3.19.78 \(\int \frac {52-26 x+4 x^2+e^{1+2 x^2+x^4} (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5)}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} (-26+2 x^2)} \, dx\)

Optimal. Leaf size=27 \[ \frac {(4-x) x}{13-e^{\left (1+x^2\right )^2}-x^2} \]

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Rubi [F]  time = 2.17, 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 {52-26 x+4 x^2+e^{1+2 x^2+x^4} \left (-4+2 x+16 x^2-4 x^3+16 x^4-4 x^5\right )}{169+e^{2+4 x^2+2 x^4}-26 x^2+x^4+e^{1+2 x^2+x^4} \left (-26+2 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(52 - 26*x + 4*x^2 + E^(1 + 2*x^2 + x^4)*(-4 + 2*x + 16*x^2 - 4*x^3 + 16*x^4 - 4*x^5))/(169 + E^(2 + 4*x^2
+ 2*x^4) - 26*x^2 + x^4 + E^(1 + 2*x^2 + x^4)*(-26 + 2*x^2)),x]

[Out]

216*Defer[Int][x^2/(-13 + E^(1 + x^2)^2 + x^2)^2, x] + 192*Defer[Int][x^4/(-13 + E^(1 + x^2)^2 + x^2)^2, x] -
16*Defer[Int][x^6/(-13 + E^(1 + x^2)^2 + x^2)^2, x] - 4*Defer[Int][(-13 + E^(1 + x^2)^2 + x^2)^(-1), x] + 16*D
efer[Int][x^2/(-13 + E^(1 + x^2)^2 + x^2), x] + 16*Defer[Int][x^4/(-13 + E^(1 + x^2)^2 + x^2), x] - 27*Defer[S
ubst][Defer[Int][x/(-13 + E^(1 + x)^2 + x)^2, x], x, x^2] - 24*Defer[Subst][Defer[Int][x^2/(-13 + E^(1 + x)^2
+ x)^2, x], x, x^2] + 2*Defer[Subst][Defer[Int][x^3/(-13 + E^(1 + x)^2 + x)^2, x], x, x^2] + Defer[Subst][Defe
r[Int][(-13 + E^(1 + x)^2 + x)^(-1), x], x, x^2] - 2*Defer[Subst][Defer[Int][x/(-13 + E^(1 + x)^2 + x), x], x,
x^2] - 2*Defer[Subst][Defer[Int][x^2/(-13 + E^(1 + x)^2 + x), x], x, x^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {52-26 x+4 x^2+2 e^{\left (1+x^2\right )^2} \left (-2+x+8 x^2-2 x^3+8 x^4-2 x^5\right )}{\left (13-e^{\left (1+x^2\right )^2}-x^2\right )^2} \, dx\\ &=\int \left (\frac {2 x^2 \left (108-27 x+96 x^2-24 x^3-8 x^4+2 x^5\right )}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {2 \left (2-x-8 x^2+2 x^3-8 x^4+2 x^5\right )}{-13+e^{\left (1+x^2\right )^2}+x^2}\right ) \, dx\\ &=2 \int \frac {x^2 \left (108-27 x+96 x^2-24 x^3-8 x^4+2 x^5\right )}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx-2 \int \frac {2-x-8 x^2+2 x^3-8 x^4+2 x^5}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx\\ &=2 \int \left (\frac {108 x^2}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {27 x^3}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}+\frac {96 x^4}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {24 x^5}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}-\frac {8 x^6}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}+\frac {2 x^7}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2}\right ) \, dx-2 \int \left (\frac {2}{-13+e^{\left (1+x^2\right )^2}+x^2}-\frac {x}{-13+e^{\left (1+x^2\right )^2}+x^2}-\frac {8 x^2}{-13+e^{\left (1+x^2\right )^2}+x^2}+\frac {2 x^3}{-13+e^{\left (1+x^2\right )^2}+x^2}-\frac {8 x^4}{-13+e^{\left (1+x^2\right )^2}+x^2}+\frac {2 x^5}{-13+e^{\left (1+x^2\right )^2}+x^2}\right ) \, dx\\ &=2 \int \frac {x}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx+4 \int \frac {x^7}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx-4 \int \frac {1}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-4 \int \frac {x^3}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-4 \int \frac {x^5}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-16 \int \frac {x^6}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+16 \int \frac {x^2}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx+16 \int \frac {x^4}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-48 \int \frac {x^5}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx-54 \int \frac {x^3}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+192 \int \frac {x^4}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+216 \int \frac {x^2}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx\\ &=2 \operatorname {Subst}\left (\int \frac {x^3}{\left (-13+e^{(1+x)^2}+x\right )^2} \, dx,x,x^2\right )-2 \operatorname {Subst}\left (\int \frac {x}{-13+e^{(1+x)^2}+x} \, dx,x,x^2\right )-2 \operatorname {Subst}\left (\int \frac {x^2}{-13+e^{(1+x)^2}+x} \, dx,x,x^2\right )-4 \int \frac {1}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-16 \int \frac {x^6}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+16 \int \frac {x^2}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx+16 \int \frac {x^4}{-13+e^{\left (1+x^2\right )^2}+x^2} \, dx-24 \operatorname {Subst}\left (\int \frac {x^2}{\left (-13+e^{(1+x)^2}+x\right )^2} \, dx,x,x^2\right )-27 \operatorname {Subst}\left (\int \frac {x}{\left (-13+e^{(1+x)^2}+x\right )^2} \, dx,x,x^2\right )+192 \int \frac {x^4}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+216 \int \frac {x^2}{\left (-13+e^{\left (1+x^2\right )^2}+x^2\right )^2} \, dx+\operatorname {Subst}\left (\int \frac {1}{-13+e^{(1+x)^2}+x} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 21, normalized size = 0.78 \begin {gather*} \frac {(-4+x) x}{-13+e^{\left (1+x^2\right )^2}+x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(52 - 26*x + 4*x^2 + E^(1 + 2*x^2 + x^4)*(-4 + 2*x + 16*x^2 - 4*x^3 + 16*x^4 - 4*x^5))/(169 + E^(2 +
4*x^2 + 2*x^4) - 26*x^2 + x^4 + E^(1 + 2*x^2 + x^4)*(-26 + 2*x^2)),x]

