 x} (-4+4 x)+e^{\frac {e^x+4 x}{2 x}} (e^{3 x} (1-x)+e^{2 x} (2 x-4 x^2))}{2 x^3} \, dx\)

Optimal. Leaf size=35 \[ 1-\frac {e^{2+\frac {e^x}{2 x}+2 x}-\frac {e^{2 x}}{x}}{x} \]

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Rubi [F]  time = 1.34, 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 {e^{2 x} (-4+4 x)+e^{\frac {e^x+4 x}{2 x}} \left (e^{3 x} (1-x)+e^{2 x} \left (2 x-4 x^2\right )\right )}{2 x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(-4 + 4*x) + E^((E^x + 4*x)/(2*x))*(E^(3*x)*(1 - x) + E^(2*x)*(2*x - 4*x^2)))/(2*x^3),x]

[Out]

E^(2*x)/x^2 + Defer[Int][E^(2 + E^x/(2*x) + 3*x)/x^3, x]/2 - Defer[Int][E^(2 + E^x/(2*x) + 3*x)/x^2, x]/2 + De
fer[Int][E^((E^x + 4*x + 4*x^2)/(2*x))/x^2, x] - 2*Defer[Int][E^((E^x + 4*x + 4*x^2)/(2*x))/x, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{2} \int \frac {e^{2 x} (-4+4 x)+e^{\frac {e^x+4 x}{2 x}} \left (e^{3 x} (1-x)+e^{2 x} \left (2 x-4 x^2\right )\right )}{x^3} \, dx\\ &=\frac {1}{2} \int \left (-\frac {e^{2+\frac {e^x}{2 x}+3 x} (-1+x)}{x^3}-\frac {2 e^{2 x} \left (2-2 x-e^{2+\frac {e^x}{2 x}} x+2 e^{2+\frac {e^x}{2 x}} x^2\right )}{x^3}\right ) \, dx\\ &=-\left (\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x} (-1+x)}{x^3} \, dx\right )-\int \frac {e^{2 x} \left (2-2 x-e^{2+\frac {e^x}{2 x}} x+2 e^{2+\frac {e^x}{2 x}} x^2\right )}{x^3} \, dx\\ &=-\left (\frac {1}{2} \int \left (-\frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^3}+\frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^2}\right ) \, dx\right )-\int \left (-\frac {2 e^{2 x} (-1+x)}{x^3}+\frac {e^{2 x+\frac {e^x+4 x}{2 x}} (-1+2 x)}{x^2}\right ) \, dx\\ &=\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^3} \, dx-\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^2} \, dx+2 \int \frac {e^{2 x} (-1+x)}{x^3} \, dx-\int \frac {e^{2 x+\frac {e^x+4 x}{2 x}} (-1+2 x)}{x^2} \, dx\\ &=\frac {e^{2 x}}{x^2}+\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^3} \, dx-\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^2} \, dx-\int \frac {e^{\frac {e^x+4 x+4 x^2}{2 x}} (-1+2 x)}{x^2} \, dx\\ &=\frac {e^{2 x}}{x^2}+\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^3} \, dx-\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^2} \, dx-\int \left (-\frac {e^{\frac {e^x+4 x+4 x^2}{2 x}}}{x^2}+\frac {2 e^{\frac {e^x+4 x+4 x^2}{2 x}}}{x}\right ) \, dx\\ &=\frac {e^{2 x}}{x^2}+\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^3} \, dx-\frac {1}{2} \int \frac {e^{2+\frac {e^x}{2 x}+3 x}}{x^2} \, dx-2 \int \frac {e^{\frac {e^x+4 x+4 x^2}{2 x}}}{x} \, dx+\int \frac {e^{\frac {e^x+4 x+4 x^2}{2 x}}}{x^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.38, size = 28, normalized size = 0.80 \begin {gather*} -\frac {e^{2 x} \left (-1+e^{2+\frac {e^x}{2 x}} x\right )}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-4 + 4*x) + E^((E^x + 4*x)/(2*x))*(E^(3*x)*(1 - x) + E^(2*x)*(2*x - 4*x^2)))/(2*x^3),x]

