3.81.12 \(\int \frac {-4 x+3 x^2+e^3 (2 x-x^2)+e^5 (4-12 x+13 x^2-6 x^3+x^4+e^6 (1-2 x+x^2)+e^3 (-4+10 x-8 x^2+2 x^3))}{4-12 x+13 x^2-6 x^3+x^4+e^6 (1-2 x+x^2)+e^3 (-4+10 x-8 x^2+2 x^3)} \, dx\)

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

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Rubi [A]  time = 0.18, antiderivative size = 49, normalized size of antiderivative = 1.58, number of steps used = 4, number of rules used = 4, integrand size = 127, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {1680, 1814, 21, 8} \begin {gather*} e^5 x+\frac {4 \left (\left (3-e^3\right ) x+e^3-2\right )}{\left (e^3-1\right )^2-4 \left (x+\frac {1}{4} \left (2 e^3-6\right )\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x + 3*x^2 + E^3*(2*x - x^2) + E^5*(4 - 12*x + 13*x^2 - 6*x^3 + x^4 + E^6*(1 - 2*x + x^2) + E^3*(-4 + 1
0*x - 8*x^2 + 2*x^3)))/(4 - 12*x + 13*x^2 - 6*x^3 + x^4 + E^6*(1 - 2*x + x^2) + E^3*(-4 + 10*x - 8*x^2 + 2*x^3
)),x]

[Out]

E^5*x + (4*(-2 + E^3 + (3 - E^3)*x))/((-1 + E^3)^2 - 4*((-6 + 2*E^3)/4 + x)^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

Rule 1680

Int[(Pq_)*(Q4_)^(p_), x_Symbol] :> With[{a = Coeff[Q4, x, 0], b = Coeff[Q4, x, 1], c = Coeff[Q4, x, 2], d = Co
eff[Q4, x, 3], e = Coeff[Q4, x, 4]}, Subst[Int[SimplifyIntegrand[(Pq /. x -> -(d/(4*e)) + x)*(a + d^4/(256*e^3
) - (b*d)/(8*e) + (c - (3*d^2)/(8*e))*x^2 + e*x^4)^p, x], x], x, d/(4*e) + x] /; EqQ[d^3 - 4*c*d*e + 8*b*e^2,
0] && NeQ[d, 0]] /; FreeQ[p, x] && PolyQ[Pq, x] && PolyQ[Q4, x, 4] &&  !IGtQ[p, 0]

Rule 1814

Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x^2, x], f = Coeff[P
olynomialRemainder[Pq, a + b*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b*x^2, x], x, 1]}, Simp[((a
*g - b*f*x)*(a + b*x^2)^(p + 1))/(2*a*b*(p + 1)), x] + Dist[1/(2*a*(p + 1)), Int[(a + b*x^2)^(p + 1)*ExpandToS
um[2*a*(p + 1)*Q + f*(2*p + 3), x], x], x]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && LtQ[p, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\operatorname {Subst}\left (\int \frac {\left (1-e^3\right )^2 \left (12-4 e^3+e^5-2 e^8+e^{11}\right )+16 \left (5-4 e^3+e^6\right ) x+8 \left (6-2 e^3-e^5+2 e^8-e^{11}\right ) x^2+16 e^5 x^4}{\left (1-2 e^3+e^6-4 x^2\right )^2} \, dx,x,\frac {1}{4} \left (-6+2 e^3\right )+x\right )\\ &=\frac {4 \left (-2+e^3+\left (3-e^3\right ) x\right )}{\left (-1+e^3\right )^2-4 \left (\frac {1}{4} \left (-6+2 e^3\right )+x\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-2 e^5 \left (1-e^3\right )^4+8 e^5 \left (1-e^3\right )^2 x^2}{1-2 e^3+e^6-4 x^2} \, dx,x,\frac {1}{4} \left (-6+2 e^3\right )+x\right )}{2 \left (1-e^3\right )^2}\\ &=\frac {4 \left (-2+e^3+\left (3-e^3\right ) x\right )}{\left (-1+e^3\right )^2-4 \left (\frac {1}{4} \left (-6+2 e^3\right )+x\right )^2}+e^5 \operatorname {Subst}\left (\int 1 \, dx,x,\frac {1}{4} \left (-6+2 e^3\right )+x\right )\\ &=e^5 x+\frac {4 \left (-2+e^3+\left (3-e^3\right ) x\right )}{\left (-1+e^3\right )^2-4 \left (\frac {1}{4} \left (-6+2 e^3\right )+x\right )^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 46, normalized size = 1.48 \begin {gather*} \frac {1}{\left (1-e^3\right ) (-1+x)}+e^5 (-1+x)+\frac {\left (-2+e^3\right )^2}{\left (-1+e^3\right ) \left (-2+e^3+x\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x + 3*x^2 + E^3*(2*x - x^2) + E^5*(4 - 12*x + 13*x^2 - 6*x^3 + x^4 + E^6*(1 - 2*x + x^2) + E^3*(
-4 + 10*x - 8*x^2 + 2*x^3)))/(4 - 12*x + 13*x^2 - 6*x^3 + x^4 + E^6*(1 - 2*x + x^2) + E^3*(-4 + 10*x - 8*x^2 +
2*x^3)),x]

