3.25.69 \(\int \frac {2-4 e^9 x+2 e^{18} x^2+e^{\frac {e^9 x+(-1+e^9 x) \log (2) \log (3)}{-1+e^9 x}} (-2+2 e^9 x-2 e^{18} x^2)}{1-2 x+x^2+e^{\frac {2 (e^9 x+(-1+e^9 x) \log (2) \log (3))}{-1+e^9 x}} (1-2 e^9 x+e^{18} x^2)+e^9 (-2 x+4 x^2-2 x^3)+e^{18} (x^2-2 x^3+x^4)+e^{\frac {e^9 x+(-1+e^9 x) \log (2) \log (3)}{-1+e^9 x}} (-2+2 x+e^9 (4 x-4 x^2)+e^{18} (-2 x^2+2 x^3))} \, dx\)

Optimal. Leaf size=34 \[ -5+\frac {2 x}{1-e^{-\frac {x}{\frac {1}{e^9}-x}+\log (2) \log (3)}-x} \]

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Rubi [F]  time = 11.59, 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 {2-4 e^9 x+2 e^{18} x^2+\exp \left (\frac {e^9 x+\left (-1+e^9 x\right ) \log (2) \log (3)}{-1+e^9 x}\right ) \left (-2+2 e^9 x-2 e^{18} x^2\right )}{1-2 x+x^2+\exp \left (\frac {2 \left (e^9 x+\left (-1+e^9 x\right ) \log (2) \log (3)\right )}{-1+e^9 x}\right ) \left (1-2 e^9 x+e^{18} x^2\right )+e^9 \left (-2 x+4 x^2-2 x^3\right )+e^{18} \left (x^2-2 x^3+x^4\right )+\exp \left (\frac {e^9 x+\left (-1+e^9 x\right ) \log (2) \log (3)}{-1+e^9 x}\right ) \left (-2+2 x+e^9 \left (4 x-4 x^2\right )+e^{18} \left (-2 x^2+2 x^3\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(2 - 4*E^9*x + 2*E^18*x^2 + E^((E^9*x + (-1 + E^9*x)*Log[2]*Log[3])/(-1 + E^9*x))*(-2 + 2*E^9*x - 2*E^18*x
^2))/(1 - 2*x + x^2 + E^((2*(E^9*x + (-1 + E^9*x)*Log[2]*Log[3]))/(-1 + E^9*x))*(1 - 2*E^9*x + E^18*x^2) + E^9
*(-2*x + 4*x^2 - 2*x^3) + E^18*(x^2 - 2*x^3 + x^4) + E^((E^9*x + (-1 + E^9*x)*Log[2]*Log[3])/(-1 + E^9*x))*(-2
+ 2*x + E^9*(4*x - 4*x^2) + E^18*(-2*x^2 + 2*x^3))),x]

[Out]

