3.8.7 \(\int \frac {e^{\frac {-2+e^x}{15-18 x+3 x^2}} (-12+4 x+e^x (11-8 x+x^2))}{75-180 x+138 x^2-36 x^3+3 x^4} \, dx\)

Optimal. Leaf size=24 \[ e^{\frac {2-e^x}{(5-x) (-3+3 x)}} \]

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Rubi [F]  time = 3.92, 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 {-2+e^x}{15-18 x+3 x^2}} \left (-12+4 x+e^x \left (11-8 x+x^2\right )\right )}{75-180 x+138 x^2-36 x^3+3 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^((-2 + E^x)/(15 - 18*x + 3*x^2))*(-12 + 4*x + E^x*(11 - 8*x + x^2)))/(75 - 180*x + 138*x^2 - 36*x^3 + 3
*x^4),x]

[Out]

Defer[Int][E^((-2 + E^x)/(15 - 18*x + 3*x^2))/(-5 + x)^2, x]/6 - Defer[Int][E^(x + (-2 + E^x)/(15 - 18*x + 3*x
^2))/(-5 + x)^2, x]/12 + Defer[Int][E^(x + (-2 + E^x)/(15 - 18*x + 3*x^2))/(-5 + x), x]/12 - Defer[Int][E^((-2
+ E^x)/(15 - 18*x + 3*x^2))/(-1 + x)^2, x]/6 + Defer[Int][E^(x + (-2 + E^x)/(15 - 18*x + 3*x^2))/(-1 + x)^2,
x]/12 - Defer[Int][E^(x + (-2 + E^x)/(15 - 18*x + 3*x^2))/(-1 + x), x]/12

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 e^{\frac {-2+e^x}{15-18 x+3 x^2}} (-3+x)}{3 (-5+x)^2 (-1+x)^2}+\frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}} \left (11-8 x+x^2\right )}{3 (-5+x)^2 (-1+x)^2}\right ) \, dx\\ &=\frac {1}{3} \int \frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}} \left (11-8 x+x^2\right )}{(-5+x)^2 (-1+x)^2} \, dx+\frac {4}{3} \int \frac {e^{\frac {-2+e^x}{15-18 x+3 x^2}} (-3+x)}{(-5+x)^2 (-1+x)^2} \, dx\\ &=\frac {1}{3} \int \left (-\frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{4 (-5+x)^2}+\frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{4 (-5+x)}+\frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{4 (-1+x)^2}-\frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{4 (-1+x)}\right ) \, dx+\frac {4}{3} \int \left (\frac {e^{\frac {-2+e^x}{15-18 x+3 x^2}}}{8 (-5+x)^2}-\frac {e^{\frac {-2+e^x}{15-18 x+3 x^2}}}{8 (-1+x)^2}\right ) \, dx\\ &=-\left (\frac {1}{12} \int \frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{(-5+x)^2} \, dx\right )+\frac {1}{12} \int \frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{-5+x} \, dx+\frac {1}{12} \int \frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{(-1+x)^2} \, dx-\frac {1}{12} \int \frac {e^{x+\frac {-2+e^x}{15-18 x+3 x^2}}}{-1+x} \, dx+\frac {1}{6} \int \frac {e^{\frac {-2+e^x}{15-18 x+3 x^2}}}{(-5+x)^2} \, dx-\frac {1}{6} \int \frac {e^{\frac {-2+e^x}{15-18 x+3 x^2}}}{(-1+x)^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.56, size = 21, normalized size = 0.88 \begin {gather*} e^{\frac {-2+e^x}{3 \left (5-6 x+x^2\right )}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^((-2 + E^x)/(15 - 18*x + 3*x^2))*(-12 + 4*x + E^x*(11 - 8*x + x^2)))/(75 - 180*x + 138*x^2 - 36*x
^3 + 3*x^4),x]

[Out]

E^((-2 + E^x)/(3*(5 - 6*x + x^2)))

