3.49.46 \(\int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} (45+e^{e^{-4+x}} (-81+3 e+e^{-4+x} (180+9 x-9 x^2+e (-15+3 x))))}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx\)

Optimal. Leaf size=28 \[ e^{\frac {-5+e^{e^{-4+x}} (5-x)}{4-\frac {e}{3}+x}} \]

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Rubi [F]  time = 12.38, 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^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{144+e^2+e (-24-6 x)+72 x+9 x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))*(45 + E^E^(-4 + x)*(-81 + 3*E + E^(-4 + x)*(180 + 9*x
- 9*x^2 + E*(-15 + 3*x)))))/(144 + E^2 + E*(-24 - 6*x) + 72*x + 9*x^2),x]

[Out]

-Defer[Int][E^(-4 + E^(-4 + x) + x + (15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x)), x] + 45*Defer[Int][E^((
15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))/(-12 + E - 3*x)^2, x] - 3*(27 - E)*Defer[Int][E^(E^(-4 + x) +
(15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))/(-12 + E - 3*x)^2, x] - (27 - E)*Defer[Int][E^(-4 + E^(-4 + x
) + x + (15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))/(-12 + E - 3*x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{(-12+e)^2+6 (12-e) x+9 x^2} \, dx\\ &=\int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (45+e^{e^{-4+x}} \left (-81+3 e+e^{-4+x} \left (180+9 x-9 x^2+e (-15+3 x)\right )\right )\right )}{(-12+e-3 x)^2} \, dx\\ &=\int \left (\frac {3 \exp \left (-4+e^{-4+x}+x+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right ) (-5+x)}{-12+e-3 x}+\frac {3 e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (15-27 \left (1-\frac {e}{27}\right ) e^{e^{-4+x}}\right )}{(12-e+3 x)^2}\right ) \, dx\\ &=3 \int \frac {\exp \left (-4+e^{-4+x}+x+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right ) (-5+x)}{-12+e-3 x} \, dx+3 \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}} \left (15-27 \left (1-\frac {e}{27}\right ) e^{e^{-4+x}}\right )}{(12-e+3 x)^2} \, dx\\ &=3 \int \left (\frac {15 e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}}}{(-12+e-3 x)^2}+\frac {(-27+e) \exp \left (e^{-4+x}+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right )}{(-12+e-3 x)^2}\right ) \, dx+3 \int \left (-\frac {1}{3} \exp \left (-4+e^{-4+x}+x+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right )+\frac {(-27+e) \exp \left (-4+e^{-4+x}+x+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right )}{3 (-12+e-3 x)}\right ) \, dx\\ &=45 \int \frac {e^{\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}}}{(-12+e-3 x)^2} \, dx-(3 (27-e)) \int \frac {\exp \left (e^{-4+x}+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right )}{(-12+e-3 x)^2} \, dx+(-27+e) \int \frac {\exp \left (-4+e^{-4+x}+x+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right )}{-12+e-3 x} \, dx-\int \exp \left (-4+e^{-4+x}+x+\frac {15+e^{e^{-4+x}} (-15+3 x)}{-12+e-3 x}\right ) \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 3.53, size = 26, normalized size = 0.93 \begin {gather*} e^{\frac {3 \left (5+e^{e^{-4+x}} (-5+x)\right )}{e-3 (4+x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((15 + E^E^(-4 + x)*(-15 + 3*x))/(-12 + E - 3*x))*(45 + E^E^(-4 + x)*(-81 + 3*E + E^(-4 + x)*(180
+ 9*x - 9*x^2 + E*(-15 + 3*x)))))/(144 + E^2 + E*(-24 - 6*x) + 72*x + 9*x^2),x]

[Out]

E^((3*(5 + E^E^(-4 + x)*(-5 + x)))/(E - 3*(4 + x)))

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fricas [A]  time = 0.66, size = 25, normalized size = 0.89 \begin {gather*} e^{\left (-\frac {3 \, {\left ({\left (x - 5\right )} e^{\left (e^{\left (x - 4\right )}\right )} + 5\right )}}{3 \, x - e + 12}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(x-4)+3*exp(1)-81)*exp(exp(x-4))+45)*exp(((3*x-15)*exp(exp(x-4)
)+15)/(exp(1)-3*x-12))/(exp(1)^2+(-6*x-24)*exp(1)+9*x^2+72*x+144),x, algorithm="fricas")

[Out]

e^(-3*((x - 5)*e^(e^(x - 4)) + 5)/(3*x - e + 12))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int -\frac {3 \, {\left ({\left ({\left (3 \, x^{2} - {\left (x - 5\right )} e - 3 \, x - 60\right )} e^{\left (x - 4\right )} - e + 27\right )} e^{\left (e^{\left (x - 4\right )}\right )} - 15\right )} e^{\left (-\frac {3 \, {\left ({\left (x - 5\right )} e^{\left (e^{\left (x - 4\right )}\right )} + 5\right )}}{3 \, x - e + 12}\right )}}{9 \, x^{2} - 6 \, {\left (x + 4\right )} e + 72 \, x + e^{2} + 144}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(x-4)+3*exp(1)-81)*exp(exp(x-4))+45)*exp(((3*x-15)*exp(exp(x-4)
)+15)/(exp(1)-3*x-12))/(exp(1)^2+(-6*x-24)*exp(1)+9*x^2+72*x+144),x, algorithm="giac")

