\(\int \frac {(-e^5+x)^{\frac {20 x}{-x^2+\log ^4(7)}} (20 x^3-20 x \log ^4(7)+(20 e^5 x^2-20 x^3+(20 e^5-20 x) \log ^4(7)) \log (-e^5+x))}{e^5 x^4-x^5+(-2 e^5 x^2+2 x^3) \log ^4(7)+(e^5-x) \log ^8(7)} \, dx\) [6696]

Optimal result

Integrand size = 120, antiderivative size = 24 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\left (-e^5+x\right )^{\frac {20}{-x+\frac {\log ^4(7)}{x}}} \]

[Out]

exp(5*ln(-exp(5)+x)/(1/x*ln(7)^4-x))^4

Rubi [F]

\[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx \]

[In]

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)*Lo
g[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]

[Out]

(Hypergeometric2F1[1, (-20*x)/(x^2 - Log[7]^4), 1 - (20*x)/(x^2 - Log[7]^4), (E^5 - x)/(E^5 - Log[7]^2)]*(x^2
- Log[7]^4))/(2*x*(-E^5 + x)^((20*x)/(x^2 - Log[7]^4))*(E^5 - Log[7]^2)) + (Hypergeometric2F1[1, (-20*x)/(x^2
- Log[7]^4), 1 - (20*x)/(x^2 - Log[7]^4), (E^5 - x)/(E^5 + Log[7]^2)]*(x^2 - Log[7]^4))/(2*x*(-E^5 + x)^((20*x
)/(x^2 - Log[7]^4))*(E^5 + Log[7]^2)) + 10*Defer[Int][((-E^5 + x)^((20*x)/(-x^2 + Log[7]^4))*Log[-E^5 + x])/(x
- Log[7]^2)^2, x] + 10*Defer[Int][((-E^5 + x)^((20*x)/(-x^2 + Log[7]^4))*Log[-E^5 + x])/(x + Log[7]^2)^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {20 \left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}} \left (-x^3+x \log ^4(7)-\left (e^5-x\right ) \left (x^2+\log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{\left (x^2-\log ^4(7)\right )^2} \, dx \\ & = 20 \int \frac {\left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}} \left (-x^3+x \log ^4(7)-\left (e^5-x\right ) \left (x^2+\log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{\left (x^2-\log ^4(7)\right )^2} \, dx \\ & = 20 \int \left (-\frac {x \left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}}}{x^2-\log ^4(7)}+\frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (x^2+\log ^4(7)\right ) \log \left (-e^5+x\right )}{\left (x^2-\log ^4(7)\right )^2}\right ) \, dx \\ & = -\left (20 \int \frac {x \left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}}}{x^2-\log ^4(7)} \, dx\right )+20 \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (x^2+\log ^4(7)\right ) \log \left (-e^5+x\right )}{\left (x^2-\log ^4(7)\right )^2} \, dx \\ & = -\left (20 \int \left (\frac {\left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}}}{2 \left (x-\log ^2(7)\right )}+\frac {\left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}}}{2 \left (x+\log ^2(7)\right )}\right ) \, dx\right )+20 \int \left (\frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \log \left (-e^5+x\right )}{2 \left (x-\log ^2(7)\right )^2}+\frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \log \left (-e^5+x\right )}{2 \left (x+\log ^2(7)\right )^2}\right ) \, dx \\ & = -\left (10 \int \frac {\left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}}}{x-\log ^2(7)} \, dx\right )-10 \int \frac {\left (-e^5+x\right )^{-1+\frac {20 x}{-x^2+\log ^4(7)}}}{x+\log ^2(7)} \, dx+10 \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \log \left (-e^5+x\right )}{\left (x-\log ^2(7)\right )^2} \, dx+10 \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \log \left (-e^5+x\right )}{\left (x+\log ^2(7)\right )^2} \, dx \\ & = \frac {\left (-e^5+x\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \, _2F_1\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 ) \left (x^2-\log ^4(7)\right )}{2 x \left (e^5-\log ^2(7)\right )}+\frac {\left (-e^5+x\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \, _2F_1\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 ) \left (x^2-\log ^4(7)\right )}{2 x \left (e^5+\log ^2(7)\right )}+10 \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \log \left (-e^5+x\right )}{\left (x-\log ^2(7)\right )^2} \, dx+10 \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \log \left (-e^5+x\right )}{\left (x+\log ^2(7)\right )^2} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.53 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\left (-e^5+x\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \]

