3.41.84 \(\int \frac {e^{\frac {1}{\log ^4(2 x^5-x^6+(2 x-x^2) \log (36))}} (40 x^4-24 x^5+(8-8 x) \log (36))}{(-2 x^5+x^6+(-2 x+x^2) \log (36)) \log ^5(2 x^5-x^6+(2 x-x^2) \log (36))} \, dx\)

Optimal. Leaf size=18 \[ e^{\frac {1}{\log ^4\left ((2-x) x \left (x^4+\log (36)\right )\right )}} \]

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Rubi [F]  time = 5.06, 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 {e^{\frac {1}{\log ^4\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )}} \left (40 x^4-24 x^5+(8-8 x) \log (36)\right )}{\left (-2 x^5+x^6+\left (-2 x+x^2\right ) \log (36)\right ) \log ^5\left (2 x^5-x^6+\left (2 x-x^2\right ) \log (36)\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^(-4)*(40*x^4 - 24*x^5 + (8 - 8*x)*Log[36]))/((-2*x^5 + x^6 + (-2
*x + x^2)*Log[36])*Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^5),x]

[Out]

-4*Defer[Int][E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)/((-2 + x)*Log[-((-2 + x)*x*(x^4 + Log[36]))]^5), x] -
4*Defer[Int][E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)/(x*Log[-((-2 + x)*x*(x^4 + Log[36]))]^5), x] + 4*Defer[
Int][E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)/((-x + (-Log[36])^(1/4))*Log[-((-2 + x)*x*(x^4 + Log[36]))]^5),
x] - 4*Defer[Int][E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)/((x + (-Log[36])^(1/4))*Log[-((-2 + x)*x*(x^4 + L
og[36]))]^5), x] + 4*Defer[Int][E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)/((-x - (-1)^(3/4)*Log[36]^(1/4))*Log
[-((-2 + x)*x*(x^4 + Log[36]))]^5), x] - 4*Defer[Int][E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)/((x - (-1)^(3/
4)*Log[36]^(1/4))*Log[-((-2 + x)*x*(x^4 + Log[36]))]^5), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {8 e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} \left (-5 x^4+3 x^5-\log (36)+x \log (36)\right )}{(2-x) x \left (x^4+\log (36)\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\\ &=8 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} \left (-5 x^4+3 x^5-\log (36)+x \log (36)\right )}{(2-x) x \left (x^4+\log (36)\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\\ &=8 \int \left (-\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{2 (-2+x) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}-\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{2 x \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}-\frac {2 e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} x^3}{\left (x^4+\log (36)\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{(-2+x) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\right )-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-16 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} x^3}{\left (x^4+\log (36)\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{(-2+x) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\right )-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-16 \int \left (\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} x}{2 \left (x^2-i \sqrt {\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}+\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} x}{2 \left (x^2+i \sqrt {\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{(-2+x) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\right )-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-8 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} x}{\left (x^2-i \sqrt {\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-8 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} x}{\left (x^2+i \sqrt {\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{(-2+x) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\right )-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-8 \int \left (-\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{2 \left (-x+\sqrt [4]{-\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}+\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{2 \left (x+\sqrt [4]{-\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}\right ) \, dx-8 \int \left (-\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{2 \left (-x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}+\frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{2 \left (x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}\right ) \, dx\\ &=-\left (4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{(-2+x) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\right )-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{x \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx+4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (-x+\sqrt [4]{-\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (x+\sqrt [4]{-\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx+4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (-x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx-4 \int \frac {e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}}}{\left (x-(-1)^{3/4} \sqrt [4]{\log (36)}\right ) \log ^5\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.88, size = 17, normalized size = 0.94 \begin {gather*} e^{\frac {1}{\log ^4\left (-\left ((-2+x) x \left (x^4+\log (36)\right )\right )\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^(-4)*(40*x^4 - 24*x^5 + (8 - 8*x)*Log[36]))/((-2*x^5 + x^6
+ (-2*x + x^2)*Log[36])*Log[2*x^5 - x^6 + (2*x - x^2)*Log[36]]^5),x]

[Out]

E^Log[-((-2 + x)*x*(x^4 + Log[36]))]^(-4)

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fricas [A]  time = 0.55, size = 26, normalized size = 1.44 \begin {gather*} e^{\left (\frac {1}{\log \left (-x^{6} + 2 \, x^{5} - 2 \, {\left (x^{2} - 2 \, x\right )} \log \relax (6)\right )^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^
6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^5,x, algorithm="fricas")

[Out]

e^(log(-x^6 + 2*x^5 - 2*(x^2 - 2*x)*log(6))^(-4))

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giac [A]  time = 0.13, size = 27, normalized size = 1.50 \begin {gather*} e^{\left (\frac {1}{\log \left (-x^{6} + 2 \, x^{5} - 2 \, x^{2} \log \relax (6) + 4 \, x \log \relax (6)\right )^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^
6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^5,x, algorithm="giac")

[Out]

e^(log(-x^6 + 2*x^5 - 2*x^2*log(6) + 4*x*log(6))^(-4))

