3.5.76 \(\int \frac {48+896 x-16 x^2+e^{2 x^2} (-12-864 x+48 x^2+192 x^3)}{144+9 e^{4 x^2}+96 x^2+16 x^4+e^{2 x^2} (-72-24 x^2)} \, dx\)

Optimal. Leaf size=31 \[ \frac {x+4 \left (-4+x^2\right )}{\frac {3}{4} \left (4-e^{2 x^2}\right )+x^2} \]

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Rubi [F]  time = 1.45, 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 {48+896 x-16 x^2+e^{2 x^2} \left (-12-864 x+48 x^2+192 x^3\right )}{144+9 e^{4 x^2}+96 x^2+16 x^4+e^{2 x^2} \left (-72-24 x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(48 + 896*x - 16*x^2 + E^(2*x^2)*(-12 - 864*x + 48*x^2 + 192*x^3))/(144 + 9*E^(4*x^2) + 96*x^2 + 16*x^4 +
E^(2*x^2)*(-72 - 24*x^2)),x]

[Out]

160*Defer[Int][x^2/(-12 + 3*E^(2*x^2) - 4*x^2)^2, x] + 64*Defer[Int][x^4/(-12 + 3*E^(2*x^2) - 4*x^2)^2, x] - 4
*Defer[Int][(-12 + 3*E^(2*x^2) - 4*x^2)^(-1), x] - 16*Defer[Int][x^2/(12 - 3*E^(2*x^2) + 4*x^2), x] - 640*Defe
r[Subst][Defer[Int][(12 - 3*E^x + 2*x)^(-2), x], x, 2*x^2] + 72*Defer[Subst][Defer[Int][(12 - 3*E^x + 2*x)^(-1
), x], x, 2*x^2] - 32*Defer[Subst][Defer[Int][x/(12 - 3*E^(2*x) + 4*x), x], x, x^2] - 192*Defer[Subst][Defer[I
nt][x/(3*E^(2*x) - 4*(3 + x))^2, x], x, x^2] + 128*Defer[Subst][Defer[Int][x^2/(3*E^(2*x) - 4*(3 + x))^2, x],
x, x^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {48+896 x-16 x^2+e^{2 x^2} \left (-12-864 x+48 x^2+192 x^3\right )}{\left (12-3 e^{2 x^2}+4 x^2\right )^2} \, dx\\ &=\int \left (\frac {4 \left (-1-72 x+4 x^2+16 x^3\right )}{-12+3 e^{2 x^2}-4 x^2}+\frac {32 x \left (-80+5 x-12 x^2+2 x^3+8 x^4\right )}{\left (12-3 e^{2 x^2}+4 x^2\right )^2}\right ) \, dx\\ &=4 \int \frac {-1-72 x+4 x^2+16 x^3}{-12+3 e^{2 x^2}-4 x^2} \, dx+32 \int \frac {x \left (-80+5 x-12 x^2+2 x^3+8 x^4\right )}{\left (12-3 e^{2 x^2}+4 x^2\right )^2} \, dx\\ &=4 \int \left (-\frac {1}{-12+3 e^{2 x^2}-4 x^2}+\frac {72 x}{12-3 e^{2 x^2}+4 x^2}-\frac {4 x^2}{12-3 e^{2 x^2}+4 x^2}-\frac {16 x^3}{12-3 e^{2 x^2}+4 x^2}\right ) \, dx+32 \int \left (\frac {5 x^2}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2}+\frac {2 x^4}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2}-\frac {80 x}{\left (12-3 e^{2 x^2}+4 x^2\right )^2}-\frac {12 x^3}{\left (12-3 e^{2 x^2}+4 x^2\right )^2}+\frac {8 x^5}{\left (12-3 e^{2 x^2}+4 x^2\right )^2}\right ) \, dx\\ &=-\left (4 \int \frac {1}{-12+3 e^{2 x^2}-4 x^2} \, dx\right )-16 \int \frac {x^2}{12-3 e^{2 x^2}+4 x^2} \, dx+64 \int \frac {x^4}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2} \, dx-64 \int \frac {x^3}{12-3 e^{2 x^2}+4 x^2} \, dx+160 \int \frac {x^2}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2} \, dx+256 \int \frac {x^5}{\left (12-3 e^{2 x^2}+4 x^2\right )^2} \, dx+288 \int \frac {x}{12-3 e^{2 x^2}+4 x^2} \, dx-384 \int \frac {x^3}{\left (12-3 e^{2 x^2}+4 x^2\right )^2} \, dx-2560 \int \frac {x}{\left (12-3 e^{2 x^2}+4 x^2\right )^2} \, dx\\ &=-\left (4 \int \frac {1}{-12+3 e^{2 x^2}-4 x^2} \, dx\right )-16 \int \frac {x^2}{12-3 e^{2 x^2}+4 x^2} \, dx-32 \operatorname {Subst}\left (\int \frac {x}{12-3 e^{2 x}+4 x} \, dx,x,x^2\right )+64 \int \frac {x^4}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2} \, dx+128 \operatorname {Subst}\left (\int \frac {x^2}{\left (3 e^{2 x}-4 (3+x)\right )^2} \, dx,x,x^2\right )+144 \operatorname {Subst}\left (\int \frac {1}{12-3 e^{2 x}+4 x} \, dx,x,x^2\right )+160 \int \frac {x^2}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2} \, dx-192 \operatorname {Subst}\left (\int \frac {x}{\left (3 e^{2 x}-4 (3+x)\right )^2} \, dx,x,x^2\right )-1280 \operatorname {Subst}\left (\int \frac {1}{\left (12-3 e^{2 x}+4 x\right )^2} \, dx,x,x^2\right )\\ &=-\left (4 \int \frac {1}{-12+3 e^{2 x^2}-4 x^2} \, dx\right )-16 \int \frac {x^2}{12-3 e^{2 x^2}+4 x^2} \, dx-32 \operatorname {Subst}\left (\int \frac {x}{12-3 e^{2 x}+4 x} \, dx,x,x^2\right )+64 \int \frac {x^4}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2} \, dx+72 \operatorname {Subst}\left (\int \frac {1}{12-3 e^x+2 x} \, dx,x,2 x^2\right )+128 \operatorname {Subst}\left (\int \frac {x^2}{\left (3 e^{2 x}-4 (3+x)\right )^2} \, dx,x,x^2\right )+160 \int \frac {x^2}{\left (-12+3 e^{2 x^2}-4 x^2\right )^2} \, dx-192 \operatorname {Subst}\left (\int \frac {x}{\left (3 e^{2 x}-4 (3+x)\right )^2} \, dx,x,x^2\right )-640 \operatorname {Subst}\left (\int \frac {1}{\left (12-3 e^x+2 x\right )^2} \, dx,x,2 x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 30, normalized size = 0.97 \begin {gather*} \frac {4 \left (16-x-4 x^2\right )}{-12+3 e^{2 x^2}-4 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48 + 896*x - 16*x^2 + E^(2*x^2)*(-12 - 864*x + 48*x^2 + 192*x^3))/(144 + 9*E^(4*x^2) + 96*x^2 + 16*
x^4 + E^(2*x^2)*(-72 - 24*x^2)),x]

