3.41.50 \(\int \frac {-9 x^4-6 x^5+6 x^6+e^{16 e^x} (-4+4 x-x^2)+e^{8 e^x} (12 x^2+6 x^3-12 x^4+3 x^5+e^x (-32 x^4+32 x^5-8 x^6))}{9 x^5+e^{16 e^x} (4 x-4 x^2+x^3)+e^{8 e^x} (-12 x^3+6 x^4)} \, dx\)

Optimal. Leaf size=28 \[ \frac {x}{\frac {3}{-2+x}+\frac {e^{8 e^x}}{x^2}}-\log (x) \]

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Rubi [F]  time = 6.39, 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 {-9 x^4-6 x^5+6 x^6+e^{16 e^x} \left (-4+4 x-x^2\right )+e^{8 e^x} \left (12 x^2+6 x^3-12 x^4+3 x^5+e^x \left (-32 x^4+32 x^5-8 x^6\right )\right )}{9 x^5+e^{16 e^x} \left (4 x-4 x^2+x^3\right )+e^{8 e^x} \left (-12 x^3+6 x^4\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-9*x^4 - 6*x^5 + 6*x^6 + E^(16*E^x)*(-4 + 4*x - x^2) + E^(8*E^x)*(12*x^2 + 6*x^3 - 12*x^4 + 3*x^5 + E^x*(
-32*x^4 + 32*x^5 - 8*x^6)))/(9*x^5 + E^(16*E^x)*(4*x - 4*x^2 + x^3) + E^(8*E^x)*(-12*x^3 + 6*x^4)),x]

[Out]

