3.64.85 \(\int \frac {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} (e^x+10 x)}{e^{-e^x+x}-x+2 x^3-x^4} \, dx\)

Optimal. Leaf size=30 \[ x+5 x^2-\log \left (-e^{-e^x+x}+x+(-2+x) x^3\right ) \]

________________________________________________________________________________________

Rubi [F]  time = 106.66, 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 {1-x-16 x^2+6 x^3+19 x^4-10 x^5+e^{-e^x+x} \left (e^x+10 x\right )}{e^{-e^x+x}-x+2 x^3-x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - x - 16*x^2 + 6*x^3 + 19*x^4 - 10*x^5 + E^(-E^x + x)*(E^x + 10*x))/(E^(-E^x + x) - x + 2*x^3 - x^4),x]

[Out]

E^x + 5*x^2 + Defer[Int][E^E^x*x, x] - 2*Defer[Int][E^E^x*x^3, x] + Defer[Int][E^E^x*x^4, x] + Defer[Int][E^E^
x/(E^x - E^E^x*(x - 2*x^3 + x^4)), x] + Defer[Int][(E^E^x*x)/(-E^x + E^E^x*(x - 2*x^3 + x^4)), x] + 6*Defer[In
t][(E^E^x*x^2)/(-E^x + E^E^x*(x - 2*x^3 + x^4)), x] - Defer[Int][(E^(2*E^x)*x^2)/(-E^x + E^E^x*(x - 2*x^3 + x^
4)), x] - 6*Defer[Int][(E^E^x*x^3)/(-E^x + E^E^x*(x - 2*x^3 + x^4)), x] + Defer[Int][(E^E^x*x^4)/(-E^x + E^E^x
*(x - 2*x^3 + x^4)), x] + 4*Defer[Int][(E^(2*E^x)*x^4)/(-E^x + E^E^x*(x - 2*x^3 + x^4)), x] - 2*Defer[Int][(E^
(2*E^x)*x^5)/(-E^x + E^E^x*(x - 2*x^3 + x^4)), x] - 4*Defer[Int][(E^(2*E^x)*x^6)/(-E^x + E^E^x*(x - 2*x^3 + x^
4)), x] + 4*Defer[Int][(E^(2*E^x)*x^7)/(-E^x + E^E^x*(x - 2*x^3 + x^4)), x] - Defer[Int][(E^(2*E^x)*x^8)/(-E^x
+ E^E^x*(x - 2*x^3 + x^4)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (e^x+x \left (10+e^{e^x}-2 e^{e^x} x^2+e^{e^x} x^3\right )-\frac {e^{e^x} \left (1-x-6 x^2+e^{e^x} x^2+6 x^3-x^4-4 e^{e^x} x^4+2 e^{e^x} x^5+4 e^{e^x} x^6-4 e^{e^x} x^7+e^{e^x} x^8\right )}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}\right ) \, dx\\ &=\int e^x \, dx+\int x \left (10+e^{e^x}-2 e^{e^x} x^2+e^{e^x} x^3\right ) \, dx-\int \frac {e^{e^x} \left (1-x-6 x^2+e^{e^x} x^2+6 x^3-x^4-4 e^{e^x} x^4+2 e^{e^x} x^5+4 e^{e^x} x^6-4 e^{e^x} x^7+e^{e^x} x^8\right )}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx\\ &=e^x+\int \left (10 x+e^{e^x} x \left (1-2 x^2+x^3\right )\right ) \, dx-\int \left (-\frac {e^{e^x}}{e^x-e^{e^x} x+2 e^{e^x} x^3-e^{e^x} x^4}-\frac {e^{e^x} x}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}-\frac {6 e^{e^x} x^2}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}+\frac {e^{2 e^x} x^2}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}+\frac {6 e^{e^x} x^3}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}-\frac {e^{e^x} x^4}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}-\frac {4 e^{2 e^x} x^4}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}+\frac {2 e^{2 e^x} x^5}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}+\frac {4 e^{2 e^x} x^6}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}-\frac {4 e^{2 e^x} x^7}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}+\frac {e^{2 e^x} x^8}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4}\right ) \, dx\\ &=e^x+5 x^2-2 \int \frac {e^{2 e^x} x^5}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx+4 \int \frac {e^{2 e^x} x^4}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx-4 \int \frac {e^{2 e^x} x^6}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx+4 \int \frac {e^{2 e^x} x^7}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx+6 \int \frac {e^{e^x} x^2}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx-6 \int \frac {e^{e^x} x^3}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx+\int e^{e^x} x \left (1-2 x^2+x^3\right ) \, dx+\int \frac {e^{e^x}}{e^x-e^{e^x} x+2 e^{e^x} x^3-e^{e^x} x^4} \, dx+\int \frac {e^{e^x} x}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx-\int \frac {e^{2 e^x} x^2}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx+\int \frac {e^{e^x} x^4}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx-\int \frac {e^{2 e^x} x^8}{-e^x+e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4} \, dx\\ &=e^x+5 x^2-2 \int \frac {e^{2 e^x} x^5}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+4 \int \frac {e^{2 e^x} x^4}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-4 \int \frac {e^{2 e^x} x^6}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+4 \int \frac {e^{2 e^x} x^7}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+6 \int \frac {e^{e^x} x^2}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-6 \int \frac {e^{e^x} x^3}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+\int \left (e^{e^x} x-2 e^{e^x} x^3+e^{e^x} x^4\right ) \, dx+\int \frac {e^{e^x}}{e^x-e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+\int \frac {e^{e^x} x}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-\int \frac {e^{2 e^x} x^2}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+\int \frac {e^{e^x} x^4}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-\int \frac {e^{2 e^x} x^8}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx\\ &=e^x+5 x^2-2 \int e^{e^x} x^3 \, dx-2 \int \frac {e^{2 e^x} x^5}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+4 \int \frac {e^{2 e^x} x^4}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-4 \int \frac {e^{2 e^x} x^6}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+4 \int \frac {e^{2 e^x} x^7}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+6 \int \frac {e^{e^x} x^2}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-6 \int \frac {e^{e^x} x^3}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+\int e^{e^x} x \, dx+\int e^{e^x} x^4 \, dx+\int \frac {e^{e^x}}{e^x-e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+\int \frac {e^{e^x} x}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-\int \frac {e^{2 e^x} x^2}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx+\int \frac {e^{e^x} x^4}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx-\int \frac {e^{2 e^x} x^8}{-e^x+e^{e^x} \left (x-2 x^3+x^4\right )} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 5.08, size = 35, normalized size = 1.17 \begin {gather*} e^x+x+5 x^2-\log \left (e^x-e^{e^x} x \left (1-2 x^2+x^3\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(1 - x - 16*x^2 + 6*x^3 + 19*x^4 - 10*x^5 + E^(-E^x + x)*(E^x + 10*x))/(E^(-E^x + x) - x + 2*x^3 - x
^4),x]

