∫ −9x4−6x5+6x6+e16ex (−4+4x−x2)+e8ex (12x2+6x3−12x4+3x5+ex(−32x4+32x5−8x6))_______________________________________________________________

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

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, $\frac{number of rules}{integrand size}$ = 0.000, Rules used = {} $\begin{matrix} \int \frac{- 9 x^{4} - 6 x^{5} + 6 x^{6} + e^{16 e^{x}} (- 4 + 4 x - x^{2}) + e^{8 e^{x}} (12 x^{2} + 6 x^{3} - 12 x^{4} + 3 x^{5} + e^{x} (- 32 x^{4} + 32 x^{5} - 8 x^{6}))}{9 x^{5} + e^{16 e^{x}} (4 x - 4 x^{2} + x^{3}) + e^{8 e^{x}} (- 12 x^{3} + 6 x^{4})} d x \end{matrix}$

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{matrix} \begin{aligned} integral & = \int \frac{- e^{16 e^{x}} (- 2 + x)^{2} - 8 e^{8 e^{x} + x} (- 2 + x)^{2} x^{4} + 3 x^{4} (- 3 - 2 x + 2 x^{2}) + 3 e^{8 e^{x}} x^{2} (4 + 2 x - 4 x^{2} + x^{3})}{x {(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \\ = \int (- \frac{8 e^{8 e^{x} + x} (- 2 + x)^{2} x^{3}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{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 {(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}}) d x \\ = - (8 \int \frac{e^{8 e^{x} + x} (- 2 + x)^{2} x^{3}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x) + \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 {(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x \\ = - (8 \int (\frac{4 e^{8 e^{x} + x} x^{3}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} - \frac{4 e^{8 e^{x} + x} x^{4}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} + \frac{e^{8 e^{x} + x} x^{5}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}}) d x) + \int \frac{- e^{16 e^{x}} (- 2 + x)^{2} + 3 x^{4} (- 3 - 2 x + 2 x^{2}) + 3 e^{8 e^{x}} x^{2} (4 + 2 x - 4 x^{2} + x^{3})}{x {(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \\ = - (8 \int \frac{e^{8 e^{x} + x} x^{5}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x) - 32 \int \frac{e^{8 e^{x} + x} x^{3}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x + 32 \int \frac{e^{8 e^{x} + x} x^{4}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x + \int (- \frac{1}{x} - \frac{3 (- 4 + x) x^{4}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} + \frac{3 (- 2 + x) x^{2}}{- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2}}) d x \\ = - \log (x) - 3 \int \frac{(- 4 + x) x^{4}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x + 3 \int \frac{(- 2 + x) x^{2}}{- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2}} d x - 8 \int \frac{e^{8 e^{x} + x} x^{5}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x - 32 \int \frac{e^{8 e^{x} + x} x^{3}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 32 \int \frac{e^{8 e^{x} + x} x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \\ = - \log (x) - 3 \int \frac{(- 4 + x) x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 3 \int \frac{(- 2 + x) x^{2}}{e^{8 e^{x}} (- 2 + x) + 3 x^{2}} d x - 8 \int \frac{e^{8 e^{x} + x} x^{5}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x - 32 \int \frac{e^{8 e^{x} + x} x^{3}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 32 \int \frac{e^{8 e^{x} + x} x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \\ = - \log (x) - 3 \int (- \frac{4 x^{4}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} + \frac{x^{5}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}}) d x + 3 \int (- \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}}) d x - 8 \int \frac{e^{8 e^{x} + x} x^{5}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x - 32 \int \frac{e^{8 e^{x} + x} x^{3}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 32 \int \frac{e^{8 e^{x} + x} x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \\ = - \log (x) - 3 \int \frac{x^{5}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x + 3 \int \frac{x^{3}}{- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2}} d x - 6 \int \frac{x^{2}}{- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2}} d x - 8 \int \frac{e^{8 e^{x} + x} x^{5}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 12 \int \frac{x^{4}}{{(- 2 e^{8 e^{x}} + e^{8 e^{x}} x + 3 x^{2})}^{2}} d x - 32 \int \frac{e^{8 e^{x} + x} x^{3}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 32 \int \frac{e^{8 e^{x} + x} x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \\ = - \log (x) - 3 \int \frac{x^{5}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 3 \int \frac{x^{3}}{e^{8 e^{x}} (- 2 + x) + 3 x^{2}} d x - 6 \int \frac{x^{2}}{e^{8 e^{x}} (- 2 + x) + 3 x^{2}} d x - 8 \int \frac{e^{8 e^{x} + x} x^{5}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 12 \int \frac{x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x - 32 \int \frac{e^{8 e^{x} + x} x^{3}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x + 32 \int \frac{e^{8 e^{x} + x} x^{4}}{{(e^{8 e^{x}} (- 2 + x) + 3 x^{2})}^{2}} d x \end{aligned} \end{matrix}$

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

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 $\begin{matrix} \frac{x^{4} - 2 x^{3} - 3 x^{2} \log (x) - (x - 2) e^{(8 e^{x})} \log (x)}{3 x^{2} + (x - 2) e^{(8 e^{x})}} \end{matrix}$

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 $\begin{matrix} \frac{x^{4} - 2 x^{3} - 3 x^{2} \log (x) - x e^{(8 e^{x})} \log (x) + 2 e^{(8 e^{x})} \log (x)}{3 x^{2} + x e^{(8 e^{x})} - 2 e^{(8 e^{x})}} \end{matrix}$

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	$- \ln (x) + \frac{(x - 2) x^{3}}{e^{8 e^{x}} x + 3 x^{2} - 2 e^{8 e^{x}}}$	$35$

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 $\begin{matrix} \frac{x^{4} - 2 x^{3}}{3 x^{2} + (x - 2) e^{(8 e^{x})}} - \log (x) \end{matrix}$

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 $\begin{matrix} - \frac{3 x^{2} \ln (x) + 2 x^{3} - x^{4} - 2 e^{8 e^{x}} \ln (x) + x e^{8 e^{x}} \ln (x)}{x e^{8 e^{x}} - 2 e^{8 e^{x}} + 3 x^{2}} \end{matrix}$

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 $\begin{matrix} - \log (x) + \frac{x^{4} - 2 x^{3}}{3 x^{2} + (x - 2) e^{8 e^{x}}} \end{matrix}$

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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3.41.50 ∫−9x4−6x5+6x6+e16ex(−4+4x−x2)+e8ex(12x2+6x3−12x4+3x5+ex(−32x4+32x5−8x6))9x5+e16ex(4x−4x2+x3)+e8ex(−12x3+6x4)dx

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