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

Optimal. Leaf size=24 \[ \left (-e^5+x\right )^{\frac {20}{-x+\frac {\log ^4(7)}{x}}} \]

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Rubi [F]  time = 2.61, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 23, normalized size = 0.96 \begin {gather*} \left (-e^5+x\right )^{-\frac {20 x}{x^2-\log ^4(7)}} \end {gather*}

Antiderivative was successfully verified.

[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))

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fricas [A]  time = 0.64, size = 22, normalized size = 0.92 \begin {gather*} {\left (x - e^{5}\right )}^{\frac {20 \, x}{\log \relax (7)^{4} - x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [B]  time = 0.83, size = 89, normalized size = 3.71 \begin {gather*} {\left (x - e^{5}\right )}^{\frac {20 \, {\left (x - e^{5}\right )}}{\log \relax (7)^{4} - {\left (x - e^{5}\right )}^{2} - 2 \, {\left (x - e^{5}\right )} e^{5} - e^{10}}} {\left (x - e^{5}\right )}^{\frac {20 \, e^{5}}{\log \relax (7)^{4} - {\left (x - e^{5}\right )}^{2} - 2 \, {\left (x - e^{5}\right )} e^{5} - e^{10}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

(x - e^5)^(20*(x - e^5)/(log(7)^4 - (x - e^5)^2 - 2*(x - e^5)*e^5 - e^10))*(x - e^5)^(20*e^5/(log(7)^4 - (x -
e^5)^2 - 2*(x - e^5)*e^5 - e^10))

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maple [A]  time = 0.13, size = 25, normalized size = 1.04

method	result	size
risch	\(\left (-{\mathrm e}^{5}+x \right )^{\frac {20 x}{\ln \relax (7)^{4}-x^{2}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.60, size = 38, normalized size = 1.58 \begin {gather*} e^{\left (-\frac {10 \, \log \left (x - e^{5}\right )}{\log \relax (7)^{2} + x} + \frac {10 \, \log \left (x - e^{5}\right )}{\log \relax (7)^{2} - x}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 4.98, size = 22, normalized size = 0.92 \begin {gather*} {\left (x-{\mathrm {e}}^5\right )}^{\frac {20\,x}{{\ln \relax (7)}^4-x^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.62, size = 19, normalized size = 0.79 \begin {gather*} e^{\frac {20 x \log {\left (x - e^{5} \right )}}{- x^{2} + \log {\relax (7 )}^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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