Integrand size = 135, antiderivative size = 25 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=e^{e^{\frac {x}{-\frac {17}{4}+e^4+x}}}-\frac {e^x}{x} \]
Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.12 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=e^{e^{\frac {4 x}{-17+4 e^4+4 x}}}-\frac {e^x}{x} \]
Integrate[(E^(E^((4*x)/(-17 + 4*E^4 + 4*x)) + (4*x)/(-17 + 4*E^4 + 4*x))*( -68*x^2 + 16*E^4*x^2) + E^x*(289 + E^8*(16 - 16*x) - 425*x + 152*x^2 - 16* x^3 + E^4*(-136 + 168*x - 32*x^2)))/(289*x^2 + 16*E^8*x^2 - 136*x^3 + 16*x ^4 + E^4*(-136*x^2 + 32*x^3)),x]
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (16 e^4 x^2-68 x^2\right ) \exp \left (\frac {4 x}{4 x+4 e^4-17}+e^{\frac {4 x}{4 x+4 e^4-17}}\right )+e^x \left (-16 x^3+152 x^2+e^4 \left (-32 x^2+168 x-136\right )-425 x+e^8 (16-16 x)+289\right )}{16 x^4-136 x^3+16 e^8 x^2+289 x^2+e^4 \left (32 x^3-136 x^2\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (16 e^4 x^2-68 x^2\right ) \exp \left (\frac {4 x}{4 x+4 e^4-17}+e^{\frac {4 x}{4 x+4 e^4-17}}\right )+e^x \left (-16 x^3+152 x^2+e^4 \left (-32 x^2+168 x-136\right )-425 x+e^8 (16-16 x)+289\right )}{16 x^4-136 x^3+\left (289+16 e^8\right ) x^2+e^4 \left (32 x^3-136 x^2\right )}dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (16 e^4 x^2-68 x^2\right ) \exp \left (\frac {4 x}{4 x+4 e^4-17}+e^{\frac {4 x}{4 x+4 e^4-17}}\right )+e^x \left (-16 x^3+152 x^2+e^4 \left (-32 x^2+168 x-136\right )-425 x+e^8 (16-16 x)+289\right )}{x^2 \left (16 x^2-8 \left (17-4 e^4\right ) x+\left (17-4 e^4\right )^2\right )}dx\) |
\(\Big \downarrow \) 2007 |
\(\displaystyle \int \frac {\left (16 e^4 x^2-68 x^2\right ) \exp \left (\frac {4 x}{4 x+4 e^4-17}+e^{\frac {4 x}{4 x+4 e^4-17}}\right )+e^x \left (-16 x^3+152 x^2+e^4 \left (-32 x^2+168 x-136\right )-425 x+e^8 (16-16 x)+289\right )}{x^2 \left (4 x+4 e^4-17\right )^2}dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {4 \left (4 e^4-17\right ) \exp \left (\frac {4 x}{4 x+4 e^4-17}+e^{\frac {4 x}{4 x+4 e^4-17}}\right )}{\left (4 x+4 e^4-17\right )^2}-\frac {e^x (x-1)}{x^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -4 \left (17-4 e^4\right ) \int \frac {\exp \left (\frac {4 x}{4 x+4 e^4-17}+e^{\frac {4 x}{4 x+4 e^4-17}}\right )}{\left (4 x+4 e^4-17\right )^2}dx-\frac {e^x}{x}\) |
Int[(E^(E^((4*x)/(-17 + 4*E^4 + 4*x)) + (4*x)/(-17 + 4*E^4 + 4*x))*(-68*x^ 2 + 16*E^4*x^2) + E^x*(289 + E^8*(16 - 16*x) - 425*x + 152*x^2 - 16*x^3 + E^4*(-136 + 168*x - 32*x^2)))/(289*x^2 + 16*E^8*x^2 - 136*x^3 + 16*x^4 + E ^4*(-136*x^2 + 32*x^3)),x]
3.8.14.3.1 Defintions of rubi rules used
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Int[(u_.)*(Px_)^(p_), x_Symbol] :> With[{a = Rt[Coeff[Px, x, 0], Expon[Px, x]], b = Rt[Coeff[Px, x, Expon[Px, x]], Expon[Px, x]]}, Int[u*(a + b*x)^(Ex pon[Px, x]*p), x] /; EqQ[Px, (a + b*x)^Expon[Px, x]]] /; IntegerQ[p] && Pol yQ[Px, x] && GtQ[Expon[Px, x], 1] && NeQ[Coeff[Px, x, 0], 0]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Time = 2.22 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00
method | result | size |
risch | \(-\frac {{\mathrm e}^{x}}{x}+{\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}}\) | \(25\) |
parallelrisch | \(\frac {256 \,{\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}} x -256 \,{\mathrm e}^{x}}{256 x}\) | \(30\) |
parts | \(-\frac {{\mathrm e}^{x}}{x}+\frac {\left (4 \,{\mathrm e}^{4}-17\right ) {\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}}+4 \,{\mathrm e}^{{\mathrm e}^{\frac {4 x}{4 \,{\mathrm e}^{4}+4 x -17}}} x}{4 \,{\mathrm e}^{4}+4 x -17}\) | \(64\) |
