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

Optimal. Leaf size=26 \[ e^{5-x}+\frac {3}{\left (e^{2 x}+x\right ) \left (\frac {1}{x}+x\right )} \]

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Rubi [F]  time = 3.20, 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 {-6 x^3+e^{5+3 x} \left (-1-2 x^2-x^4\right )+e^{5-x} \left (-x^2-2 x^4-x^6\right )+e^{2 x} \left (3-6 x-3 x^2-6 x^3+e^{5-x} \left (-2 x-4 x^3-2 x^5\right )\right )}{x^2+2 x^4+x^6+e^{4 x} \left (1+2 x^2+x^4\right )+e^{2 x} \left (2 x+4 x^3+2 x^5\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.08, size = 27, normalized size = 1.04 \begin {gather*} e^{5-x}+\frac {3 x}{\left (e^{2 x}+x\right ) \left (1+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.80, size = 58, normalized size = 2.23 \begin {gather*} \frac {{\left (x^{2} + 1\right )} e^{\left (-x + 15\right )} + 3 \, x e^{\left (-2 \, x + 10\right )} + {\left (x^{3} + x\right )} e^{\left (-3 \, x + 15\right )}}{{\left (x^{2} + 1\right )} e^{10} + {\left (x^{3} + x\right )} e^{\left (-2 \, x + 10\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((x^2 + 1)*e^(-x + 15) + 3*x*e^(-2*x + 10) + (x^3 + x)*e^(-3*x + 15))/((x^2 + 1)*e^10 + (x^3 + x)*e^(-2*x + 10
))

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giac [B]  time = 0.39, size = 58, normalized size = 2.23 \begin {gather*} \frac {x^{3} e^{5} + x^{2} e^{\left (2 \, x + 5\right )} + x e^{5} + 3 \, x e^{x} + e^{\left (2 \, x + 5\right )}}{x^{3} e^{x} + x^{2} e^{\left (3 \, x\right )} + x e^{x} + e^{\left (3 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^3*e^5 + x^2*e^(2*x + 5) + x*e^5 + 3*x*e^x + e^(2*x + 5))/(x^3*e^x + x^2*e^(3*x) + x*e^x + e^(3*x))

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maple [A]  time = 0.06, size = 26, normalized size = 1.00

method	result	size
risch	\({\mathrm e}^{5-x}+\frac {3 x}{\left (x^{2}+1\right ) \left ({\mathrm e}^{2 x}+x \right )}\)	\(26\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [B]  time = 1.12, size = 52, normalized size = 2.00 \begin {gather*} \frac {x^{3} e^{5} + x e^{5} + {\left (x^{2} e^{5} + e^{5}\right )} e^{\left (2 \, x\right )} + 3 \, x e^{x}}{{\left (x^{2} + 1\right )} e^{\left (3 \, x\right )} + {\left (x^{3} + x\right )} e^{x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.03, size = 60, normalized size = 2.31 \begin {gather*} \frac {{\mathrm {e}}^{5-x}\,\left (x+3\,x\,{\mathrm {e}}^{x-5}+{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,x-10}+x^3+x^2\,{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,x-10}\right )}{\left (x^2+1\right )\,\left (x+{\mathrm {e}}^{10}\,{\mathrm {e}}^{2\,x-10}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(exp(5 - x)*(x + 3*x*exp(x - 5) + exp(10)*exp(2*x - 10) + x^3 + x^2*exp(10)*exp(2*x - 10)))/((x^2 + 1)*(x + ex
p(10)*exp(2*x - 10)))

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sympy [A]  time = 0.26, size = 29, normalized size = 1.12 \begin {gather*} \frac {3 x}{x^{3} + x + \left (x^{2} + 1\right ) e^{2 x}} + \frac {e^{5}}{\sqrt {e^{2 x}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3*x/(x**3 + x + (x**2 + 1)*exp(2*x)) + exp(5)/sqrt(exp(2*x))

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