3.31.16 \(\int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} (-7+2 x^2)}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} (8-2 x-8 x^2-2 x^3)} \, dx\)

Optimal. Leaf size=31 \[ \frac {x}{x^2+x \left (3+\frac {5-e^{2 x}+x}{-3+x+x^2}\right )} \]

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Rubi [F]  time = 1.09, 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 {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{16+e^{4 x}-8 x-31 x^2+18 x^4+8 x^5+x^6+e^{2 x} \left (8-2 x-8 x^2-2 x^3\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-1 + 16*x + 6*x^2 - 2*x^3 - x^4 + E^(2*x)*(-7 + 2*x^2))/(16 + E^(4*x) - 8*x - 31*x^2 + 18*x^4 + 8*x^5 + x
^6 + E^(2*x)*(8 - 2*x - 8*x^2 - 2*x^3)),x]

[Out]

27*Defer[Int][(4 + E^(2*x) - x - 4*x^2 - x^3)^(-2), x] - 7*Defer[Int][(4 + E^(2*x) - x - 4*x^2 - x^3)^(-1), x]
+ 9*Defer[Int][x/(-4 - E^(2*x) + x + 4*x^2 + x^3)^2, x] - 30*Defer[Int][x^2/(-4 - E^(2*x) + x + 4*x^2 + x^3)^
2, x] - 7*Defer[Int][x^3/(-4 - E^(2*x) + x + 4*x^2 + x^3)^2, x] + 7*Defer[Int][x^4/(-4 - E^(2*x) + x + 4*x^2 +
x^3)^2, x] + 2*Defer[Int][x^5/(-4 - E^(2*x) + x + 4*x^2 + x^3)^2, x] - 2*Defer[Int][x^2/(-4 - E^(2*x) + x + 4
*x^2 + x^3), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-1+16 x+6 x^2-2 x^3-x^4+e^{2 x} \left (-7+2 x^2\right )}{\left (4+e^{2 x}-x-4 x^2-x^3\right )^2} \, dx\\ &=\int \left (-\frac {-7+2 x^2}{-4-e^{2 x}+x+4 x^2+x^3}+\frac {27+9 x-30 x^2-7 x^3+7 x^4+2 x^5}{\left (4+e^{2 x}-x-4 x^2-x^3\right )^2}\right ) \, dx\\ &=-\int \frac {-7+2 x^2}{-4-e^{2 x}+x+4 x^2+x^3} \, dx+\int \frac {27+9 x-30 x^2-7 x^3+7 x^4+2 x^5}{\left (4+e^{2 x}-x-4 x^2-x^3\right )^2} \, dx\\ &=\int \left (\frac {27}{\left (4+e^{2 x}-x-4 x^2-x^3\right )^2}+\frac {9 x}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2}-\frac {30 x^2}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2}-\frac {7 x^3}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2}+\frac {7 x^4}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2}+\frac {2 x^5}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2}\right ) \, dx-\int \left (\frac {7}{4+e^{2 x}-x-4 x^2-x^3}+\frac {2 x^2}{-4-e^{2 x}+x+4 x^2+x^3}\right ) \, dx\\ &=2 \int \frac {x^5}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2} \, dx-2 \int \frac {x^2}{-4-e^{2 x}+x+4 x^2+x^3} \, dx-7 \int \frac {1}{4+e^{2 x}-x-4 x^2-x^3} \, dx-7 \int \frac {x^3}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2} \, dx+7 \int \frac {x^4}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2} \, dx+9 \int \frac {x}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2} \, dx+27 \int \frac {1}{\left (4+e^{2 x}-x-4 x^2-x^3\right )^2} \, dx-30 \int \frac {x^2}{\left (-4-e^{2 x}+x+4 x^2+x^3\right )^2} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.33, size = 27, normalized size = 0.87 \begin {gather*} \frac {-3+x+x^2}{-4-e^{2 x}+x+4 x^2+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + 16*x + 6*x^2 - 2*x^3 - x^4 + E^(2*x)*(-7 + 2*x^2))/(16 + E^(4*x) - 8*x - 31*x^2 + 18*x^4 + 8*x
^5 + x^6 + E^(2*x)*(8 - 2*x - 8*x^2 - 2*x^3)),x]

[Out]

(-3 + x + x^2)/(-4 - E^(2*x) + x + 4*x^2 + x^3)

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fricas [A]  time = 0.67, size = 26, normalized size = 0.84 \begin {gather*} \frac {x^{2} + x - 3}{x^{3} + 4 \, x^{2} + x - e^{\left (2 \, x\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3-8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x
^4-31*x^2-8*x+16),x, algorithm="fricas")

[Out]

(x^2 + x - 3)/(x^3 + 4*x^2 + x - e^(2*x) - 4)

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giac [A]  time = 0.24, size = 26, normalized size = 0.84 \begin {gather*} \frac {x^{2} + x - 3}{x^{3} + 4 \, x^{2} + x - e^{\left (2 \, x\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3-8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x
^4-31*x^2-8*x+16),x, algorithm="giac")

[Out]

(x^2 + x - 3)/(x^3 + 4*x^2 + x - e^(2*x) - 4)

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maple [A]  time = 0.12, size = 27, normalized size = 0.87

method	result	size
norman	\(\frac {x^{2}+x -3}{x^{3}+4 x^{2}+x -{\mathrm e}^{2 x}-4}\)	\(27\)
risch	\(\frac {x^{2}+x -3}{x^{3}+4 x^{2}+x -{\mathrm e}^{2 x}-4}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3-8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x^4-31*
x^2-8*x+16),x,method=_RETURNVERBOSE)

[Out]

(x^2+x-3)/(x^3+4*x^2+x-exp(2*x)-4)

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maxima [A]  time = 0.46, size = 26, normalized size = 0.84 \begin {gather*} \frac {x^{2} + x - 3}{x^{3} + 4 \, x^{2} + x - e^{\left (2 \, x\right )} - 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^2-7)*exp(2*x)-x^4-2*x^3+6*x^2+16*x-1)/(exp(2*x)^2+(-2*x^3-8*x^2-2*x+8)*exp(2*x)+x^6+8*x^5+18*x
^4-31*x^2-8*x+16),x, algorithm="maxima")

[Out]

(x^2 + x - 3)/(x^3 + 4*x^2 + x - e^(2*x) - 4)

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mupad [B]  time = 1.84, size = 26, normalized size = 0.84 \begin {gather*} \frac {x^2+x-3}{x-{\mathrm {e}}^{2\,x}+4\,x^2+x^3-4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((16*x + exp(2*x)*(2*x^2 - 7) + 6*x^2 - 2*x^3 - x^4 - 1)/(exp(4*x) - 8*x - exp(2*x)*(2*x + 8*x^2 + 2*x^3 -
8) - 31*x^2 + 18*x^4 + 8*x^5 + x^6 + 16),x)

[Out]

(x + x^2 - 3)/(x - exp(2*x) + 4*x^2 + x^3 - 4)

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sympy [A]  time = 0.14, size = 22, normalized size = 0.71 \begin {gather*} \frac {- x^{2} - x + 3}{- x^{3} - 4 x^{2} - x + e^{2 x} + 4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**2-7)*exp(2*x)-x**4-2*x**3+6*x**2+16*x-1)/(exp(2*x)**2+(-2*x**3-8*x**2-2*x+8)*exp(2*x)+x**6+8*
x**5+18*x**4-31*x**2-8*x+16),x)

[Out]

(-x**2 - x + 3)/(-x**3 - 4*x**2 - x + exp(2*x) + 4)

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