3.55.21 \(\int \frac {e^{2 x} (-3+6 e^5+e^{-x} (3+e^5 (-3+x)))+e^x (-36+e^5 (36-12 x))}{144 e^5+e^x (-24 e^{5-x}+e^5 (-48-24 x))+e^{2 x} (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 (4+4 x+x^2))} \, dx\)

Optimal. Leaf size=21 \[ \frac {-4+\frac {3}{e^5}+x}{2-11 e^{-x}+x} \]

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Rubi [F]  time = 1.55, 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 {e^{2 x} \left (-3+6 e^5+e^{-x} \left (3+e^5 (-3+x)\right )\right )+e^x \left (-36+e^5 (36-12 x)\right )}{144 e^5+e^x \left (-24 e^{5-x}+e^5 (-48-24 x)\right )+e^{2 x} \left (e^{5-2 x}+e^{5-x} (4+2 x)+e^5 \left (4+4 x+x^2\right )\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(2*x)*(-3 + 6*E^5 + (3 + E^5*(-3 + x))/E^x) + E^x*(-36 + E^5*(36 - 12*x)))/(144*E^5 + E^x*(-24*E^(5 - x
) + E^5*(-48 - 24*x)) + E^(2*x)*(E^(5 - 2*x) + E^(5 - x)*(4 + 2*x) + E^5*(4 + 4*x + x^2))),x]

[Out]

-33*(1 - E^5)*Defer[Int][E^(-5 + x)/(-11 + 2*E^x + E^x*x)^2, x] - 11*Defer[Int][(E^x*x)/(-11 + 2*E^x + E^x*x)^
2, x] - 33*(1 - 2*E^5)*Defer[Int][E^(-5 + x)/((2 + x)*(-11 + 2*E^x + E^x*x)^2), x] - 3*(1 - 2*E^5)*Defer[Int][
E^(-5 + x)/((2 + x)*(-11 + 2*E^x + E^x*x)), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-5+x} \left (-33-3 e^x \left (1-2 e^5\right )-11 e^5 (-3+x)\right )}{\left (11-e^x (2+x)\right )^2} \, dx\\ &=\int \left (\frac {3 e^{-5+x} \left (-1+2 e^5\right )}{(2+x) \left (-11+2 e^x+e^x x\right )}+\frac {11 e^{-5+x} \left (-3 \left (3-4 e^5\right )-\left (3-e^5\right ) x-e^5 x^2\right )}{(2+x) \left (11-2 e^x-e^x x\right )^2}\right ) \, dx\\ &=11 \int \frac {e^{-5+x} \left (-3 \left (3-4 e^5\right )-\left (3-e^5\right ) x-e^5 x^2\right )}{(2+x) \left (11-2 e^x-e^x x\right )^2} \, dx-\left (3 \left (1-2 e^5\right )\right ) \int \frac {e^{-5+x}}{(2+x) \left (-11+2 e^x+e^x x\right )} \, dx\\ &=11 \int \left (\frac {3 e^{-5+x} \left (-1+e^5\right )}{\left (-11+2 e^x+e^x x\right )^2}-\frac {e^x x}{\left (-11+2 e^x+e^x x\right )^2}+\frac {3 e^{-5+x} \left (-1+2 e^5\right )}{(2+x) \left (-11+2 e^x+e^x x\right )^2}\right ) \, dx-\left (3 \left (1-2 e^5\right )\right ) \int \frac {e^{-5+x}}{(2+x) \left (-11+2 e^x+e^x x\right )} \, dx\\ &=-\left (11 \int \frac {e^x x}{\left (-11+2 e^x+e^x x\right )^2} \, dx\right )-\left (3 \left (1-2 e^5\right )\right ) \int \frac {e^{-5+x}}{(2+x) \left (-11+2 e^x+e^x x\right )} \, dx-\left (33 \left (1-2 e^5\right )\right ) \int \frac {e^{-5+x}}{(2+x) \left (-11+2 e^x+e^x x\right )^2} \, dx-\left (33 \left (1-e^5\right )\right ) \int \frac {e^{-5+x}}{\left (-11+2 e^x+e^x x\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 1.03, size = 33, normalized size = 1.57 \begin {gather*} \frac {11 e^5+3 e^x-6 e^{5+x}}{e^5 \left (-11+e^x (2+x)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(2*x)*(-3 + 6*E^5 + (3 + E^5*(-3 + x))/E^x) + E^x*(-36 + E^5*(36 - 12*x)))/(144*E^5 + E^x*(-24*E^
(5 - x) + E^5*(-48 - 24*x)) + E^(2*x)*(E^(5 - 2*x) + E^(5 - x)*(4 + 2*x) + E^5*(4 + 4*x + x^2))),x]

[Out]

(11*E^5 + 3*E^x - 6*E^(5 + x))/(E^5*(-11 + E^x*(2 + x)))