(2*Defer[Int][(-1 + E^((E^9*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x)^(-2), x])/E^9 + 2*Defer[Int][x/(-1 + E^((E^9
*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x)^2, x] - 2*Defer[Int][(-1 + E^((E^9*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x
)^(-1), x] - 2*(1 - E^(-9))*Defer[Int][1/((-1 + E^((E^9*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x)^2*(-1 + E^9*x)^2
), x] - 2*Defer[Int][1/((-1 + E^((E^9*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x)*(-1 + E^9*x)^2), x] + (2*(2 - E^9)
*Defer[Int][1/((-1 + E^((E^9*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x)^2*(-1 + E^9*x)), x])/E^9 - 2*Defer[Int][1/(
(-1 + E^((E^9*x)/(-1 + E^9*x) + Log[2]*Log[3]) + x)*(-1 + E^9*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {2 \left (1-e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}-2 e^9 x+e^{9+\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)} x+e^{18} x^2-e^{18+\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)} x^2\right )}{\left (1-e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}-x\right )^2 \left (1-e^9 x\right )^2} \, dx\\ &=2 \int \frac {1-e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}-2 e^9 x+e^{9+\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)} x+e^{18} x^2-e^{18+\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)} x^2}{\left (1-e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}-x\right )^2 \left (1-e^9 x\right )^2} \, dx\\ &=2 \int \left (-\frac {1-e^9 x+e^{18} x^2}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right ) \left (-1+e^9 x\right )^2}+\frac {x \left (1-e^9-e^9 x+e^{18} x^2\right )}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2 \left (-1+e^9 x\right )^2}\right ) \, dx\\ &=-\left (2 \int \frac {1-e^9 x+e^{18} x^2}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right ) \left (-1+e^9 x\right )^2} \, dx\right )+2 \int \frac {x \left (1-e^9-e^9 x+e^{18} x^2\right )}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2 \left (-1+e^9 x\right )^2} \, dx\\ &=2 \int \left (\frac {1}{e^9 \left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2}+\frac {x}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2}-\frac {-1+e^9}{e^9 \left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2 \left (-1+e^9 x\right )^2}-\frac {-2+e^9}{e^9 \left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2 \left (-1+e^9 x\right )}\right ) \, dx-2 \int \left (\frac {1}{-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x}+\frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right ) \left (-1+e^9 x\right )^2}+\frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right ) \left (-1+e^9 x\right )}\right ) \, dx\\ &=2 \int \frac {x}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2} \, dx-2 \int \frac {1}{-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x} \, dx-2 \int \frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right ) \left (-1+e^9 x\right )^2} \, dx-2 \int \frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right ) \left (-1+e^9 x\right )} \, dx+\frac {2 \int \frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2} \, dx}{e^9}-\frac {\left (2 \left (-2+e^9\right )\right ) \int \frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2 \left (-1+e^9 x\right )} \, dx}{e^9}-\frac {\left (2 \left (-1+e^9\right )\right ) \int \frac {1}{\left (-1+e^{\frac {e^9 x}{-1+e^9 x}+\log (2) \log (3)}+x\right )^2 \left (-1+e^9 x\right )^2} \, dx}{e^9}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.25, size = 26, normalized size = 0.76 \begin {gather*} -\frac {2 x}{-1+e^{1+\frac {1}{-1+e^9 x}+\log (2) \log (3)}+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 4*E^9*x + 2*E^18*x^2 + E^((E^9*x + (-1 + E^9*x)*Log[2]*Log[3])/(-1 + E^9*x))*(-2 + 2*E^9*x - 2*
E^18*x^2))/(1 - 2*x + x^2 + E^((2*(E^9*x + (-1 + E^9*x)*Log[2]*Log[3]))/(-1 + E^9*x))*(1 - 2*E^9*x + E^18*x^2)
+ E^9*(-2*x + 4*x^2 - 2*x^3) + E^18*(x^2 - 2*x^3 + x^4) + E^((E^9*x + (-1 + E^9*x)*Log[2]*Log[3])/(-1 + E^9*x
))*(-2 + 2*x + E^9*(4*x - 4*x^2) + E^18*(-2*x^2 + 2*x^3))),x]

[Out]

(-2*x)/(-1 + E^(1 + (-1 + E^9*x)^(-1) + Log[2]*Log[3]) + x)

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fricas [A]  time = 1.03, size = 34, normalized size = 1.00 \begin {gather*} -\frac {2 \, x}{x + e^{\left (\frac {{\left (x e^{9} - 1\right )} \log \relax (3) \log \relax (2) + x e^{9}}{x e^{9} - 1}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(9)^2+2*x*exp(9)-2)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))+2*x^2*exp(9)
^2-4*x*exp(9)+2)/((x^2*exp(9)^2-2*x*exp(9)+1)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))^2+((2*x^
3-2*x^2)*exp(9)^2+(-4*x^2+4*x)*exp(9)+2*x-2)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))+(x^4-2*x^
3+x^2)*exp(9)^2+(-2*x^3+4*x^2-2*x)*exp(9)+x^2-2*x+1),x, algorithm="fricas")

[Out]

-2*x/(x + e^(((x*e^9 - 1)*log(3)*log(2) + x*e^9)/(x*e^9 - 1)) - 1)

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giac [A]  time = 8.40, size = 27, normalized size = 0.79 \begin {gather*} -\frac {2 \, x}{x + e^{\left (\log \relax (3) \log \relax (2) + \frac {x e^{9}}{x e^{9} - 1}\right )} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(9)^2+2*x*exp(9)-2)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))+2*x^2*exp(9)
^2-4*x*exp(9)+2)/((x^2*exp(9)^2-2*x*exp(9)+1)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))^2+((2*x^
3-2*x^2)*exp(9)^2+(-4*x^2+4*x)*exp(9)+2*x-2)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))+(x^4-2*x^
3+x^2)*exp(9)^2+(-2*x^3+4*x^2-2*x)*exp(9)+x^2-2*x+1),x, algorithm="giac")

[Out]

-2*x/(x + e^(log(3)*log(2) + x*e^9/(x*e^9 - 1)) - 1)

