\(\int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx\) [9912]

Optimal result

Integrand size = 39, antiderivative size = 25 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {1}{4} \left (e^{3+e^4+2 e^{2/x} x}+3 x\right ) \]

[Out]

1/4*exp(2*x*exp(2/x)+exp(4)+3)+3/4*x

Rubi [F]

\[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.08 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {1}{4} e^{3+e^4+2 e^{2/x} x}+\frac {3 x}{4} \]

[In]

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

[Out]

E^(3 + E^4 + 2*E^(2/x)*x)/4 + (3*x)/4

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84

method	result	size
norman	\(\frac {{\mathrm e}^{2 x \,{\mathrm e}^{\frac {2}{x}}+{\mathrm e}^{4}+3}}{4}+\frac {3 x}{4}\)	\(21\)
risch	\(\frac {{\mathrm e}^{2 x \,{\mathrm e}^{\frac {2}{x}}+{\mathrm e}^{4}+3}}{4}+\frac {3 x}{4}\)	\(21\)
parallelrisch	\(\frac {{\mathrm e}^{2 x \,{\mathrm e}^{\frac {2}{x}}+{\mathrm e}^{4}+3}}{4}+\frac {3 x}{4}\)	\(21\)
parts	\(\frac {{\mathrm e}^{2 x \,{\mathrm e}^{\frac {2}{x}}+{\mathrm e}^{4}+3}}{4}+\frac {3 x}{4}\)	\(21\)

[In]

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

[Out]

1/4*exp(2*x*exp(2/x)+exp(4)+3)+3/4*x

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).

Time = 0.27 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {1}{4} \, {\left (3 \, x e^{\frac {2}{x}} + e^{\left (\frac {2 \, x^{2} e^{\frac {2}{x}} + x e^{4} + 3 \, x + 2}{x}\right )}\right )} e^{\left (-\frac {2}{x}\right )} \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {3 x}{4} + \frac {e^{2 x e^{\frac {2}{x}} + 3 + e^{4}}}{4} \]

[In]

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

[Out]

3*x/4 + exp(2*x*exp(2/x) + 3 + exp(4))/4

Maxima [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {3}{4} \, x + \frac {1}{4} \, e^{\left (2 \, x e^{\frac {2}{x}} + e^{4} + 3\right )} \]

[In]

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

[Out]

3/4*x + 1/4*e^(2*x*e^(2/x) + e^4 + 3)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 43 vs. \(2 (20) = 40\).

Time = 0.28 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {1}{4} \, {\left (3 \, x e^{\frac {2}{x}} + e^{\left (\frac {2 \, x^{2} e^{\frac {2}{x}} + x e^{4} + 3 \, x + 2}{x}\right )}\right )} e^{\left (-\frac {2}{x}\right )} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 16.66 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.84 \[ \int \frac {3 x+e^{3+e^4+\frac {2}{x}+2 e^{2/x} x} (-4+2 x)}{4 x} \, dx=\frac {3\,x}{4}+\frac {{\mathrm {e}}^3\,{\mathrm {e}}^{{\mathrm {e}}^4}\,{\mathrm {e}}^{2\,x\,{\mathrm {e}}^{2/x}}}{4} \]

[In]

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

[Out]

(3*x)/4 + (exp(3)*exp(exp(4))*exp(2*x*exp(2/x)))/4