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fricas [A]  time = 0.63, size = 32, normalized size = 1.52 \begin {gather*} -\frac {3 \, {\left (2 \, e^{5} - 1\right )} e^{x} - 11 \, e^{5}}{{\left (x + 2\right )} e^{\left (x + 5\right )} - 11 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp(5)-36)*exp(x))/((exp(5)*exp(-x)^2+(2
*x+4)*exp(5)*exp(-x)+(x^2+4*x+4)*exp(5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x,
algorithm="fricas")

[Out]

-(3*(2*e^5 - 1)*e^x - 11*e^5)/((x + 2)*e^(x + 5) - 11*e^5)

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giac [A]  time = 0.23, size = 35, normalized size = 1.67 \begin {gather*} \frac {11 \, e^{5} - 6 \, e^{\left (x + 5\right )} + 3 \, e^{x}}{x e^{\left (x + 5\right )} - 11 \, e^{5} + 2 \, e^{\left (x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp(5)-36)*exp(x))/((exp(5)*exp(-x)^2+(2
*x+4)*exp(5)*exp(-x)+(x^2+4*x+4)*exp(5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x,
algorithm="giac")

[Out]

(11*e^5 - 6*e^(x + 5) + 3*e^x)/(x*e^(x + 5) - 11*e^5 + 2*e^(x + 5))

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maple [A]  time = 0.18, size = 41, normalized size = 1.95

method	result	size
norman	\(\frac {\left (11 \,{\mathrm e}^{3 x}-3 \left (2 \,{\mathrm e}^{5}-1\right ) {\mathrm e}^{-5} {\mathrm e}^{4 x}\right ) {\mathrm e}^{-3 x}}{{\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-11}\)	\(41\)
risch	\(-\frac {6 \,{\mathrm e}^{-5} {\mathrm e}^{5}}{2+x}+\frac {3 \,{\mathrm e}^{-5}}{2+x}+\frac {11 \left (x \,{\mathrm e}^{5}-4 \,{\mathrm e}^{5}+3\right ) {\mathrm e}^{-5}}{\left (2+x \right ) \left ({\mathrm e}^{x} x +2 \,{\mathrm e}^{x}-11\right )}\)	\(53\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((x-3)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp(5)-36)*exp(x))/((exp(5)*exp(-x)^2+(2*x+4)*
exp(5)*exp(-x)+(x^2+4*x+4)*exp(5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x,method
=_RETURNVERBOSE)

[Out]

(11*exp(x)^3-3*(2*exp(5)-1)/exp(5)*exp(x)^4)/exp(x)^3/(exp(x)*x+2*exp(x)-11)

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maxima [A]  time = 0.42, size = 36, normalized size = 1.71 \begin {gather*} -\frac {3 \, {\left (2 \, e^{5} - 1\right )} e^{x} - 11 \, e^{5}}{{\left (x e^{5} + 2 \, e^{5}\right )} e^{x} - 11 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)^2+((-12*x+36)*exp(5)-36)*exp(x))/((exp(5)*exp(-x)^2+(2
*x+4)*exp(5)*exp(-x)+(x^2+4*x+4)*exp(5))*exp(x)^2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)),x,
algorithm="maxima")

[Out]

-(3*(2*e^5 - 1)*e^x - 11*e^5)/((x*e^5 + 2*e^5)*e^x - 11*e^5)

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mupad [B]  time = 3.59, size = 27, normalized size = 1.29 \begin {gather*} \frac {{\mathrm {e}}^{x-5}\,\left (x\,{\mathrm {e}}^5-4\,{\mathrm {e}}^5+3\right )}{2\,{\mathrm {e}}^x+x\,{\mathrm {e}}^x-11} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x)*(6*exp(5) + exp(-x)*(exp(5)*(x - 3) + 3) - 3) - exp(x)*(exp(5)*(12*x - 36) + 36))/(144*exp(5) +
exp(2*x)*(exp(-2*x)*exp(5) + exp(5)*(4*x + x^2 + 4) + exp(-x)*exp(5)*(2*x + 4)) - exp(x)*(24*exp(-x)*exp(5) +
exp(5)*(24*x + 48))),x)

[Out]

(exp(x - 5)*(x*exp(5) - 4*exp(5) + 3))/(2*exp(x) + x*exp(x) - 11)

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sympy [A]  time = 0.17, size = 29, normalized size = 1.38 \begin {gather*} \frac {- x e^{5} - 3 + 4 e^{5}}{- x e^{5} - 2 e^{5} + 11 e^{5} e^{- x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((x-3)*exp(5)+3)*exp(-x)+6*exp(5)-3)*exp(x)**2+((-12*x+36)*exp(5)-36)*exp(x))/((exp(5)*exp(-x)**2+
(2*x+4)*exp(5)*exp(-x)+(x**2+4*x+4)*exp(5))*exp(x)**2+(-24*exp(5)*exp(-x)+(-24*x-48)*exp(5))*exp(x)+144*exp(5)
),x)

[Out]

(-x*exp(5) - 3 + 4*exp(5))/(-x*exp(5) - 2*exp(5) + 11*exp(5)*exp(-x))

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