[In]

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]

[Out]

(-E^5 + x)^((-20*x)/(x^2 - Log[7]^4))

Maple [A] (verified)

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	\({\mathrm e}^{\frac {20 x \ln \left (-{\mathrm e}^{5}+x \right )}{\ln \left (7\right )^{4}-x^{2}}}\)	\(26\)

[In]

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)

[Out]

((-exp(5)+x)^(5*x/(ln(7)^4-x^2)))^4

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx={\left (x - e^{5}\right )}^{\frac {20 \, x}{\log \left (7\right )^{4} - x^{2}}} \]

[In]

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="
fricas")

[Out]

(x - e^5)^(20*x/(log(7)^4 - x^2))

Sympy [A] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.79 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=e^{\frac {20 x \log {\left (x - e^{5} \right )}}{- x^{2} + \log {\left (7 \right )}^{4}}} \]

[In]

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)

[Out]

exp(20*x*log(x - exp(5))/(-x**2 + log(7)**4))

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 38, normalized size of antiderivative = 1.58 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=e^{\left (-\frac {10 \, \log \left (x - e^{5}\right )}{\log \left (7\right )^{2} + x} + \frac {10 \, \log \left (x - e^{5}\right )}{\log \left (7\right )^{2} - x}\right )} \]

[In]

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="
maxima")

[Out]

e^(-10*log(x - e^5)/(log(7)^2 + x) + 10*log(x - e^5)/(log(7)^2 - x))

Giac [F]

\[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx=\int { \frac {20 \, {\left (x \log \left (7\right )^{4} - x^{3} + {\left ({\left (x - e^{5}\right )} \log \left (7\right )^{4} + x^{3} - x^{2} e^{5}\right )} \log \left (x - e^{5}\right )\right )} {\left (x - e^{5}\right )}^{\frac {20 \, x}{\log \left (7\right )^{4} - x^{2}}}}{{\left (x - e^{5}\right )} \log \left (7\right )^{8} + x^{5} - x^{4} e^{5} - 2 \, {\left (x^{3} - x^{2} e^{5}\right )} \log \left (7\right )^{4}} \,d x } \]

[In]

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="
giac")

[Out]

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)

Mupad [B] (verification not implemented)

Time = 12.00 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-e^5+x\right )^{\frac {20 x}{-x^2+\log ^4(7)}} \left (20 x^3-20 x \log ^4(7)+\left (20 e^5 x^2-20 x^3+\left (20 e^5-20 x\right ) \log ^4(7)\right ) \log \left (-e^5+x\right )\right )}{e^5 x^4-x^5+\left (-2 e^5 x^2+2 x^3\right ) \log ^4(7)+\left (e^5-x\right ) \log ^8(7)} \, dx={\left (x-{\mathrm {e}}^5\right )}^{\frac {20\,x}{{\ln \left (7\right )}^4-x^2}} \]

[In]

int((exp((20*x*log(x - exp(5)))/(log(7)^4 - x^2))*(20*x*log(7)^4 + log(x - exp(5))*(log(7)^4*(20*x - 20*exp(5)
) - 20*x^2*exp(5) + 20*x^3) - 20*x^3))/(log(7)^4*(2*x^2*exp(5) - 2*x^3) - x^4*exp(5) + log(7)^8*(x - exp(5)) +
x^5),x)

[Out]

(x - exp(5))^((20*x)/(log(7)^4 - x^2))