[Out]

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

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fricas [A]  time = 0.49, size = 30, normalized size = 0.86 \begin {gather*} -\frac {x e^{\left (2 \, x + \frac {4 \, x + e^{x}}{2 \, x}\right )} - e^{\left (2 \, x\right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x+1)*exp(x)^3+(-4*x^2+2*x)*exp(x)^2)*exp(1/2*(4*x+exp(x))/x)+(4*x-4)*exp(x)^2)/x^3,x, algori
thm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {4 \, {\left (x - 1\right )} e^{\left (2 \, x\right )} - {\left ({\left (x - 1\right )} e^{\left (3 \, x\right )} + 2 \, {\left (2 \, x^{2} - x\right )} e^{\left (2 \, x\right )}\right )} e^{\left (\frac {4 \, x + e^{x}}{2 \, x}\right )}}{2 \, x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x+1)*exp(x)^3+(-4*x^2+2*x)*exp(x)^2)*exp(1/2*(4*x+exp(x))/x)+(4*x-4)*exp(x)^2)/x^3,x, algori
thm="giac")

[Out]

integrate(1/2*(4*(x - 1)*e^(2*x) - ((x - 1)*e^(3*x) + 2*(2*x^2 - x)*e^(2*x))*e^(1/2*(4*x + e^x)/x))/x^3, x)

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maple [A]  time = 0.06, size = 32, normalized size = 0.91

method	result	size
risch	\(\frac {{\mathrm e}^{2 x}}{x^{2}}-\frac {{\mathrm e}^{\frac {4 x^{2}+{\mathrm e}^{x}+4 x}{2 x}}}{x}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/2*(((1-x)*exp(x)^3+(-4*x^2+2*x)*exp(x)^2)*exp(1/2*(4*x+exp(x))/x)+(4*x-4)*exp(x)^2)/x^3,x,method=_RETURN
VERBOSE)

[Out]

exp(2*x)/x^2-1/x*exp(1/2*(4*x^2+exp(x)+4*x)/x)

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maxima [C]  time = 0.43, size = 33, normalized size = 0.94 \begin {gather*} -\frac {e^{\left (2 \, x + \frac {e^{x}}{2 \, x} + 2\right )}}{x} + 4 \, \Gamma \left (-1, -2 \, x\right ) + 8 \, \Gamma \left (-2, -2 \, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x+1)*exp(x)^3+(-4*x^2+2*x)*exp(x)^2)*exp(1/2*(4*x+exp(x))/x)+(4*x-4)*exp(x)^2)/x^3,x, algori
thm="maxima")

[Out]

-e^(2*x + 1/2*e^x/x + 2)/x + 4*gamma(-1, -2*x) + 8*gamma(-2, -2*x)

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mupad [B]  time = 7.26, size = 25, normalized size = 0.71 \begin {gather*} \frac {{\mathrm {e}}^{2\,x}-x\,{\mathrm {e}}^{2\,x+\frac {{\mathrm {e}}^x}{2\,x}+2}}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp((2*x + exp(x)/2)/x)*(exp(2*x)*(2*x - 4*x^2) - exp(3*x)*(x - 1)))/2 + (exp(2*x)*(4*x - 4))/2)/x^3,x)

[Out]

(exp(2*x) - x*exp(2*x + exp(x)/(2*x) + 2))/x^2

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sympy [A]  time = 0.20, size = 26, normalized size = 0.74 \begin {gather*} - \frac {e^{2 x} e^{\frac {2 x + \frac {e^{x}}{2}}{x}}}{x} + \frac {e^{2 x}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/2*(((-x+1)*exp(x)**3+(-4*x**2+2*x)*exp(x)**2)*exp(1/2*(4*x+exp(x))/x)+(4*x-4)*exp(x)**2)/x**3,x)

[Out]

-exp(2*x)*exp((2*x + exp(x)/2)/x)/x + exp(2*x)/x**2

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