[Out]

E^x + x + 5*x^2 - Log[E^x - E^E^x*x*(1 - 2*x^2 + x^3)]

________________________________________________________________________________________

fricas [A]  time = 0.73, size = 31, normalized size = 1.03 \begin {gather*} 5 \, x^{2} + x - \log \left (-x^{4} + 2 \, x^{3} - x + e^{\left (x - e^{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(exp(x-exp(x))-x^4+2*x^3-x),x, algorith
m="fricas")

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.32, size = 31, normalized size = 1.03 \begin {gather*} 5 \, x^{2} + x - \log \left (-x^{4} + 2 \, x^{3} - x + e^{\left (x - e^{x}\right )}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(exp(x-exp(x))-x^4+2*x^3-x),x, algorith
m="giac")

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.08, size = 30, normalized size = 1.00

method	result	size
norman	\(5 x^{2}+x -\ln \left (x^{4}-2 x^{3}+x -{\mathrm e}^{x -{\mathrm e}^{x}}\right )\)	\(30\)
risch	\(5 x^{2}+x -\ln \left ({\mathrm e}^{x -{\mathrm e}^{x}}-x^{4}+2 x^{3}-x \right )\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(exp(x-exp(x))-x^4+2*x^3-x),x,method=_RETURNV
ERBOSE)

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 0.57, size = 65, normalized size = 2.17 \begin {gather*} 5 \, x^{2} + x + e^{x} - \log \left (x^{2} - x - 1\right ) - \log \left (x - 1\right ) - \log \relax (x) - \log \left (\frac {{\left (x^{4} - 2 \, x^{3} + x\right )} e^{\left (e^{x}\right )} - e^{x}}{x^{4} - 2 \, x^{3} + x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x^5+19*x^4+6*x^3-16*x^2-x+1)/(exp(x-exp(x))-x^4+2*x^3-x),x, algorith
m="maxima")

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 4.39, size = 29, normalized size = 0.97 \begin {gather*} x-\ln \left (x-{\mathrm {e}}^{x-{\mathrm {e}}^x}-2\,x^3+x^4\right )+5\,x^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(6*x^3 - 16*x^2 - x + 19*x^4 - 10*x^5 + exp(x - exp(x))*(10*x + exp(x)) + 1)/(x - exp(x - exp(x)) - 2*x^3
+ x^4),x)

[Out]

x - log(x - exp(x - exp(x)) - 2*x^3 + x^4) + 5*x^2

________________________________________________________________________________________

sympy [A]  time = 0.19, size = 24, normalized size = 0.80 \begin {gather*} 5 x^{2} + x - \log {\left (- x^{4} + 2 x^{3} - x + e^{x - e^{x}} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((exp(x)+10*x)*exp(x-exp(x))-10*x**5+19*x**4+6*x**3-16*x**2-x+1)/(exp(x-exp(x))-x**4+2*x**3-x),x)

[Out]

5*x**2 + x - log(-x**4 + 2*x**3 - x + exp(x - exp(x)))

________________________________________________________________________________________