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fricas [A]  time = 0.60, size = 17, normalized size = 0.71 \begin {gather*} e^{\left (\frac {e^{x} - 2}{3 \, {\left (x^{2} - 6 \, x + 5\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-8*x+11)*exp(x)+4*x-12)*exp((exp(x)-2)/(3*x^2-18*x+15))/(3*x^4-36*x^3+138*x^2-180*x+75),x, algo
rithm="fricas")

[Out]

e^(1/3*(e^x - 2)/(x^2 - 6*x + 5))

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giac [A]  time = 0.35, size = 28, normalized size = 1.17 \begin {gather*} e^{\left (\frac {e^{x}}{3 \, {\left (x^{2} - 6 \, x + 5\right )}} - \frac {2}{3 \, {\left (x^{2} - 6 \, x + 5\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-8*x+11)*exp(x)+4*x-12)*exp((exp(x)-2)/(3*x^2-18*x+15))/(3*x^4-36*x^3+138*x^2-180*x+75),x, algo
rithm="giac")

[Out]

e^(1/3*e^x/(x^2 - 6*x + 5) - 2/3/(x^2 - 6*x + 5))

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maple [A]  time = 0.10, size = 18, normalized size = 0.75

method	result	size
risch	\({\mathrm e}^{\frac {{\mathrm e}^{x}-2}{3 \left (x -1\right ) \left (x -5\right )}}\)	\(18\)
norman	\(\frac {x^{2} {\mathrm e}^{\frac {{\mathrm e}^{x}-2}{3 x^{2}-18 x +15}}-6 x \,{\mathrm e}^{\frac {{\mathrm e}^{x}-2}{3 x^{2}-18 x +15}}+5 \,{\mathrm e}^{\frac {{\mathrm e}^{x}-2}{3 x^{2}-18 x +15}}}{x^{2}-6 x +5}\)	\(76\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((x^2-8*x+11)*exp(x)+4*x-12)*exp((exp(x)-2)/(3*x^2-18*x+15))/(3*x^4-36*x^3+138*x^2-180*x+75),x,method=_RET
URNVERBOSE)

[Out]

exp(1/3*(exp(x)-2)/(x-1)/(x-5))

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maxima [A]  time = 1.05, size = 34, normalized size = 1.42 \begin {gather*} e^{\left (-\frac {e^{x}}{12 \, {\left (x - 1\right )}} + \frac {e^{x}}{12 \, {\left (x - 5\right )}} + \frac {1}{6 \, {\left (x - 1\right )}} - \frac {1}{6 \, {\left (x - 5\right )}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x^2-8*x+11)*exp(x)+4*x-12)*exp((exp(x)-2)/(3*x^2-18*x+15))/(3*x^4-36*x^3+138*x^2-180*x+75),x, algo
rithm="maxima")

[Out]

e^(-1/12*e^x/(x - 1) + 1/12*e^x/(x - 5) + 1/6/(x - 1) - 1/6/(x - 5))

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mupad [B]  time = 0.73, size = 18, normalized size = 0.75 \begin {gather*} {\mathrm {e}}^{\frac {{\mathrm {e}}^x-2}{3\,x^2-18\,x+15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((exp(x) - 2)/(3*x^2 - 18*x + 15))*(4*x + exp(x)*(x^2 - 8*x + 11) - 12))/(138*x^2 - 180*x - 36*x^3 + 3
*x^4 + 75),x)

[Out]

exp((exp(x) - 2)/(3*x^2 - 18*x + 15))

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sympy [A]  time = 0.32, size = 15, normalized size = 0.62 \begin {gather*} e^{\frac {e^{x} - 2}{3 x^{2} - 18 x + 15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((x**2-8*x+11)*exp(x)+4*x-12)*exp((exp(x)-2)/(3*x**2-18*x+15))/(3*x**4-36*x**3+138*x**2-180*x+75),x)

[Out]

exp((exp(x) - 2)/(3*x**2 - 18*x + 15))

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