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maple [A]  time = 0.04, size = 32, normalized size = 1.78

method	result	size
risch	\({\mathrm e}^{\frac {1}{\ln \left (2 \left (-x^{2}+2 x \right ) \left (\ln \relax (2)+\ln \relax (3)\right )-x^{6}+2 x^{5}\right )^{4}}}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*(-8*x+8)*ln(6)-24*x^5+40*x^4)*exp(1/ln(2*(-x^2+2*x)*ln(6)-x^6+2*x^5)^4)/(2*(x^2-2*x)*ln(6)+x^6-2*x^5)/l
n(2*(-x^2+2*x)*ln(6)-x^6+2*x^5)^5,x,method=_RETURNVERBOSE)

[Out]

exp(1/ln(2*(-x^2+2*x)*(ln(2)+ln(3))-x^6+2*x^5)^4)

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maxima [B]  time = 0.76, size = 1148, normalized size = 63.78 result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x+8)*log(6)-24*x^5+40*x^4)*exp(1/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^4)/(2*(x^2-2*x)*log(6)+x^
6-2*x^5)/log(2*(-x^2+2*x)*log(6)-x^6+2*x^5)^5,x, algorithm="maxima")

[Out]

3*x^5*e^(1/(log(x^4 + 2*log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*log(x)^3 + log
(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*
log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^3 + log(-x + 2)^4 + 6*(log(x^4 + 2*log(3) + 2*log(2))^2 + 2*log(x^4
+ 2*log(3) + 2*log(2))*log(-x + 2) + log(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 2*log(2))^3 + 3*log(x^
4 + 2*log(3) + 2*log(2))^2*log(-x + 2) + 3*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^2 + log(-x + 2)^3)*log(x
)))/(3*x^5 - 5*x^4 + 2*x*(log(3) + log(2)) - 2*log(3) - 2*log(2)) - 5*x^4*e^(1/(log(x^4 + 2*log(3) + 2*log(2))
^4 + 4*(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*log(x)^3 + log(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2))^3
*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)
^3 + log(-x + 2)^4 + 6*(log(x^4 + 2*log(3) + 2*log(2))^2 + 2*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2) + log(
-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 2*log(2))^3 + 3*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2) + 3
*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^2 + log(-x + 2)^3)*log(x)))/(3*x^5 - 5*x^4 + 2*x*(log(3) + log(2))
- 2*log(3) - 2*log(2)) + 2*x*e^(1/(log(x^4 + 2*log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*log(2)) + log
(-x + 2))*log(x)^3 + log(x)^4 + 4*log(x^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log(
2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^3 + log(-x + 2)^4 + 6*(log(x^4 + 2*log(3) +
2*log(2))^2 + 2*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2) + log(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3)
+ 2*log(2))^3 + 3*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2) + 3*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^
2 + log(-x + 2)^3)*log(x)))*log(6)/(3*x^5 - 5*x^4 + 2*x*(log(3) + log(2)) - 2*log(3) - 2*log(2)) - 2*e^(1/(log
(x^4 + 2*log(3) + 2*log(2))^4 + 4*(log(x^4 + 2*log(3) + 2*log(2)) + log(-x + 2))*log(x)^3 + log(x)^4 + 4*log(x
^4 + 2*log(3) + 2*log(2))^3*log(-x + 2) + 6*log(x^4 + 2*log(3) + 2*log(2))^2*log(-x + 2)^2 + 4*log(x^4 + 2*log
(3) + 2*log(2))*log(-x + 2)^3 + log(-x + 2)^4 + 6*(log(x^4 + 2*log(3) + 2*log(2))^2 + 2*log(x^4 + 2*log(3) + 2
*log(2))*log(-x + 2) + log(-x + 2)^2)*log(x)^2 + 4*(log(x^4 + 2*log(3) + 2*log(2))^3 + 3*log(x^4 + 2*log(3) +
2*log(2))^2*log(-x + 2) + 3*log(x^4 + 2*log(3) + 2*log(2))*log(-x + 2)^2 + log(-x + 2)^3)*log(x)))*log(6)/(3*x
^5 - 5*x^4 + 2*x*(log(3) + log(2)) - 2*log(3) - 2*log(2))

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mupad [B]  time = 3.84, size = 27, normalized size = 1.50 \begin {gather*} {\mathrm {e}}^{\frac {1}{{\ln \left (-x^6+2\,x^5-2\,\ln \relax (6)\,x^2+4\,\ln \relax (6)\,x\right )}^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(1/log(2*log(6)*(2*x - x^2) + 2*x^5 - x^6)^4)*(2*log(6)*(8*x - 8) - 40*x^4 + 24*x^5))/(log(2*log(6)*(2
*x - x^2) + 2*x^5 - x^6)^5*(2*log(6)*(2*x - x^2) + 2*x^5 - x^6)),x)

[Out]

exp(1/log(4*x*log(6) - 2*x^2*log(6) + 2*x^5 - x^6)^4)

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sympy [A]  time = 0.65, size = 26, normalized size = 1.44 \begin {gather*} e^{\frac {1}{\log {\left (- x^{6} + 2 x^{5} + \left (- 2 x^{2} + 4 x\right ) \log {\relax (6 )} \right )}^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*(-8*x+8)*ln(6)-24*x**5+40*x**4)*exp(1/ln(2*(-x**2+2*x)*ln(6)-x**6+2*x**5)**4)/(2*(x**2-2*x)*ln(6)
+x**6-2*x**5)/ln(2*(-x**2+2*x)*ln(6)-x**6+2*x**5)**5,x)

[Out]

exp(log(-x**6 + 2*x**5 + (-2*x**2 + 4*x)*log(6))**(-4))

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