3.57.77 \(\int \frac {9+e^{2 x} (1+e^4-e^9)+6 x+x^2+e^9 (-9-6 x-x^2)+e^4 (13+6 x+x^2)+e^x (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x))}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 (9+6 x+x^2)} \, dx\)

Optimal. Leaf size=24 \[ 1+x+\frac {x}{e^4}-e^5 x-\frac {4}{3+e^x+x} \]

________________________________________________________________________________________

Rubi [F]  time = 0.55, 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+e^{2 x} \left (1+e^4-e^9\right )+6 x+x^2+e^9 \left (-9-6 x-x^2\right )+e^4 \left (13+6 x+x^2\right )+e^x \left (6+e^9 (-6-2 x)+2 x+e^4 (10+2 x)\right )}{e^{4+2 x}+e^{4+x} (6+2 x)+e^4 \left (9+6 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(9 + E^(2*x)*(1 + E^4 - E^9) + 6*x + x^2 + E^9*(-9 - 6*x - x^2) + E^4*(13 + 6*x + x^2) + E^x*(6 + E^9*(-6
- 2*x) + 2*x + E^4*(10 + 2*x)))/(E^(4 + 2*x) + E^(4 + x)*(6 + 2*x) + E^4*(9 + 6*x + x^2)),x]

[Out]

((1 + E^4 - E^9)*x)/E^4 - 8*Defer[Int][(3 + E^x + x)^(-2), x] - 4*Defer[Int][x/(3 + E^x + x)^2, x] + 4*Defer[I
nt][(3 + E^x + x)^(-1), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{2 x} \left (1+e^4-e^9\right )+2 e^x \left (1-e^9\right ) (3+x)+\left (1-e^9\right ) (3+x)^2+2 e^{4+x} (5+x)+e^4 \left (13+6 x+x^2\right )}{e^4 \left (3+e^x+x\right )^2} \, dx\\ &=\frac {\int \frac {e^{2 x} \left (1+e^4-e^9\right )+2 e^x \left (1-e^9\right ) (3+x)+\left (1-e^9\right ) (3+x)^2+2 e^{4+x} (5+x)+e^4 \left (13+6 x+x^2\right )}{\left (3+e^x+x\right )^2} \, dx}{e^4}\\ &=\frac {\int \left (1+e^4-e^9-\frac {4 e^4 (2+x)}{\left (3+e^x+x\right )^2}+\frac {4 e^4}{3+e^x+x}\right ) \, dx}{e^4}\\ &=\frac {\left (1+e^4-e^9\right ) x}{e^4}-4 \int \frac {2+x}{\left (3+e^x+x\right )^2} \, dx+4 \int \frac {1}{3+e^x+x} \, dx\\ &=\frac {\left (1+e^4-e^9\right ) x}{e^4}+4 \int \frac {1}{3+e^x+x} \, dx-4 \int \left (\frac {2}{\left (3+e^x+x\right )^2}+\frac {x}{\left (3+e^x+x\right )^2}\right ) \, dx\\ &=\frac {\left (1+e^4-e^9\right ) x}{e^4}-4 \int \frac {x}{\left (3+e^x+x\right )^2} \, dx+4 \int \frac {1}{3+e^x+x} \, dx-8 \int \frac {1}{\left (3+e^x+x\right )^2} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.05, size = 30, normalized size = 1.25 \begin {gather*} \frac {\left (1+e^4-e^9\right ) x-\frac {4 e^4}{3+e^x+x}}{e^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(9 + E^(2*x)*(1 + E^4 - E^9) + 6*x + x^2 + E^9*(-9 - 6*x - x^2) + E^4*(13 + 6*x + x^2) + E^x*(6 + E^
9*(-6 - 2*x) + 2*x + E^4*(10 + 2*x)))/(E^(4 + 2*x) + E^(4 + x)*(6 + 2*x) + E^4*(9 + 6*x + x^2)),x]

[Out]

((1 + E^4 - E^9)*x - (4*E^4)/(3 + E^x + x))/E^4

________________________________________________________________________________________

fricas [B]  time = 0.49, size = 67, normalized size = 2.79 \begin {gather*} -\frac {{\left (x^{2} + 3 \, x\right )} e^{13} - {\left (x^{2} + 3 \, x - 4\right )} e^{8} - {\left (x^{2} + 3 \, x\right )} e^{4} + {\left (x e^{9} - x e^{4} - x\right )} e^{\left (x + 4\right )}}{{\left (x + 3\right )} e^{8} + e^{\left (x + 8\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-
9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x,
algorithm="fricas")

[Out]

-((x^2 + 3*x)*e^13 - (x^2 + 3*x - 4)*e^8 - (x^2 + 3*x)*e^4 + (x*e^9 - x*e^4 - x)*e^(x + 4))/((x + 3)*e^8 + e^(
x + 8))

