3.99.91 \(\int \frac {-5+10 x-15 x^2+e^{2+x+e^4 x^2} (-5-5 x-10 e^4 x^2)}{1+2 x-x^2+e^{4+2 x+2 e^4 x^2} x^2+3 x^4-2 x^5+x^6+e^{2+x+e^4 x^2} (2 x+2 x^2-2 x^3+2 x^4)} \, dx\)

Optimal. Leaf size=33 \[ \frac {5}{1+x^2 \left (\frac {1+e^{2+x+e^4 x^2}-x}{x}+x\right )} \]

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Rubi [A]  time = 0.34, antiderivative size = 29, normalized size of antiderivative = 0.88, number of steps used = 3, number of rules used = 3, integrand size = 113, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.027, Rules used = {6688, 12, 6686} \begin {gather*} \frac {5}{x^3-x^2+e^{e^4 x^2+x+2} x+x+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-5 + 10*x - 15*x^2 + E^(2 + x + E^4*x^2)*(-5 - 5*x - 10*E^4*x^2))/(1 + 2*x - x^2 + E^(4 + 2*x + 2*E^4*x^2
)*x^2 + 3*x^4 - 2*x^5 + x^6 + E^(2 + x + E^4*x^2)*(2*x + 2*x^2 - 2*x^3 + 2*x^4)),x]

[Out]

5/(1 + x + E^(2 + x + E^4*x^2)*x - x^2 + x^3)

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

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Mathematica [A]  time = 0.04, size = 29, normalized size = 0.88 \begin {gather*} \frac {5}{1+x+e^{2+x+e^4 x^2} x-x^2+x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-5 + 10*x - 15*x^2 + E^(2 + x + E^4*x^2)*(-5 - 5*x - 10*E^4*x^2))/(1 + 2*x - x^2 + E^(4 + 2*x + 2*E
^4*x^2)*x^2 + 3*x^4 - 2*x^5 + x^6 + E^(2 + x + E^4*x^2)*(2*x + 2*x^2 - 2*x^3 + 2*x^4)),x]

[Out]

5/(1 + x + E^(2 + x + E^4*x^2)*x - x^2 + x^3)

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fricas [A]  time = 0.86, size = 27, normalized size = 0.82 \begin {gather*} \frac {5}{x^{3} - x^{2} + x e^{\left (x^{2} e^{4} + x + 2\right )} + x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2*exp(4)-5*x-5)*exp(x^2*exp(4)+2+x)-15*x^2+10*x-5)/(x^2*exp(x^2*exp(4)+2+x)^2+(2*x^4-2*x^3+2
*x^2+2*x)*exp(x^2*exp(4)+2+x)+x^6-2*x^5+3*x^4-x^2+2*x+1),x, algorithm="fricas")

[Out]

5/(x^3 - x^2 + x*e^(x^2*e^4 + x + 2) + x + 1)

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giac [B]  time = 2.32, size = 253, normalized size = 7.67 \begin {gather*} \frac {5 \, {\left (2 \, x^{5} e^{4} - 2 \, x^{4} e^{4} + x^{4} + 2 \, x^{3} e^{4} - 3 \, x^{3} + 2 \, x^{2} e^{4} + 2 \, x^{2} + x + 1\right )}}{2 \, x^{8} e^{4} - 4 \, x^{7} e^{4} + x^{7} + 6 \, x^{6} e^{4} + 2 \, x^{6} e^{\left (x^{2} e^{4} + x + 6\right )} - 4 \, x^{6} - 2 \, x^{5} e^{\left (x^{2} e^{4} + x + 6\right )} + x^{5} e^{\left (x^{2} e^{4} + x + 2\right )} + 6 \, x^{5} - 2 \, x^{4} e^{4} + 2 \, x^{4} e^{\left (x^{2} e^{4} + x + 6\right )} - 3 \, x^{4} e^{\left (x^{2} e^{4} + x + 2\right )} - 3 \, x^{4} + 4 \, x^{3} e^{4} + 2 \, x^{3} e^{\left (x^{2} e^{4} + x + 6\right )} + 2 \, x^{3} e^{\left (x^{2} e^{4} + x + 2\right )} - x^{3} + 2 \, x^{2} e^{4} + x^{2} e^{\left (x^{2} e^{4} + x + 2\right )} + 2 \, x^{2} + x e^{\left (x^{2} e^{4} + x + 2\right )} + 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2*exp(4)-5*x-5)*exp(x^2*exp(4)+2+x)-15*x^2+10*x-5)/(x^2*exp(x^2*exp(4)+2+x)^2+(2*x^4-2*x^3+2
*x^2+2*x)*exp(x^2*exp(4)+2+x)+x^6-2*x^5+3*x^4-x^2+2*x+1),x, algorithm="giac")

