Integrand size = 236, antiderivative size = 31 \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx=e^{\frac {x}{4 \log (x)}} \left (e^{3+\left (2-\frac {5}{x}+x\right )^4}+x+x^2\right ) \]
Time = 0.46 (sec) , antiderivative size = 62, normalized size of antiderivative = 2.00 \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx=e^{-71+\frac {x}{4 \log (x)}} \left (e^{\frac {625-1000 x+100 x^2+440 x^3-88 x^5+4 x^6+8 x^7+x^8}{x^4}}+e^{71} x (1+x)\right ) \]
Integrate[(E^(x/(4*Log[x]))*(-(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x ^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5) - x^6 - x^7 + (E^((625 - 1000 *x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^6 + x^7)*Log[x] + (4*x^5 + 8*x^6 + E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*(-10000 + 12000*x - 800*x^2 - 1760*x^3 - 352*x^5 + 32*x^6 + 96*x^7 + 16*x^8))*Log[x]^2))/(4*x^5*Log[x ]^2),x]
E^(-71 + x/(4*Log[x]))*(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 88*x^5 + 4* x^6 + 8*x^7 + x^8)/x^4) + E^71*x*(1 + x))
Leaf count is larger than twice the leaf count of optimal. \(146\) vs. \(2(31)=62\).
Time = 3.98 (sec) , antiderivative size = 146, normalized size of antiderivative = 4.71, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.013, Rules used = {27, 25, 2726}
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.
\(\displaystyle \int \frac {e^{\frac {x}{4 \log (x)}} \left (-x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )+\left (\left (16 x^8+96 x^7+32 x^6-352 x^5-1760 x^3-800 x^2+12000 x-10000\right ) \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )+8 x^6+4 x^5\right ) \log ^2(x)+\left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )+x^7+x^6\right ) \log (x)-x^7-x^6\right )}{4 x^5 \log ^2(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{4} \int -\frac {e^{\frac {x}{4 \log (x)}} \left (x^7+x^6+\exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right ) x^5-4 \left (2 x^6+x^5-4 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right ) \left (-x^8-6 x^7-2 x^6+22 x^5+110 x^3+50 x^2-750 x+625\right )\right ) \log ^2(x)-\left (x^7+x^6+\exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right ) x^5\right ) \log (x)\right )}{x^5 \log ^2(x)}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {1}{4} \int \frac {e^{\frac {x}{4 \log (x)}} \left (x^7+x^6+\exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right ) x^5-4 \left (2 x^6+x^5-4 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right ) \left (-x^8-6 x^7-2 x^6+22 x^5+110 x^3+50 x^2-750 x+625\right )\right ) \log ^2(x)-\left (x^7+x^6+\exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right ) x^5\right ) \log (x)\right )}{x^5 \log ^2(x)}dx\) |
\(\Big \downarrow \) 2726 |
\(\displaystyle \frac {e^{\frac {x}{4 \log (x)}} \left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )-\left (x^5 \exp \left (\frac {x^8+8 x^7+4 x^6-88 x^5-71 x^4+440 x^3+100 x^2-1000 x+625}{x^4}\right )+x^7+x^6\right ) \log (x)+x^7+x^6\right )}{x^5 \left (\frac {1}{\log ^2(x)}-\frac {1}{\log (x)}\right ) \log ^2(x)}\) |
Int[(E^(x/(4*Log[x]))*(-(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 8 8*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5) - x^6 - x^7 + (E^((625 - 1000*x + 1 00*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^6 + x^7)*Log[x] + (4*x^5 + 8*x^6 + E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71* x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*(-10000 + 12000*x - 800*x^2 - 176 0*x^3 - 352*x^5 + 32*x^6 + 96*x^7 + 16*x^8))*Log[x]^2))/(4*x^5*Log[x]^2),x ]
(E^(x/(4*Log[x]))*(E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^6 + x^7 - (E^((625 - 1000*x + 100*x^2 + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8)/x^4)*x^5 + x^6 + x^7)*L og[x]))/(x^5*(Log[x]^(-2) - Log[x]^(-1))*Log[x]^2)
3.27.88.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z, x], w*y]] /; FreeQ[F, x]
Leaf count of result is larger than twice the leaf count of optimal. \(64\) vs. \(2(27)=54\).
