3.83.96 \(\int \frac {-12800+1920 x-12800 e^{e^{x^2}+x^2} x}{625+6400 e^{2 e^{x^2}}+8000 x+25000 x^2-3840 x^3+144 x^4+e^{e^{x^2}} (4000+25600 x-1920 x^2)} \, dx\)

Optimal. Leaf size=35 \[ \frac {4}{\frac {5}{4}+x+4 \left (e^{e^{x^2}}+x-\frac {3}{4} \left (-x+\frac {x^2}{5}\right )\right )} \]

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Rubi [A]  time = 0.26, antiderivative size = 23, normalized size of antiderivative = 0.66, number of steps used = 3, number of rules used = 3, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {6688, 12, 6686} \begin {gather*} \frac {80}{-12 x^2+80 e^{e^{x^2}}+160 x+25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-12800 + 1920*x - 12800*E^(E^x^2 + x^2)*x)/(625 + 6400*E^(2*E^x^2) + 8000*x + 25000*x^2 - 3840*x^3 + 144*
x^4 + E^E^x^2*(4000 + 25600*x - 1920*x^2)),x]

[Out]

80/(25 + 80*E^E^x^2 + 160*x - 12*x^2)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rule 6688

Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; SimplerIntegrandQ[v, u, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {640 \left (-20-\left (-3+20 e^{e^{x^2}+x^2}\right ) x\right )}{\left (25+80 e^{e^{x^2}}+160 x-12 x^2\right )^2} \, dx\\ &=640 \int \frac {-20-\left (-3+20 e^{e^{x^2}+x^2}\right ) x}{\left (25+80 e^{e^{x^2}}+160 x-12 x^2\right )^2} \, dx\\ &=\frac {80}{25+80 e^{e^{x^2}}+160 x-12 x^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 23, normalized size = 0.66 \begin {gather*} \frac {80}{25+80 e^{e^{x^2}}+160 x-12 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-12800 + 1920*x - 12800*E^(E^x^2 + x^2)*x)/(625 + 6400*E^(2*E^x^2) + 8000*x + 25000*x^2 - 3840*x^3
+ 144*x^4 + E^E^x^2*(4000 + 25600*x - 1920*x^2)),x]

[Out]

80/(25 + 80*E^E^x^2 + 160*x - 12*x^2)

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fricas [A]  time = 0.79, size = 35, normalized size = 1.00 \begin {gather*} -\frac {80 \, e^{\left (x^{2}\right )}}{{\left (12 \, x^{2} - 160 \, x - 25\right )} e^{\left (x^{2}\right )} - 80 \, e^{\left (x^{2} + e^{\left (x^{2}\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12800*x*exp(x^2)*exp(exp(x^2))+1920*x-12800)/(6400*exp(exp(x^2))^2+(-1920*x^2+25600*x+4000)*exp(ex
p(x^2))+144*x^4-3840*x^3+25000*x^2+8000*x+625),x, algorithm="fricas")

[Out]

-80*e^(x^2)/((12*x^2 - 160*x - 25)*e^(x^2) - 80*e^(x^2 + e^(x^2)))

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giac [A]  time = 0.24, size = 21, normalized size = 0.60 \begin {gather*} -\frac {80}{12 \, x^{2} - 160 \, x - 80 \, e^{\left (e^{\left (x^{2}\right )}\right )} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12800*x*exp(x^2)*exp(exp(x^2))+1920*x-12800)/(6400*exp(exp(x^2))^2+(-1920*x^2+25600*x+4000)*exp(ex
p(x^2))+144*x^4-3840*x^3+25000*x^2+8000*x+625),x, algorithm="giac")

[Out]

-80/(12*x^2 - 160*x - 80*e^(e^(x^2)) - 25)

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maple [A]  time = 0.12, size = 22, normalized size = 0.63

method	result	size
norman	\(-\frac {80}{12 x^{2}-160 x -80 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}-25}\)	\(22\)
risch	\(-\frac {80}{12 x^{2}-160 x -80 \,{\mathrm e}^{{\mathrm e}^{x^{2}}}-25}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-12800*x*exp(x^2)*exp(exp(x^2))+1920*x-12800)/(6400*exp(exp(x^2))^2+(-1920*x^2+25600*x+4000)*exp(exp(x^2)
)+144*x^4-3840*x^3+25000*x^2+8000*x+625),x,method=_RETURNVERBOSE)

[Out]

-80/(12*x^2-160*x-80*exp(exp(x^2))-25)

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maxima [A]  time = 0.40, size = 21, normalized size = 0.60 \begin {gather*} -\frac {80}{12 \, x^{2} - 160 \, x - 80 \, e^{\left (e^{\left (x^{2}\right )}\right )} - 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12800*x*exp(x^2)*exp(exp(x^2))+1920*x-12800)/(6400*exp(exp(x^2))^2+(-1920*x^2+25600*x+4000)*exp(ex
p(x^2))+144*x^4-3840*x^3+25000*x^2+8000*x+625),x, algorithm="maxima")

[Out]

-80/(12*x^2 - 160*x - 80*e^(e^(x^2)) - 25)

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mupad [B]  time = 5.26, size = 21, normalized size = 0.60 \begin {gather*} \frac {80}{160\,x+80\,{\mathrm {e}}^{{\mathrm {e}}^{x^2}}-12\,x^2+25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(12800*x*exp(x^2)*exp(exp(x^2)) - 1920*x + 12800)/(8000*x + 6400*exp(2*exp(x^2)) + exp(exp(x^2))*(25600*x
- 1920*x^2 + 4000) + 25000*x^2 - 3840*x^3 + 144*x^4 + 625),x)

[Out]

80/(160*x + 80*exp(exp(x^2)) - 12*x^2 + 25)

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sympy [A]  time = 0.39, size = 19, normalized size = 0.54 \begin {gather*} \frac {80}{- 12 x^{2} + 160 x + 80 e^{e^{x^{2}}} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-12800*x*exp(x**2)*exp(exp(x**2))+1920*x-12800)/(6400*exp(exp(x**2))**2+(-1920*x**2+25600*x+4000)*e
xp(exp(x**2))+144*x**4-3840*x**3+25000*x**2+8000*x+625),x)

[Out]

80/(-12*x**2 + 160*x + 80*exp(exp(x**2)) + 25)

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