[Out]

(4*(16 - x - 4*x^2))/(-12 + 3*E^(2*x^2) - 4*x^2)

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fricas [A]  time = 0.57, size = 27, normalized size = 0.87 \begin {gather*} \frac {4 \, {\left (4 \, x^{2} + x - 16\right )}}{4 \, x^{2} - 3 \, e^{\left (2 \, x^{2}\right )} + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^3+48*x^2-864*x-12)*exp(x^2)^2-16*x^2+896*x+48)/(9*exp(x^2)^4+(-24*x^2-72)*exp(x^2)^2+16*x^4+
96*x^2+144),x, algorithm="fricas")

[Out]

4*(4*x^2 + x - 16)/(4*x^2 - 3*e^(2*x^2) + 12)

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giac [A]  time = 0.36, size = 27, normalized size = 0.87 \begin {gather*} \frac {4 \, {\left (4 \, x^{2} + x - 16\right )}}{4 \, x^{2} - 3 \, e^{\left (2 \, x^{2}\right )} + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^3+48*x^2-864*x-12)*exp(x^2)^2-16*x^2+896*x+48)/(9*exp(x^2)^4+(-24*x^2-72)*exp(x^2)^2+16*x^4+
96*x^2+144),x, algorithm="giac")

[Out]

4*(4*x^2 + x - 16)/(4*x^2 - 3*e^(2*x^2) + 12)

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maple [A]  time = 0.08, size = 28, normalized size = 0.90

method	result	size
risch	\(\frac {16 x^{2}+4 x -64}{4 x^{2}-3 \,{\mathrm e}^{2 x^{2}}+12}\)	\(28\)
norman	\(\frac {16 x^{2}+4 x -64}{4 x^{2}-3 \,{\mathrm e}^{2 x^{2}}+12}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((192*x^3+48*x^2-864*x-12)*exp(x^2)^2-16*x^2+896*x+48)/(9*exp(x^2)^4+(-24*x^2-72)*exp(x^2)^2+16*x^4+96*x^2
+144),x,method=_RETURNVERBOSE)

[Out]

4*(4*x^2+x-16)/(4*x^2-3*exp(2*x^2)+12)

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maxima [A]  time = 0.64, size = 27, normalized size = 0.87 \begin {gather*} \frac {4 \, {\left (4 \, x^{2} + x - 16\right )}}{4 \, x^{2} - 3 \, e^{\left (2 \, x^{2}\right )} + 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x^3+48*x^2-864*x-12)*exp(x^2)^2-16*x^2+896*x+48)/(9*exp(x^2)^4+(-24*x^2-72)*exp(x^2)^2+16*x^4+
96*x^2+144),x, algorithm="maxima")

[Out]

4*(4*x^2 + x - 16)/(4*x^2 - 3*e^(2*x^2) + 12)

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mupad [B]  time = 0.54, size = 28, normalized size = 0.90 \begin {gather*} \frac {16\,x^2+4\,x-64}{4\,x^2-3\,{\mathrm {e}}^{2\,x^2}+12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((896*x - exp(2*x^2)*(864*x - 48*x^2 - 192*x^3 + 12) - 16*x^2 + 48)/(9*exp(4*x^2) - exp(2*x^2)*(24*x^2 + 72
) + 96*x^2 + 16*x^4 + 144),x)

[Out]

(4*x + 16*x^2 - 64)/(4*x^2 - 3*exp(2*x^2) + 12)

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sympy [A]  time = 0.15, size = 24, normalized size = 0.77 \begin {gather*} \frac {- 16 x^{2} - 4 x + 64}{- 4 x^{2} + 3 e^{2 x^{2}} - 12} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((192*x**3+48*x**2-864*x-12)*exp(x**2)**2-16*x**2+896*x+48)/(9*exp(x**2)**4+(-24*x**2-72)*exp(x**2)*
*2+16*x**4+96*x**2+144),x)

[Out]

(-16*x**2 - 4*x + 64)/(-4*x**2 + 3*exp(2*x**2) - 12)

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