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maple [A]  time = 3.44, size = 38, normalized size = 1.12

method	result	size
risch	\(-\frac {2 x}{x +{\mathrm e}^{\frac {\ln \relax (2) \ln \relax (3) {\mathrm e}^{9} x -\ln \relax (2) \ln \relax (3)+x \,{\mathrm e}^{9}}{x \,{\mathrm e}^{9}-1}}-1}\)	\(38\)
norman	\(\frac {2 x -2 x^{2} {\mathrm e}^{9}}{\left (x +{\mathrm e}^{\frac {\left (x \,{\mathrm e}^{9}-1\right ) \ln \relax (2) \ln \relax (3)+x \,{\mathrm e}^{9}}{x \,{\mathrm e}^{9}-1}}-1\right ) \left (x \,{\mathrm e}^{9}-1\right )}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-2*x^2*exp(9)^2+2*x*exp(9)-2)*exp(((x*exp(9)-1)*ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))+2*x^2*exp(9)^2-4*x*e
xp(9)+2)/((x^2*exp(9)^2-2*x*exp(9)+1)*exp(((x*exp(9)-1)*ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))^2+((2*x^3-2*x^2)*e
xp(9)^2+(-4*x^2+4*x)*exp(9)+2*x-2)*exp(((x*exp(9)-1)*ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))+(x^4-2*x^3+x^2)*exp(9
)^2+(-2*x^3+4*x^2-2*x)*exp(9)+x^2-2*x+1),x,method=_RETURNVERBOSE)

[Out]

-2*x/(x+exp((ln(2)*ln(3)*exp(9)*x-ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))-1)

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maxima [A]  time = 1.09, size = 24, normalized size = 0.71 \begin {gather*} -\frac {2 \, x}{2^{\log \relax (3)} e^{\left (\frac {1}{x e^{9} - 1} + 1\right )} + x - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x^2*exp(9)^2+2*x*exp(9)-2)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))+2*x^2*exp(9)
^2-4*x*exp(9)+2)/((x^2*exp(9)^2-2*x*exp(9)+1)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))^2+((2*x^
3-2*x^2)*exp(9)^2+(-4*x^2+4*x)*exp(9)+2*x-2)*exp(((x*exp(9)-1)*log(2)*log(3)+x*exp(9))/(x*exp(9)-1))+(x^4-2*x^
3+x^2)*exp(9)^2+(-2*x^3+4*x^2-2*x)*exp(9)+x^2-2*x+1),x, algorithm="maxima")

[Out]

-2*x/(2^log(3)*e^(1/(x*e^9 - 1) + 1) + x - 1)

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mupad [B]  time = 1.74, size = 26, normalized size = 0.76 \begin {gather*} -\frac {2\,x}{x+2^{\ln \relax (3)}\,{\mathrm {e}}^{\frac {x\,{\mathrm {e}}^9}{x\,{\mathrm {e}}^9-1}}-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(4*x*exp(9) - 2*x^2*exp(18) + exp((x*exp(9) + log(2)*log(3)*(x*exp(9) - 1))/(x*exp(9) - 1))*(2*x^2*exp(18
) - 2*x*exp(9) + 2) - 2)/(exp(18)*(x^2 - 2*x^3 + x^4) - 2*x - exp(9)*(2*x - 4*x^2 + 2*x^3) + exp((x*exp(9) + l
og(2)*log(3)*(x*exp(9) - 1))/(x*exp(9) - 1))*(2*x + exp(9)*(4*x - 4*x^2) - exp(18)*(2*x^2 - 2*x^3) - 2) + exp(
(2*(x*exp(9) + log(2)*log(3)*(x*exp(9) - 1)))/(x*exp(9) - 1))*(x^2*exp(18) - 2*x*exp(9) + 1) + x^2 + 1),x)

[Out]

-(2*x)/(x + 2^log(3)*exp((x*exp(9))/(x*exp(9) - 1)) - 1)

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sympy [A]  time = 0.35, size = 34, normalized size = 1.00 \begin {gather*} - \frac {2 x}{x + e^{\frac {x e^{9} + \left (x e^{9} - 1\right ) \log {\relax (2 )} \log {\relax (3 )}}{x e^{9} - 1}} - 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-2*x**2*exp(9)**2+2*x*exp(9)-2)*exp(((x*exp(9)-1)*ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))+2*x**2*exp(9
)**2-4*x*exp(9)+2)/((x**2*exp(9)**2-2*x*exp(9)+1)*exp(((x*exp(9)-1)*ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))**2+((2
*x**3-2*x**2)*exp(9)**2+(-4*x**2+4*x)*exp(9)+2*x-2)*exp(((x*exp(9)-1)*ln(2)*ln(3)+x*exp(9))/(x*exp(9)-1))+(x**
4-2*x**3+x**2)*exp(9)**2+(-2*x**3+4*x**2-2*x)*exp(9)+x**2-2*x+1),x)

[Out]

-2*x/(x + exp((x*exp(9) + (x*exp(9) - 1)*log(2)*log(3))/(x*exp(9) - 1)) - 1)

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