-Log[x] - 32*Defer[Int][(E^(8*E^x + x)*x^3)/(E^(8*E^x)*(-2 + x) + 3*x^2)^2, x] + 12*Defer[Int][x^4/(E^(8*E^x)*
(-2 + x) + 3*x^2)^2, x] + 32*Defer[Int][(E^(8*E^x + x)*x^4)/(E^(8*E^x)*(-2 + x) + 3*x^2)^2, x] - 3*Defer[Int][
x^5/(E^(8*E^x)*(-2 + x) + 3*x^2)^2, x] - 8*Defer[Int][(E^(8*E^x + x)*x^5)/(E^(8*E^x)*(-2 + x) + 3*x^2)^2, x] -
6*Defer[Int][x^2/(E^(8*E^x)*(-2 + x) + 3*x^2), x] + 3*Defer[Int][x^3/(E^(8*E^x)*(-2 + x) + 3*x^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-e^{16 e^x} (-2+x)^2-8 e^{8 e^x+x} (-2+x)^2 x^4+3 x^4 \left (-3-2 x+2 x^2\right )+3 e^{8 e^x} x^2 \left (4+2 x-4 x^2+x^3\right )}{x \left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ &=\int \left (-\frac {8 e^{8 e^x+x} (-2+x)^2 x^3}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}+\frac {-4 e^{16 e^x}+4 e^{16 e^x} x+12 e^{8 e^x} x^2-e^{16 e^x} x^2+6 e^{8 e^x} x^3-9 x^4-12 e^{8 e^x} x^4-6 x^5+3 e^{8 e^x} x^5+6 x^6}{x \left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}\right ) \, dx\\ &=-\left (8 \int \frac {e^{8 e^x+x} (-2+x)^2 x^3}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx\right )+\int \frac {-4 e^{16 e^x}+4 e^{16 e^x} x+12 e^{8 e^x} x^2-e^{16 e^x} x^2+6 e^{8 e^x} x^3-9 x^4-12 e^{8 e^x} x^4-6 x^5+3 e^{8 e^x} x^5+6 x^6}{x \left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx\\ &=-\left (8 \int \left (\frac {4 e^{8 e^x+x} x^3}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}-\frac {4 e^{8 e^x+x} x^4}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}+\frac {e^{8 e^x+x} x^5}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}\right ) \, dx\right )+\int \frac {-e^{16 e^x} (-2+x)^2+3 x^4 \left (-3-2 x+2 x^2\right )+3 e^{8 e^x} x^2 \left (4+2 x-4 x^2+x^3\right )}{x \left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ &=-\left (8 \int \frac {e^{8 e^x+x} x^5}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx\right )-32 \int \frac {e^{8 e^x+x} x^3}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx+32 \int \frac {e^{8 e^x+x} x^4}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx+\int \left (-\frac {1}{x}-\frac {3 (-4+x) x^4}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}+\frac {3 (-2+x) x^2}{-2 e^{8 e^x}+e^{8 e^x} x+3 x^2}\right ) \, dx\\ &=-\log (x)-3 \int \frac {(-4+x) x^4}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx+3 \int \frac {(-2+x) x^2}{-2 e^{8 e^x}+e^{8 e^x} x+3 x^2} \, dx-8 \int \frac {e^{8 e^x+x} x^5}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx-32 \int \frac {e^{8 e^x+x} x^3}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+32 \int \frac {e^{8 e^x+x} x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ &=-\log (x)-3 \int \frac {(-4+x) x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+3 \int \frac {(-2+x) x^2}{e^{8 e^x} (-2+x)+3 x^2} \, dx-8 \int \frac {e^{8 e^x+x} x^5}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx-32 \int \frac {e^{8 e^x+x} x^3}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+32 \int \frac {e^{8 e^x+x} x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ &=-\log (x)-3 \int \left (-\frac {4 x^4}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}+\frac {x^5}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2}\right ) \, dx+3 \int \left (-\frac {2 x^2}{-2 e^{8 e^x}+e^{8 e^x} x+3 x^2}+\frac {x^3}{-2 e^{8 e^x}+e^{8 e^x} x+3 x^2}\right ) \, dx-8 \int \frac {e^{8 e^x+x} x^5}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx-32 \int \frac {e^{8 e^x+x} x^3}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+32 \int \frac {e^{8 e^x+x} x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ &=-\log (x)-3 \int \frac {x^5}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx+3 \int \frac {x^3}{-2 e^{8 e^x}+e^{8 e^x} x+3 x^2} \, dx-6 \int \frac {x^2}{-2 e^{8 e^x}+e^{8 e^x} x+3 x^2} \, dx-8 \int \frac {e^{8 e^x+x} x^5}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+12 \int \frac {x^4}{\left (-2 e^{8 e^x}+e^{8 e^x} x+3 x^2\right )^2} \, dx-32 \int \frac {e^{8 e^x+x} x^3}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+32 \int \frac {e^{8 e^x+x} x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ &=-\log (x)-3 \int \frac {x^5}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+3 \int \frac {x^3}{e^{8 e^x} (-2+x)+3 x^2} \, dx-6 \int \frac {x^2}{e^{8 e^x} (-2+x)+3 x^2} \, dx-8 \int \frac {e^{8 e^x+x} x^5}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+12 \int \frac {x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx-32 \int \frac {e^{8 e^x+x} x^3}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx+32 \int \frac {e^{8 e^x+x} x^4}{\left (e^{8 e^x} (-2+x)+3 x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.10, size = 31, normalized size = 1.11 \begin {gather*} \frac {(-2+x) x^3}{e^{8 e^x} (-2+x)+3 x^2}-\log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-9*x^4 - 6*x^5 + 6*x^6 + E^(16*E^x)*(-4 + 4*x - x^2) + E^(8*E^x)*(12*x^2 + 6*x^3 - 12*x^4 + 3*x^5 +
E^x*(-32*x^4 + 32*x^5 - 8*x^6)))/(9*x^5 + E^(16*E^x)*(4*x - 4*x^2 + x^3) + E^(8*E^x)*(-12*x^3 + 6*x^4)),x]

[Out]

((-2 + x)*x^3)/(E^(8*E^x)*(-2 + x) + 3*x^2) - Log[x]

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fricas [A]  time = 0.62, size = 46, normalized size = 1.64 \begin {gather*} \frac {x^{4} - 2 \, x^{3} - 3 \, x^{2} \log \relax (x) - {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )} \log \relax (x)}{3 \, x^{2} + {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x)
)+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm="fricas
")

[Out]

(x^4 - 2*x^3 - 3*x^2*log(x) - (x - 2)*e^(8*e^x)*log(x))/(3*x^2 + (x - 2)*e^(8*e^x))

