Integrand size = 120, antiderivative size = 24 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\left (-e^5+x\right )^{\frac {20}{-x+\frac {\log ^4(7)}{x}}} \]
Time = 1.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\left (-e^5+x\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \]
Integrate[((-E^5 + x)^((20*x)/(-x^2 + Log[7]^4))*(20*x^3 - 20*x*Log[7]^4 + (20*E^5*x^2 - 20*x^3 + (20*E^5 - 20*x)*Log[7]^4)*Log[-E^5 + x]))/(E^5*x^4 - x^5 + (-2*E^5*x^2 + 2*x^3)*Log[7]^4 + (E^5 - x)*Log[7]^8),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.
| \(\displaystyle \int \frac {\left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}} \left (20 x^3+\left (-20 x^3+20 e^5 x^2+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (x-e^5\right )-20 x \log ^4(7)\right )}{-x^5+e^5 x^4+\left (2 x^3-2 e^5 x^2\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx\) |
| \(\Big \downarrow \) 2463 |
| \(\displaystyle \int \left (-\frac {\left (20 x^3+\left (-20 x^3+20 e^5 x^2+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (x-e^5\right )-20 x \log ^4(7)\right ) \left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}-1}}{\left (e^{10}-\log ^4(7)\right )^2}-\frac {\left (20 x^3+\left (-20 x^3+20 e^5 x^2+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (x-e^5\right )-20 x \log ^4(7)\right ) \left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}}}{4 \log ^4(7) \left (\log ^2(7)-e^5\right ) \left (\log ^2(7)-x\right )^2}+\frac {\left (20 x^3+\left (-20 x^3+20 e^5 x^2+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (x-e^5\right )-20 x \log ^4(7)\right ) \left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}}}{4 \log ^4(7) \left (e^5+\log ^2(7)\right ) \left (x+\log ^2(7)\right )^2}+\frac {\left (e^5-2 \log ^2(7)\right ) \left (20 x^3+\left (-20 x^3+20 e^5 x^2+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (x-e^5\right )-20 x \log ^4(7)\right ) \left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}}}{4 \log ^6(7) \left (e^5-\log ^2(7)\right )^2 \left (\log ^2(7)-x\right )}+\frac {\left (e^5+2 \log ^2(7)\right ) \left (20 x^3+\left (-20 x^3+20 e^5 x^2+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (x-e^5\right )-20 x \log ^4(7)\right ) \left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}}}{4 \log ^6(7) \left (e^5+\log ^2(7)\right )^2 \left (x+\log ^2(7)\right )}\right )dx\) |
| \(\Big \downarrow \) 7239 |
| \(\displaystyle \int \frac {20 \left (x-e^5\right )^{\frac {20 x}{\log ^4(7)-x^2}-1} \left (-x^3-\left (e^5-x\right ) \left (x^2+\log ^4(7)\right ) \log \left (x-e^5\right )+x \log ^4(7)\right )}{\left (x^2-\log ^4(7)\right )^2}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 20 \int -\frac {\left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}-1} \left (x^3-\log ^4(7) x+\left (e^5-x\right ) \left (x^2+\log ^4(7)\right ) \log \left (x-e^5\right )\right )}{\left (x^2-\log ^4(7)\right )^2}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -20 \int \frac {\left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}-1} \left (x^3-\log ^4(7) x+\left (e^5-x\right ) \left (x^2+\log ^4(7)\right ) \log \left (x-e^5\right )\right )}{\left (x^2-\log ^4(7)\right )^2}dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle -20 \int \left (\frac {x \left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}-1}}{x^2-\log ^4(7)}-\frac {\left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \left (x^2+\log ^4(7)\right ) \log \left (x-e^5\right )}{\left (x^2-\log ^4(7)\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -20 \left (-\frac {1}{2} \int \frac {\left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \log \left (x-e^5\right )}{\left (x-\log ^2(7)\right )^2}dx-\frac {1}{2} \int \frac {\left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \log \left (x-e^5\right )}{\left (x+\log ^2(7)\right )^2}dx-\frac {\left (x^2-\log ^4(7)\right ) \left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \operatorname {Hypergeometric2F1}\left (1,-\frac {20 x}{x^2-\log ^4(7)},1-\frac {20 x}{x^2-\log ^4(7)},\frac {e^5-x}{e^5+\log ^2(7)}\right )}{40 x \left (e^5+\log ^2(7)\right )}-\frac {\left (x^2-\log ^4(7)\right ) \left (x-e^5\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \operatorname {Hypergeometric2F1}\left (1,-\frac {20 x}{x^2-\log ^4(7)},1-\frac {20 x}{x^2-\log ^4(7)},\frac {e^5-x}{e^5-\log ^2(7)}\right )}{40 x \left (e^5-\log ^2(7)\right )}\right )\) |
Int[((-E^5 + x)^((20*x)/(-x^2 + Log[7]^4))*(20*x^3 - 20*x*Log[7]^4 + (20*E ^5*x^2 - 20*x^3 + (20*E^5 - 20*x)*Log[7]^4)*Log[-E^5 + x]))/(E^5*x^4 - x^5 + (-2*E^5*x^2 + 2*x^3)*Log[7]^4 + (E^5 - x)*Log[7]^8),x]
3.7.96.3.1 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[(u_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 17.72 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.04
| method | result | size |
| risch | \(\left (-{\mathrm e}^{5}+x \right )^{\frac {20 x}{\ln \left (7\right )^{4}-x^{2}}}\) | \(25\) |
| parallelrisch | \({\mathrm e}^{\frac {20 x \ln \left (-{\mathrm e}^{5}+x \right )}{\ln \left (7\right )^{4}-x^{2}}}\) | \(26\) |
int((((20*exp(5)-20*x)*ln(7)^4+20*x^2*exp(5)-20*x^3)*ln(-exp(5)+x)-20*x*ln (7)^4+20*x^3)*exp(5*x*ln(-exp(5)+x)/(ln(7)^4-x^2))^4/((exp(5)-x)*ln(7)^8+( -2*x^2*exp(5)+2*x^3)*ln(7)^4+x^4*exp(5)-x^5),x,method=_RETURNVERBOSE)
Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx={\left (x - e^{5}\right )}^{\frac {20 \, x}{\log \left (7\right )^{4} - x^{2}}} \]
integrate((((20*exp(5)-20*x)*log(7)^4+20*x^2*exp(5)-20*x^3)*log(-exp(5)+x) -20*x*log(7)^4+20*x^3)*exp(5*x*log(-exp(5)+x)/(log(7)^4-x^2))^4/((exp(5)-x )*log(7)^8+(-2*x^2*exp(5)+2*x^3)*log(7)^4+x^4*exp(5)-x^5),x, algorithm=\
Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=e^{\frac {20 x \log {\left (x - e^{5} \right )}}{- x^{2} + \log {\left (7 \right )}^{4}}} \]
integrate((((20*exp(5)-20*x)*ln(7)**4+20*x**2*exp(5)-20*x**3)*ln(-exp(5)+x )-20*x*ln(7)**4+20*x**3)*exp(5*x*ln(-exp(5)+x)/(ln(7)**4-x**2))**4/((exp(5 )-x)*ln(7)**8+(-2*x**2*exp(5)+2*x**3)*ln(7)**4+x**4*exp(5)-x**5),x)
Time = 0.43 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=e^{\left (-\frac {10 \, \log \left (x - e^{5}\right )}{\log \left (7\right )^{2} + x} + \frac {10 \, \log \left (x - e^{5}\right )}{\log \left (7\right )^{2} - x}\right )} \]
integrate((((20*exp(5)-20*x)*log(7)^4+20*x^2*exp(5)-20*x^3)*log(-exp(5)+x) -20*x*log(7)^4+20*x^3)*exp(5*x*log(-exp(5)+x)/(log(7)^4-x^2))^4/((exp(5)-x )*log(7)^8+(-2*x^2*exp(5)+2*x^3)*log(7)^4+x^4*exp(5)-x^5),x, algorithm=\
\[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\int { \frac {20 \, {\left (x \log \left (7\right )^{4} - x^{3} + {\left ({\left (x - e^{5}\right )} \log \left (7\right )^{4} + x^{3} - x^{2} e^{5}\right )} \log \left (x - e^{5}\right )\right )} {\left (x - e^{5}\right )}^{\frac {20 \, x}{\log \left (7\right )^{4} - x^{2}}}}{{\left (x - e^{5}\right )} \log \left (7\right )^{8} + x^{5} - x^{4} e^{5} - 2 \, {\left (x^{3} - x^{2} e^{5}\right )} \log \left (7\right )^{4}} \,d x } \]
integrate((((20*exp(5)-20*x)*log(7)^4+20*x^2*exp(5)-20*x^3)*log(-exp(5)+x) -20*x*log(7)^4+20*x^3)*exp(5*x*log(-exp(5)+x)/(log(7)^4-x^2))^4/((exp(5)-x )*log(7)^8+(-2*x^2*exp(5)+2*x^3)*log(7)^4+x^4*exp(5)-x^5),x, algorithm=\
integrate(20*(x*log(7)^4 - x^3 + ((x - e^5)*log(7)^4 + x^3 - x^2*e^5)*log( x - e^5))*(x - e^5)^(20*x/(log(7)^4 - x^2))/((x - e^5)*log(7)^8 + x^5 - x^ 4*e^5 - 2*(x^3 - x^2*e^5)*log(7)^4), x)
Time = 12.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx={\left (x-{\mathrm {e}}^5\right )}^{\frac {20\,x}{{\ln \left (7\right )}^4-x^2}} \]