________________________________________________________________________________________

giac [B]  time = 0.16, size = 71, normalized size = 2.96 \begin {gather*} -\frac {x^{2} e^{9} - x^{2} e^{4} - x^{2} + 3 \, x e^{9} - 3 \, x e^{4} + x e^{\left (x + 9\right )} - x e^{\left (x + 4\right )} - x e^{x} - 3 \, x + 4 \, e^{4}}{x e^{4} + 3 \, e^{4} + e^{\left (x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-
9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x,
algorithm="giac")

[Out]

-(x^2*e^9 - x^2*e^4 - x^2 + 3*x*e^9 - 3*x*e^4 + x*e^(x + 9) - x*e^(x + 4) - x*e^x - 3*x + 4*e^4)/(x*e^4 + 3*e^
4 + e^(x + 4))

________________________________________________________________________________________

maple [A]  time = 0.14, size = 28, normalized size = 1.17

method	result	size
risch	\({\mathrm e}^{4} {\mathrm e}^{-4} x -x \,{\mathrm e}^{-4} {\mathrm e}^{9}+x \,{\mathrm e}^{-4}-\frac {4}{{\mathrm e}^{x}+3+x}\)	\(28\)
norman	\(\frac {3 \,{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) {\mathrm e}^{x}-{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x^{2}-{\mathrm e}^{-4} \left ({\mathrm e}^{4} {\mathrm e}^{5}-{\mathrm e}^{4}-1\right ) x \,{\mathrm e}^{x}+{\mathrm e}^{-4} \left (9 \,{\mathrm e}^{4} {\mathrm e}^{5}-13 \,{\mathrm e}^{4}-9\right )}{{\mathrm e}^{x}+3+x}\)	\(86\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-9)*exp
(4)*exp(5)+(x^2+6*x+13)*exp(4)+x^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x,method=
_RETURNVERBOSE)

[Out]

exp(4)*exp(-4)*x-x*exp(-4)*exp(9)+x*exp(-4)-4/(exp(x)+3+x)

________________________________________________________________________________________

maxima [B]  time = 0.40, size = 57, normalized size = 2.38 \begin {gather*} -\frac {x^{2} {\left (e^{9} - e^{4} - 1\right )} + x {\left (e^{9} - e^{4} - 1\right )} e^{x} + 3 \, x {\left (e^{9} - e^{4} - 1\right )} + 4 \, e^{4}}{x e^{4} + 3 \, e^{4} + e^{\left (x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)^2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*exp(4)+2*x+6)*exp(x)+(-x^2-6*x-
9)*exp(4)*exp(5)+(x^2+6*x+13)*exp(4)+x^2+6*x+9)/(exp(4)*exp(x)^2+(2*x+6)*exp(4)*exp(x)+(x^2+6*x+9)*exp(4)),x,
algorithm="maxima")

[Out]

-(x^2*(e^9 - e^4 - 1) + x*(e^9 - e^4 - 1)*e^x + 3*x*(e^9 - e^4 - 1) + 4*e^4)/(x*e^4 + 3*e^4 + e^(x + 4))

________________________________________________________________________________________

mupad [B]  time = 0.17, size = 30, normalized size = 1.25 \begin {gather*} x+x\,{\mathrm {e}}^{-4}-x\,{\mathrm {e}}^5-\frac {4\,{\mathrm {e}}^4}{{\mathrm {e}}^{x+4}+3\,{\mathrm {e}}^4+x\,{\mathrm {e}}^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((6*x + exp(2*x)*(exp(4) - exp(9) + 1) + exp(x)*(2*x + exp(4)*(2*x + 10) - exp(9)*(2*x + 6) + 6) + exp(4)*(
6*x + x^2 + 13) - exp(9)*(6*x + x^2 + 9) + x^2 + 9)/(exp(2*x)*exp(4) + exp(4)*(6*x + x^2 + 9) + exp(4)*exp(x)*
(2*x + 6)),x)

[Out]

x + x*exp(-4) - x*exp(5) - (4*exp(4))/(exp(x + 4) + 3*exp(4) + x*exp(4))

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 20, normalized size = 0.83 \begin {gather*} \frac {x \left (- e^{9} + 1 + e^{4}\right )}{e^{4}} - \frac {4}{x + e^{x} + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-exp(4)*exp(5)+exp(4)+1)*exp(x)**2+((-2*x-6)*exp(4)*exp(5)+(2*x+10)*exp(4)+2*x+6)*exp(x)+(-x**2-6*
x-9)*exp(4)*exp(5)+(x**2+6*x+13)*exp(4)+x**2+6*x+9)/(exp(4)*exp(x)**2+(2*x+6)*exp(4)*exp(x)+(x**2+6*x+9)*exp(4
)),x)

[Out]

x*(-exp(9) + 1 + exp(4))*exp(-4) - 4/(x + exp(x) + 3)

________________________________________________________________________________________