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giac [B]  time = 0.28, size = 58, normalized size = 2.07 \begin {gather*} \frac {x^{4} - 2 \, x^{3} - 3 \, x^{2} \log \relax (x) - x e^{\left (8 \, e^{x}\right )} \log \relax (x) + 2 \, e^{\left (8 \, e^{x}\right )} \log \relax (x)}{3 \, x^{2} + x e^{\left (8 \, e^{x}\right )} - 2 \, e^{\left (8 \, e^{x}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x)
)+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm="giac")

[Out]

(x^4 - 2*x^3 - 3*x^2*log(x) - x*e^(8*e^x)*log(x) + 2*e^(8*e^x)*log(x))/(3*x^2 + x*e^(8*e^x) - 2*e^(8*e^x))

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maple [A]  time = 0.05, size = 35, normalized size = 1.25

method	result	size
risch	\(-\ln \relax (x )+\frac {\left (x -2\right ) x^{3}}{{\mathrm e}^{8 \,{\mathrm e}^{x}} x +3 x^{2}-2 \,{\mathrm e}^{8 \,{\mathrm e}^{x}}}\)	\(35\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x))+6*x^
6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x,method=_RETURNVERBOSE)

[Out]

-ln(x)+(x-2)*x^3/(exp(8*exp(x))*x+3*x^2-2*exp(8*exp(x)))

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maxima [A]  time = 0.43, size = 32, normalized size = 1.14 \begin {gather*} \frac {x^{4} - 2 \, x^{3}}{3 \, x^{2} + {\left (x - 2\right )} e^{\left (8 \, e^{x}\right )}} - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2+4*x-4)*exp(8*exp(x))^2+((-8*x^6+32*x^5-32*x^4)*exp(x)+3*x^5-12*x^4+6*x^3+12*x^2)*exp(8*exp(x)
)+6*x^6-6*x^5-9*x^4)/((x^3-4*x^2+4*x)*exp(8*exp(x))^2+(6*x^4-12*x^3)*exp(8*exp(x))+9*x^5),x, algorithm="maxima
")

[Out]

(x^4 - 2*x^3)/(3*x^2 + (x - 2)*e^(8*e^x)) - log(x)

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mupad [B]  time = 3.10, size = 60, normalized size = 2.14 \begin {gather*} -\frac {3\,x^2\,\ln \relax (x)+2\,x^3-x^4-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\ln \relax (x)+x\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}\,\ln \relax (x)}{x\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}-2\,{\mathrm {e}}^{8\,{\mathrm {e}}^x}+3\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(16*exp(x))*(x^2 - 4*x + 4) - exp(8*exp(x))*(12*x^2 - exp(x)*(32*x^4 - 32*x^5 + 8*x^6) + 6*x^3 - 12*x
^4 + 3*x^5) + 9*x^4 + 6*x^5 - 6*x^6)/(exp(16*exp(x))*(4*x - 4*x^2 + x^3) - exp(8*exp(x))*(12*x^3 - 6*x^4) + 9*
x^5),x)

[Out]

-(3*x^2*log(x) + 2*x^3 - x^4 - 2*exp(8*exp(x))*log(x) + x*exp(8*exp(x))*log(x))/(x*exp(8*exp(x)) - 2*exp(8*exp
(x)) + 3*x^2)

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sympy [A]  time = 0.24, size = 26, normalized size = 0.93 \begin {gather*} - \log {\relax (x )} + \frac {x^{4} - 2 x^{3}}{3 x^{2} + \left (x - 2\right ) e^{8 e^{x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2+4*x-4)*exp(8*exp(x))**2+((-8*x**6+32*x**5-32*x**4)*exp(x)+3*x**5-12*x**4+6*x**3+12*x**2)*exp
(8*exp(x))+6*x**6-6*x**5-9*x**4)/((x**3-4*x**2+4*x)*exp(8*exp(x))**2+(6*x**4-12*x**3)*exp(8*exp(x))+9*x**5),x)

[Out]

-log(x) + (x**4 - 2*x**3)/(3*x**2 + (x - 2)*exp(8*exp(x)))

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