Time = 38.77 (sec) , antiderivative size = 65, normalized size of antiderivative = 2.10
method | result | size |
risch | \(\frac {\left (4 x^{2}+4 x +4 \,{\mathrm e}^{\frac {x^{8}+8 x^{7}+4 x^{6}-88 x^{5}-71 x^{4}+440 x^{3}+100 x^{2}-1000 x +625}{x^{4}}}\right ) {\mathrm e}^{\frac {x}{4 \ln \left (x \right )}}}{4}\) | \(65\) |
parallelrisch | \(x^{2} {\mathrm e}^{\frac {x}{4 \ln \left (x \right )}}+{\mathrm e}^{\frac {x}{4 \ln \left (x \right )}} x +{\mathrm e}^{\frac {x^{8}+8 x^{7}+4 x^{6}-88 x^{5}-71 x^{4}+440 x^{3}+100 x^{2}-1000 x +625}{x^{4}}} {\mathrm e}^{\frac {x}{4 \ln \left (x \right )}}\) | \(76\) |
int(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-10000)*ex p((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)+8*x^6+4* x^5)*ln(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000* x+625)/x^4)+x^7+x^6)*ln(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+ 100*x^2-1000*x+625)/x^4)-x^7-x^6)*exp(1/4*x/ln(x))/x^5/ln(x)^2,x,method=_R ETURNVERBOSE)
1/4*(4*x^2+4*x+4*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x +625)/x^4))*exp(1/4*x/ln(x))
Leaf count of result is larger than twice the leaf count of optimal. 57 vs. \(2 (27) = 54\).
Time = 0.26 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.84 \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx={\left (x^{2} + x + e^{\left (\frac {x^{8} + 8 \, x^{7} + 4 \, x^{6} - 88 \, x^{5} - 71 \, x^{4} + 440 \, x^{3} + 100 \, x^{2} - 1000 \, x + 625}{x^{4}}\right )}\right )} e^{\left (\frac {x}{4 \, \log \left (x\right )}\right )} \]
integrate(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-100 00)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)+8* x^6+4*x^5)*log(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^ 2-1000*x+625)/x^4)+x^7+x^6)*log(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+ 440*x^3+100*x^2-1000*x+625)/x^4)-x^7-x^6)*exp(1/4*x/log(x))/x^5/log(x)^2,x , algorithm=\
(x^2 + x + e^((x^8 + 8*x^7 + 4*x^6 - 88*x^5 - 71*x^4 + 440*x^3 + 100*x^2 - 1000*x + 625)/x^4))*e^(1/4*x/log(x))
Exception generated. \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx=\text {Exception raised: TypeError} \]
integrate(1/4*(((16*x**8+96*x**7+32*x**6-352*x**5-1760*x**3-800*x**2+12000 *x-10000)*exp((x**8+8*x**7+4*x**6-88*x**5-71*x**4+440*x**3+100*x**2-1000*x +625)/x**4)+8*x**6+4*x**5)*ln(x)**2+(x**5*exp((x**8+8*x**7+4*x**6-88*x**5- 71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)+x**7+x**6)*ln(x)-x**5*exp((x** 8+8*x**7+4*x**6-88*x**5-71*x**4+440*x**3+100*x**2-1000*x+625)/x**4)-x**7-x **6)*exp(1/4*x/ln(x))/x**5/ln(x)**2,x)
Exception generated. \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx=\text {Exception raised: RuntimeError} \]
integrate(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-100 00)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)+8* x^6+4*x^5)*log(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^ 2-1000*x+625)/x^4)+x^7+x^6)*log(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+ 440*x^3+100*x^2-1000*x+625)/x^4)-x^7-x^6)*exp(1/4*x/log(x))/x^5/log(x)^2,x , algorithm=\
Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is 0which is not of the expected type LIST
Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (27) = 54\).
