3.73.22 \(\int \frac {e^{-1-\frac {28-2 x+6 x^2-x^3}{2+x^2}} (20+160 x+20 x^2+5 x^4)}{4+4 x^2+x^4} \, dx\)

Optimal. Leaf size=16 \[ 5 e^{-7+x-\frac {16}{2+x^2}} \]

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Rubi [F]  time = 0.94, 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^{-1-\frac {28-2 x+6 x^2-x^3}{2+x^2}} \left (20+160 x+20 x^2+5 x^4\right )}{4+4 x^2+x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(-1 - (28 - 2*x + 6*x^2 - x^3)/(2 + x^2))*(20 + 160*x + 20*x^2 + 5*x^4))/(4 + 4*x^2 + x^4),x]

[Out]

5*Defer[Int][E^((-30 + 2*x - 7*x^2 + x^3)/(2 + x^2)), x] + 160*Defer[Int][(E^((-30 + 2*x - 7*x^2 + x^3)/(2 + x
^2))*x)/(2 + x^2)^2, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {e^{-1-\frac {28-2 x+6 x^2-x^3}{2+x^2}} \left (20+160 x+20 x^2+5 x^4\right )}{\left (2+x^2\right )^2} \, dx\\ &=\int \frac {e^{\frac {-30+2 x-7 x^2+x^3}{2+x^2}} \left (20+160 x+20 x^2+5 x^4\right )}{\left (2+x^2\right )^2} \, dx\\ &=\int \left (5 e^{\frac {-30+2 x-7 x^2+x^3}{2+x^2}}+\frac {160 e^{\frac {-30+2 x-7 x^2+x^3}{2+x^2}} x}{\left (2+x^2\right )^2}\right ) \, dx\\ &=5 \int e^{\frac {-30+2 x-7 x^2+x^3}{2+x^2}} \, dx+160 \int \frac {e^{\frac {-30+2 x-7 x^2+x^3}{2+x^2}} x}{\left (2+x^2\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.12, size = 16, normalized size = 1.00 \begin {gather*} 5 e^{-7+x-\frac {16}{2+x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(-1 - (28 - 2*x + 6*x^2 - x^3)/(2 + x^2))*(20 + 160*x + 20*x^2 + 5*x^4))/(4 + 4*x^2 + x^4),x]

[Out]

5*E^(-7 + x - 16/(2 + x^2))

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fricas [A]  time = 0.56, size = 24, normalized size = 1.50 \begin {gather*} 5 \, e^{\left (\frac {x^{3} - 7 \, x^{2} + 2 \, x - 30}{x^{2} + 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4+20*x^2+160*x+20)/(x^4+4*x^2+4)/exp(1)/exp((-x^3+6*x^2-2*x+28)/(x^2+2)),x, algorithm="fricas")

[Out]

5*e^((x^3 - 7*x^2 + 2*x - 30)/(x^2 + 2))

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giac [B]  time = 0.14, size = 46, normalized size = 2.88 \begin {gather*} 5 \, e^{\left (\frac {x^{3}}{x^{2} + 2} - \frac {7 \, x^{2}}{x^{2} + 2} + \frac {2 \, x}{x^{2} + 2} - \frac {30}{x^{2} + 2}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4+20*x^2+160*x+20)/(x^4+4*x^2+4)/exp(1)/exp((-x^3+6*x^2-2*x+28)/(x^2+2)),x, algorithm="giac")

[Out]

5*e^(x^3/(x^2 + 2) - 7*x^2/(x^2 + 2) + 2*x/(x^2 + 2) - 30/(x^2 + 2))

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maple [A]  time = 0.11, size = 25, normalized size = 1.56

method	result	size
risch	\(5 \,{\mathrm e}^{\frac {x^{3}-7 x^{2}+2 x -30}{x^{2}+2}}\)	\(25\)
gosper	\(5 \,{\mathrm e}^{-1} {\mathrm e}^{\frac {x^{3}-6 x^{2}+2 x -28}{x^{2}+2}}\)	\(32\)
norman	\(\frac {\left (10 \,{\mathrm e}^{-1}+5 x^{2} {\mathrm e}^{-1}\right ) {\mathrm e}^{-\frac {-x^{3}+6 x^{2}-2 x +28}{x^{2}+2}}}{x^{2}+2}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x^4+20*x^2+160*x+20)/(x^4+4*x^2+4)/exp(1)/exp((-x^3+6*x^2-2*x+28)/(x^2+2)),x,method=_RETURNVERBOSE)

[Out]

5*exp((x^3-7*x^2+2*x-30)/(x^2+2))

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maxima [A]  time = 0.45, size = 15, normalized size = 0.94 \begin {gather*} 5 \, e^{\left (x - \frac {16}{x^{2} + 2} - 7\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x^4+20*x^2+160*x+20)/(x^4+4*x^2+4)/exp(1)/exp((-x^3+6*x^2-2*x+28)/(x^2+2)),x, algorithm="maxima")

[Out]

5*e^(x - 16/(x^2 + 2) - 7)

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mupad [B]  time = 4.51, size = 24, normalized size = 1.50 \begin {gather*} 5\,{\mathrm {e}}^{\frac {x^3-7\,x^2+2\,x-30}{x^2+2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(-1)*exp((2*x - 6*x^2 + x^3 - 28)/(x^2 + 2))*(160*x + 20*x^2 + 5*x^4 + 20))/(4*x^2 + x^4 + 4),x)

[Out]

5*exp((2*x - 7*x^2 + x^3 - 30)/(x^2 + 2))

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sympy [A]  time = 0.31, size = 24, normalized size = 1.50 \begin {gather*} \frac {5 e^{- \frac {- x^{3} + 6 x^{2} - 2 x + 28}{x^{2} + 2}}}{e} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((5*x**4+20*x**2+160*x+20)/(x**4+4*x**2+4)/exp(1)/exp((-x**3+6*x**2-2*x+28)/(x**2+2)),x)

[Out]

5*exp(-1)*exp(-(-x**3 + 6*x**2 - 2*x + 28)/(x**2 + 2))

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