int(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/(4*exp( 4)+4*x-17)))+((-16*x+16)*exp(4)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+152*x^ 2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x^4-136*x ^3+289*x^2),x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 83 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 83, normalized size of antiderivative = 3.32 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\frac {{\left (x e^{\left (\frac {{\left (4 \, x + 4 \, e^{4} - 17\right )} e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} + 4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} - e^{\left (x + \frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}\right )} e^{\left (-\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}}{x} \]
integrate(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/( 4*exp(4)+4*x-17)))+((-16*x+16)*exp(4)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+ 152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x^4 -136*x^3+289*x^2),x, algorithm=\
(x*e^(((4*x + 4*e^4 - 17)*e^(4*x/(4*x + 4*e^4 - 17)) + 4*x)/(4*x + 4*e^4 - 17)) - e^(x + 4*x/(4*x + 4*e^4 - 17)))*e^(-4*x/(4*x + 4*e^4 - 17))/x
Time = 0.49 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.80 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=e^{e^{\frac {4 x}{4 x - 17 + 4 e^{4}}}} - \frac {e^{x}}{x} \]
integrate(((16*x**2*exp(4)-68*x**2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x /(4*exp(4)+4*x-17)))+((-16*x+16)*exp(4)**2+(-32*x**2+168*x-136)*exp(4)-16* x**3+152*x**2-425*x+289)*exp(x))/(16*x**2*exp(4)**2+(32*x**3-136*x**2)*exp (4)+16*x**4-136*x**3+289*x**2),x)
Time = 0.26 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.72 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\frac {x e^{\left (e^{\left (-\frac {4 \, e^{4}}{4 \, x + 4 \, e^{4} - 17} + \frac {17}{4 \, x + 4 \, e^{4} - 17} + 1\right )}\right )} - e^{x}}{x} \]
integrate(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/( 4*exp(4)+4*x-17)))+((-16*x+16)*exp(4)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+ 152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x^4 -136*x^3+289*x^2),x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (24) = 48\).
Time = 1.32 (sec) , antiderivative size = 112, normalized size of antiderivative = 4.48 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx=\frac {{\left (x e^{\left (\frac {4 \, x e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} + 4 \, x - 17 \, e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )} + 4 \, e^{\left (\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17} + 4\right )}}{4 \, x + 4 \, e^{4} - 17}\right )} - e^{\left (x + \frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}\right )} e^{\left (-\frac {4 \, x}{4 \, x + 4 \, e^{4} - 17}\right )}}{x} \]
integrate(((16*x^2*exp(4)-68*x^2)*exp(4*x/(4*exp(4)+4*x-17))*exp(exp(4*x/( 4*exp(4)+4*x-17)))+((-16*x+16)*exp(4)^2+(-32*x^2+168*x-136)*exp(4)-16*x^3+ 152*x^2-425*x+289)*exp(x))/(16*x^2*exp(4)^2+(32*x^3-136*x^2)*exp(4)+16*x^4 -136*x^3+289*x^2),x, algorithm=\
(x*e^((4*x*e^(4*x/(4*x + 4*e^4 - 17)) + 4*x - 17*e^(4*x/(4*x + 4*e^4 - 17) ) + 4*e^(4*x/(4*x + 4*e^4 - 17) + 4))/(4*x + 4*e^4 - 17)) - e^(x + 4*x/(4* x + 4*e^4 - 17)))*e^(-4*x/(4*x + 4*e^4 - 17))/x
Time = 8.26 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int \frac {e^{e^{\frac {4 x}{-17+4 e^4+4 x}}+\frac {4 x}{-17+4 e^4+4 x}} \left (-68 x^2+16 e^4 x^2\right )+e^x \left (289+e^8 (16-16 x)-425 x+152 x^2-16 x^3+e^4 \left (-136+168 x-32 x^2\right )\right )}{289 x^2+16 e^8 x^2-136 x^3+16 x^4+e^4 \left (-136 x^2+32 x^3\right )} \, dx={\mathrm {e}}^{{\mathrm {e}}^{\frac {4\,x}{4\,x+4\,{\mathrm {e}}^4-17}}}-\frac {{\mathrm {e}}^x}{x} \]