[Out]

integrate(-3*(((3*x^2 - (x - 5)*e - 3*x - 60)*e^(x - 4) - e + 27)*e^(e^(x - 4)) - 15)*e^(-3*((x - 5)*e^(e^(x -
4)) + 5)/(3*x - e + 12))/(9*x^2 - 6*(x + 4)*e + 72*x + e^2 + 144), x)

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maple [A]  time = 0.46, size = 29, normalized size = 1.04

method	result	size
risch	\({\mathrm e}^{\frac {3 x \,{\mathrm e}^{{\mathrm e}^{x -4}}-15 \,{\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}\)	\(29\)
norman	\(\frac {\left ({\mathrm e}-12\right ) {\mathrm e}^{\frac {\left (3 x -15\right ) {\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}-3 x \,{\mathrm e}^{\frac {\left (3 x -15\right ) {\mathrm e}^{{\mathrm e}^{x -4}}+15}{{\mathrm e}-3 x -12}}}{{\mathrm e}-3 x -12}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(x-4)+3*exp(1)-81)*exp(exp(x-4))+45)*exp(((3*x-15)*exp(exp(x-4))+15)/
(exp(1)-3*x-12))/(exp(1)^2+(-6*x-24)*exp(1)+9*x^2+72*x+144),x,method=_RETURNVERBOSE)

[Out]

exp(3*(x*exp(exp(x-4))-5*exp(exp(x-4))+5)/(exp(1)-3*x-12))

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maxima [B]  time = 0.56, size = 60, normalized size = 2.14 \begin {gather*} e^{\left (-\frac {e^{\left (e^{\left (x - 4\right )} + 1\right )}}{3 \, x - e + 12} + \frac {27 \, e^{\left (e^{\left (x - 4\right )}\right )}}{3 \, x - e + 12} - \frac {15}{3 \, x - e + 12} - e^{\left (e^{\left (x - 4\right )}\right )}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((3*x-15)*exp(1)-9*x^2+9*x+180)*exp(x-4)+3*exp(1)-81)*exp(exp(x-4))+45)*exp(((3*x-15)*exp(exp(x-4)
)+15)/(exp(1)-3*x-12))/(exp(1)^2+(-6*x-24)*exp(1)+9*x^2+72*x+144),x, algorithm="maxima")

[Out]

e^(-e^(e^(x - 4) + 1)/(3*x - e + 12) + 27*e^(e^(x - 4))/(3*x - e + 12) - 15/(3*x - e + 12) - e^(e^(x - 4)))

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mupad [B]  time = 0.48, size = 56, normalized size = 2.00 \begin {gather*} {\mathrm {e}}^{-\frac {15}{3\,x-\mathrm {e}+12}}\,{\mathrm {e}}^{\frac {15\,{\mathrm {e}}^{{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}}{3\,x-\mathrm {e}+12}}\,{\mathrm {e}}^{-\frac {3\,x\,{\mathrm {e}}^{{\mathrm {e}}^{-4}\,{\mathrm {e}}^x}}{3\,x-\mathrm {e}+12}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-(exp(exp(x - 4))*(3*x - 15) + 15)/(3*x - exp(1) + 12))*(exp(exp(x - 4))*(3*exp(1) + exp(x - 4)*(9*x
- 9*x^2 + exp(1)*(3*x - 15) + 180) - 81) + 45))/(72*x + exp(2) + 9*x^2 - exp(1)*(6*x + 24) + 144),x)

[Out]

exp(-15/(3*x - exp(1) + 12))*exp((15*exp(exp(-4)*exp(x)))/(3*x - exp(1) + 12))*exp(-(3*x*exp(exp(-4)*exp(x)))/
(3*x - exp(1) + 12))

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sympy [A]  time = 1.24, size = 22, normalized size = 0.79 \begin {gather*} e^{\frac {\left (3 x - 15\right ) e^{e^{x - 4}} + 15}{- 3 x - 12 + e}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((((3*x-15)*exp(1)-9*x**2+9*x+180)*exp(x-4)+3*exp(1)-81)*exp(exp(x-4))+45)*exp(((3*x-15)*exp(exp(x-4
))+15)/(exp(1)-3*x-12))/(exp(1)**2+(-6*x-24)*exp(1)+9*x**2+72*x+144),x)

[Out]

exp(((3*x - 15)*exp(exp(x - 4)) + 15)/(-3*x - 12 + E))

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