Time = 0.59 (sec) , antiderivative size = 95, normalized size of antiderivative = 3.06 \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx=x^{2} e^{\left (\frac {x}{4 \, \log \left (x\right )}\right )} + x e^{\left (\frac {x}{4 \, \log \left (x\right )}\right )} + e^{\left (\frac {4 \, x^{8} \log \left (x\right ) + 32 \, x^{7} \log \left (x\right ) + 16 \, x^{6} \log \left (x\right ) - 352 \, x^{5} \log \left (x\right ) + x^{5} - 284 \, x^{4} \log \left (x\right ) + 1760 \, x^{3} \log \left (x\right ) + 400 \, x^{2} \log \left (x\right ) - 4000 \, x \log \left (x\right ) + 2500 \, \log \left (x\right )}{4 \, x^{4} \log \left (x\right )}\right )} \]
integrate(1/4*(((16*x^8+96*x^7+32*x^6-352*x^5-1760*x^3-800*x^2+12000*x-100 00)*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^2-1000*x+625)/x^4)+8* x^6+4*x^5)*log(x)^2+(x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+440*x^3+100*x^ 2-1000*x+625)/x^4)+x^7+x^6)*log(x)-x^5*exp((x^8+8*x^7+4*x^6-88*x^5-71*x^4+ 440*x^3+100*x^2-1000*x+625)/x^4)-x^7-x^6)*exp(1/4*x/log(x))/x^5/log(x)^2,x , algorithm=\
x^2*e^(1/4*x/log(x)) + x*e^(1/4*x/log(x)) + e^(1/4*(4*x^8*log(x) + 32*x^7* log(x) + 16*x^6*log(x) - 352*x^5*log(x) + x^5 - 284*x^4*log(x) + 1760*x^3* log(x) + 400*x^2*log(x) - 4000*x*log(x) + 2500*log(x))/(x^4*log(x)))
Time = 11.22 (sec) , antiderivative size = 78, normalized size of antiderivative = 2.52 \[ \int \frac {e^{\frac {x}{4 \log (x)}} \left (-e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5-x^6-x^7+\left (e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} x^5+x^6+x^7\right ) \log (x)+\left (4 x^5+8 x^6+e^{\frac {625-1000 x+100 x^2+440 x^3-71 x^4-88 x^5+4 x^6+8 x^7+x^8}{x^4}} \left (-10000+12000 x-800 x^2-1760 x^3-352 x^5+32 x^6+96 x^7+16 x^8\right )\right ) \log ^2(x)\right )}{4 x^5 \log ^2(x)} \, dx=x\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (x\right )}}+x^2\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (x\right )}}+{\mathrm {e}}^{-88\,x}\,{\mathrm {e}}^{x^4}\,{\mathrm {e}}^{-71}\,{\mathrm {e}}^{\frac {x}{4\,\ln \left (x\right )}}\,{\mathrm {e}}^{4\,x^2}\,{\mathrm {e}}^{8\,x^3}\,{\mathrm {e}}^{\frac {100}{x^2}}\,{\mathrm {e}}^{440/x}\,{\mathrm {e}}^{\frac {625}{x^4}}\,{\mathrm {e}}^{-\frac {1000}{x^3}} \]
int(-(exp(x/(4*log(x)))*(x^5*exp((100*x^2 - 1000*x + 440*x^3 - 71*x^4 - 88 *x^5 + 4*x^6 + 8*x^7 + x^8 + 625)/x^4) - log(x)^2*(exp((100*x^2 - 1000*x + 440*x^3 - 71*x^4 - 88*x^5 + 4*x^6 + 8*x^7 + x^8 + 625)/x^4)*(12000*x - 80 0*x^2 - 1760*x^3 - 352*x^5 + 32*x^6 + 96*x^7 + 16*x^8 - 10000) + 4*x^5 + 8 *x^6) - log(x)*(x^5*exp((100*x^2 - 1000*x + 440*x^3 - 71*x^4 - 88*x^5 + 4* x^6 + 8*x^7 + x^8 + 625)/x^4) + x^6 + x^7) + x^6 + x^7))/(4*x^5*log(x)^2), x)