\(\int \frac {x+e^{16} x^2 (e^{e^x} (-2 e^{12}-e^{12+x} x)+e^{12+x^2} (2+2 x^2))}{x} \, dx\) [5091]

Optimal result

Integrand size = 49, antiderivative size = 22 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=x+e^{28} \left (-e^{e^x}+e^{x^2}\right ) x^2 \]

[Out]

x+(exp(x^2)-exp(exp(x)))*exp(12)*exp(ln(x^2)+16)

Rubi [F]

\[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=\int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx \]

[In]

Int[(x + E^16*x^2*(E^E^x*(-2*E^12 - E^(12 + x)*x) + E^(12 + x^2)*(2 + 2*x^2)))/x,x]

[Out]

x + E^(28 + x^2)*x^2 - 2*Defer[Int][E^(28 + E^x)*x, x] - Defer[Int][E^(28 + E^x + x)*x^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (1-2 e^{28+e^x} x-e^{28+e^x+x} x^2+2 e^{28+x^2} x \left (1+x^2\right )\right ) \, dx \\ & = x-2 \int e^{28+e^x} x \, dx+2 \int e^{28+x^2} x \left (1+x^2\right ) \, dx-\int e^{28+e^x+x} x^2 \, dx \\ & = x-2 \int e^{28+e^x} x \, dx+2 \int \left (e^{28+x^2} x+e^{28+x^2} x^3\right ) \, dx-\int e^{28+e^x+x} x^2 \, dx \\ & = x-2 \int e^{28+e^x} x \, dx+2 \int e^{28+x^2} x \, dx+2 \int e^{28+x^2} x^3 \, dx-\int e^{28+e^x+x} x^2 \, dx \\ & = e^{28+x^2}+x+e^{28+x^2} x^2-2 \int e^{28+e^x} x \, dx-2 \int e^{28+x^2} x \, dx-\int e^{28+e^x+x} x^2 \, dx \\ & = x+e^{28+x^2} x^2-2 \int e^{28+e^x} x \, dx-\int e^{28+e^x+x} x^2 \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.14 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=x-e^{28+e^x} x^2+e^{28+x^2} x^2 \]

[In]

Integrate[(x + E^16*x^2*(E^E^x*(-2*E^12 - E^(12 + x)*x) + E^(12 + x^2)*(2 + 2*x^2)))/x,x]

[Out]

x - E^(28 + E^x)*x^2 + E^(28 + x^2)*x^2

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(46\) vs. \(2(22)=44\).

Time = 9.56 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14

method	result	size
parallelrisch	\(\frac {2 \,{\mathrm e}^{x^{2}} {\mathrm e}^{12} {\mathrm e}^{\ln \left (x^{2}\right )+16} x^{2}-2 \,{\mathrm e}^{12} x^{2} {\mathrm e}^{{\mathrm e}^{x}} {\mathrm e}^{\ln \left (x^{2}\right )+16}+2 x^{3}}{2 x^{2}}\)	\(47\)
risch	\(x -x^{2} \left ({\mathrm e}^{x^{2}}-{\mathrm e}^{{\mathrm e}^{x}}\right ) {\mathrm e}^{28} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i x^{2}\right )^{3}}{2}} {\mathrm e}^{-\frac {i \pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )}{2}}\)	\(53\)

[In]

int((((-x*exp(12)*exp(x)-2*exp(12))*exp(exp(x))+(2*x^2+2)*exp(12)*exp(x^2))*exp(ln(x^2)+16)+x)/x,x,method=_RET
URNVERBOSE)

[Out]

1/2*(2*exp(x^2)*exp(12)*exp(ln(x^2)+16)*x^2-2*exp(12)*x^2*exp(exp(x))*exp(ln(x^2)+16)+2*x^3)/x^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=x^{2} e^{\left (x^{2} + 28\right )} - x^{2} e^{\left (e^{x} + 28\right )} + x \]

[In]

integrate((((-x*exp(12)*exp(x)-2*exp(12))*exp(exp(x))+(2*x^2+2)*exp(12)*exp(x^2))*exp(log(x^2)+16)+x)/x,x, alg
orithm="fricas")

[Out]

x^2*e^(x^2 + 28) - x^2*e^(e^x + 28) + x

Sympy [A] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=x^{2} e^{28} e^{x^{2}} - x^{2} e^{28} e^{e^{x}} + x \]

[In]

integrate((((-x*exp(12)*exp(x)-2*exp(12))*exp(exp(x))+(2*x**2+2)*exp(12)*exp(x**2))*exp(ln(x**2)+16)+x)/x,x)

[Out]

x**2*exp(28)*exp(x**2) - x**2*exp(28)*exp(exp(x)) + x

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.55 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=-x^{2} e^{\left (e^{x} + 28\right )} + {\left (x^{2} e^{28} - e^{28}\right )} e^{\left (x^{2}\right )} + x + e^{\left (x^{2} + 28\right )} \]

[In]

integrate((((-x*exp(12)*exp(x)-2*exp(12))*exp(exp(x))+(2*x^2+2)*exp(12)*exp(x^2))*exp(log(x^2)+16)+x)/x,x, alg
orithm="maxima")

[Out]

-x^2*e^(e^x + 28) + (x^2*e^28 - e^28)*e^(x^2) + x + e^(x^2 + 28)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.45 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx={\left (x^{2} e^{\left (x^{2} + x + 28\right )} - x^{2} e^{\left (x + e^{x} + 28\right )} + x e^{x}\right )} e^{\left (-x\right )} \]

[In]

integrate((((-x*exp(12)*exp(x)-2*exp(12))*exp(exp(x))+(2*x^2+2)*exp(12)*exp(x^2))*exp(log(x^2)+16)+x)/x,x, alg
orithm="giac")

[Out]

(x^2*e^(x^2 + x + 28) - x^2*e^(x + e^x + 28) + x*e^x)*e^(-x)

Mupad [B] (verification not implemented)

Time = 14.41 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {x+e^{16} x^2 \left (e^{e^x} \left (-2 e^{12}-e^{12+x} x\right )+e^{12+x^2} \left (2+2 x^2\right )\right )}{x} \, dx=x+x^2\,{\mathrm {e}}^{x^2}\,{\mathrm {e}}^{28}-x^2\,{\mathrm {e}}^{{\mathrm {e}}^x}\,{\mathrm {e}}^{28} \]

[In]

int((x - exp(log(x^2) + 16)*(exp(exp(x))*(2*exp(12) + x*exp(12)*exp(x)) - exp(x^2)*exp(12)*(2*x^2 + 2)))/x,x)

[Out]

x + x^2*exp(x^2)*exp(28) - x^2*exp(exp(x))*exp(28)