[Out]

5*(2*x^5*e^4 - 2*x^4*e^4 + x^4 + 2*x^3*e^4 - 3*x^3 + 2*x^2*e^4 + 2*x^2 + x + 1)/(2*x^8*e^4 - 4*x^7*e^4 + x^7 +
6*x^6*e^4 + 2*x^6*e^(x^2*e^4 + x + 6) - 4*x^6 - 2*x^5*e^(x^2*e^4 + x + 6) + x^5*e^(x^2*e^4 + x + 2) + 6*x^5 -
2*x^4*e^4 + 2*x^4*e^(x^2*e^4 + x + 6) - 3*x^4*e^(x^2*e^4 + x + 2) - 3*x^4 + 4*x^3*e^4 + 2*x^3*e^(x^2*e^4 + x
+ 6) + 2*x^3*e^(x^2*e^4 + x + 2) - x^3 + 2*x^2*e^4 + x^2*e^(x^2*e^4 + x + 2) + 2*x^2 + x*e^(x^2*e^4 + x + 2) +
2*x + 1)

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maple [A]  time = 0.18, size = 28, normalized size = 0.85

method	result	size
norman	\(\frac {5}{x^{3}+{\mathrm e}^{x^{2} {\mathrm e}^{4}+2+x} x -x^{2}+x +1}\)	\(28\)
risch	\(\frac {5}{x^{3}+{\mathrm e}^{x^{2} {\mathrm e}^{4}+2+x} x -x^{2}+x +1}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^2*exp(4)-5*x-5)*exp(x^2*exp(4)+2+x)-15*x^2+10*x-5)/(x^2*exp(x^2*exp(4)+2+x)^2+(2*x^4-2*x^3+2*x^2+2
*x)*exp(x^2*exp(4)+2+x)+x^6-2*x^5+3*x^4-x^2+2*x+1),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.39, size = 27, normalized size = 0.82 \begin {gather*} \frac {5}{x^{3} - x^{2} + x e^{\left (x^{2} e^{4} + x + 2\right )} + x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^2*exp(4)-5*x-5)*exp(x^2*exp(4)+2+x)-15*x^2+10*x-5)/(x^2*exp(x^2*exp(4)+2+x)^2+(2*x^4-2*x^3+2
*x^2+2*x)*exp(x^2*exp(4)+2+x)+x^6-2*x^5+3*x^4-x^2+2*x+1),x, algorithm="maxima")

[Out]

5/(x^3 - x^2 + x*e^(x^2*e^4 + x + 2) + x + 1)

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mupad [B]  time = 5.75, size = 28, normalized size = 0.85 \begin {gather*} \frac {5}{x-x^2+x^3+x\,{\mathrm {e}}^{x^2\,{\mathrm {e}}^4}\,{\mathrm {e}}^2\,{\mathrm {e}}^x+1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(x + x^2*exp(4) + 2)*(5*x + 10*x^2*exp(4) + 5) - 10*x + 15*x^2 + 5)/(2*x + exp(x + x^2*exp(4) + 2)*(2
*x + 2*x^2 - 2*x^3 + 2*x^4) + x^2*exp(2*x + 2*x^2*exp(4) + 4) - x^2 + 3*x^4 - 2*x^5 + x^6 + 1),x)

[Out]

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

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sympy [A]  time = 0.22, size = 24, normalized size = 0.73 \begin {gather*} \frac {5}{x^{3} - x^{2} + x e^{x^{2} e^{4} + x + 2} + x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**2*exp(4)-5*x-5)*exp(x**2*exp(4)+2+x)-15*x**2+10*x-5)/(x**2*exp(x**2*exp(4)+2+x)**2+(2*x**4-
2*x**3+2*x**2+2*x)*exp(x**2*exp(4)+2+x)+x**6-2*x**5+3*x**4-x**2+2*x+1),x)

[Out]

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

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