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

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

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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 = {} $$\begin{array}{c}\int \frac{e^{-1-\frac{28-2x+6x^2-x^3}{2+x^2}}(20+160x+20x^2+5x^4)}{4+4x^2+x^4}dx\end{array}$$

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

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{e^{-1-\frac{28-2x+6x^2-x^3}{2+x^2}}(20+160x+20x^2+5x^4)}{{(2+x^2)}^2}dx\\ & =\int \frac{e^{\frac{-30+2x-7x^2+x^3}{2+x^2}}(20+160x+20x^2+5x^4)}{{(2+x^2)}^2}dx\\ & =\int (5e^{\frac{-30+2x-7x^2+x^3}{2+x^2}}+\frac{160e^{\frac{-30+2x-7x^2+x^3}{2+x^2}}x}{{(2+x^2)}^2})dx\\ & =5\int e^{\frac{-30+2x-7x^2+x^3}{2+x^2}}dx+160\int \frac{e^{\frac{-30+2x-7x^2+x^3}{2+x^2}}x}{{(2+x^2)}^2}dx\end{array}\end{array}$$

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

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 $$\begin{array}{c}5e^{\left(\frac{x^3-7x^2+2x-30}{x^2+2}\right)}\end{array}$$

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 $$\begin{array}{c}5e^{(\frac{x^3}{x^2+2}-\frac{7x^2}{x^2+2}+\frac{2x}{x^2+2}-\frac{30}{x^2+2})}\end{array}$$

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	$5{\mathrm{e}}^{\frac{x^3-7x^2+2x-30}{x^2+2}}$	$25$
gosper	$5{\mathrm{e}}^{-1}{\mathrm{e}}^{\frac{x^3-6x^2+2x-28}{x^2+2}}$	$32$
norman	$\frac{(10{\mathrm{e}}^{-1}+5x^2{\mathrm{e}}^{-1}){\mathrm{e}}^{-\frac{-x^3+6x^2-2x+28}{x^2+2}}}{x^2+2}$	$51$

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 $$\begin{array}{c}5e^{(x-\frac{16}{x^2+2}-7)}\end{array}$$

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 $$\begin{array}{c}5{\mathrm{e}}^{\frac{x^3-7x^2+2x-30}{x^2+2}}\end{array}$$

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 $$\begin{array}{c}\frac{5e^{-\frac{-x^3+6x^2-2x+28}{x^2+2}}}{e}\end{array}$$

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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