[Out]

((-4 + x)*x)/(-13 + E^(1 + x^2)^2 + x^2)

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fricas [A]  time = 0.67, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*e
xp(x^4+2*x^2+1)+x^4-26*x^2+169),x, algorithm="fricas")

[Out]

(x^2 - 4*x)/(x^2 + e^(x^4 + 2*x^2 + 1) - 13)

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giac [A]  time = 0.31, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*e
xp(x^4+2*x^2+1)+x^4-26*x^2+169),x, algorithm="giac")

[Out]

(x^2 - 4*x)/(x^2 + e^(x^4 + 2*x^2 + 1) - 13)

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maple [A]  time = 0.06, size = 21, normalized size = 0.78

method	result	size
risch	\(\frac {\left (x -4\right ) x}{x^{2}+{\mathrm e}^{\left (x^{2}+1\right )^{2}}-13}\)	\(21\)
norman	\(\frac {x^{2}-4 x}{x^{2}+{\mathrm e}^{x^{4}+2 x^{2}+1}-13}\)	\(27\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*exp(x^4
+2*x^2+1)+x^4-26*x^2+169),x,method=_RETURNVERBOSE)

[Out]

(x-4)*x/(x^2+exp((x^2+1)^2)-13)

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maxima [A]  time = 0.71, size = 26, normalized size = 0.96 \begin {gather*} \frac {x^{2} - 4 \, x}{x^{2} + e^{\left (x^{4} + 2 \, x^{2} + 1\right )} - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x^5+16*x^4-4*x^3+16*x^2+2*x-4)*exp(x^4+2*x^2+1)+4*x^2-26*x+52)/(exp(x^4+2*x^2+1)^2+(2*x^2-26)*e
xp(x^4+2*x^2+1)+x^4-26*x^2+169),x, algorithm="maxima")

[Out]

(x^2 - 4*x)/(x^2 + e^(x^4 + 2*x^2 + 1) - 13)

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mupad [B]  time = 1.23, size = 29, normalized size = 1.07 \begin {gather*} -\frac {4\,x-x^2}{{\mathrm {e}}^{x^4+2\,x^2+1}+x^2-13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(2*x^2 + x^4 + 1)*(2*x + 16*x^2 - 4*x^3 + 16*x^4 - 4*x^5 - 4) - 26*x + 4*x^2 + 52)/(exp(4*x^2 + 2*x^4
+ 2) + exp(2*x^2 + x^4 + 1)*(2*x^2 - 26) - 26*x^2 + x^4 + 169),x)

[Out]

-(4*x - x^2)/(exp(2*x^2 + x^4 + 1) + x^2 - 13)

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sympy [A]  time = 0.15, size = 22, normalized size = 0.81 \begin {gather*} \frac {x^{2} - 4 x}{x^{2} + e^{x^{4} + 2 x^{2} + 1} - 13} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4*x**5+16*x**4-4*x**3+16*x**2+2*x-4)*exp(x**4+2*x**2+1)+4*x**2-26*x+52)/(exp(x**4+2*x**2+1)**2+(2
*x**2-26)*exp(x**4+2*x**2+1)+x**4-26*x**2+169),x)

[Out]

(x**2 - 4*x)/(x**2 + exp(x**4 + 2*x**2 + 1) - 13)

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