[Out]

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

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fricas [A]  time = 0.62, size = 53, normalized size = 1.71 \begin {gather*} \frac {{\left (x^{2} - x\right )} e^{8} + {\left (x^{3} - 3 \, x^{2} + 2 \, x\right )} e^{5} + {\left (x - 1\right )} e^{3} - 3 \, x + 2}{x^{2} + {\left (x - 1\right )} e^{3} - 3 \, x + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4)*exp(5)+(-x^2+2*x)*exp(3)
+3*x^2-4*x)/((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4),x, algorithm="fricas")

[Out]

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

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: NotImplementedError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4)*exp(5)+(-x^2+2*x)*exp(3)
+3*x^2-4*x)/((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: sageVARx*exp(5)+(-exp(6)*exp(3)+exp(6)+2
*exp(3)^2-3*exp(3)+1)/(exp(6)-2*exp(3)+1)^2/(sageVARx-1)+(-exp(6)+exp(3)^2)/(exp(6)^2-4*exp(6)*exp(3)+2*exp(6)
+4*exp(3)^2-4*exp(3)+

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maple [A]  time = 0.12, size = 37, normalized size = 1.19

method	result	size
risch	\(x \,{\mathrm e}^{5}+\frac {\left ({\mathrm e}^{3}-3\right ) x -{\mathrm e}^{3}+2}{x \,{\mathrm e}^{3}+x^{2}-{\mathrm e}^{3}-3 x +2}\)	\(37\)
norman	\(\frac {x^{3} {\mathrm e}^{5}+2+\left (-{\mathrm e}^{5} {\mathrm e}^{6}+5 \,{\mathrm e}^{3} {\mathrm e}^{5}-7 \,{\mathrm e}^{5}+{\mathrm e}^{3}-3\right ) x +{\mathrm e}^{5} {\mathrm e}^{6}-5 \,{\mathrm e}^{3} {\mathrm e}^{5}+6 \,{\mathrm e}^{5}-{\mathrm e}^{3}}{\left (x -1\right ) \left (x +{\mathrm e}^{3}-2\right )}\)	\(67\)
gosper	\(-\frac {{\mathrm e}^{5} {\mathrm e}^{6} x -x^{3} {\mathrm e}^{5}-{\mathrm e}^{5} {\mathrm e}^{6}-5 x \,{\mathrm e}^{3} {\mathrm e}^{5}+5 \,{\mathrm e}^{3} {\mathrm e}^{5}+7 x \,{\mathrm e}^{5}-x \,{\mathrm e}^{3}-6 \,{\mathrm e}^{5}+{\mathrm e}^{3}+3 x -2}{x \,{\mathrm e}^{3}+x^{2}-{\mathrm e}^{3}-3 x +2}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4)*exp(5)+(-x^2+2*x)*exp(3)+3*x^2
-4*x)/((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4),x,method=_RETURNVERBOSE)

[Out]

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

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maxima [A]  time = 0.35, size = 35, normalized size = 1.13 \begin {gather*} x e^{5} + \frac {x {\left (e^{3} - 3\right )} - e^{3} + 2}{x^{2} + x {\left (e^{3} - 3\right )} - e^{3} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4)*exp(5)+(-x^2+2*x)*exp(3)
+3*x^2-4*x)/((x^2-2*x+1)*exp(3)^2+(2*x^3-8*x^2+10*x-4)*exp(3)+x^4-6*x^3+13*x^2-12*x+4),x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 6.35, size = 35, normalized size = 1.13 \begin {gather*} \frac {x\,\left ({\mathrm {e}}^3-3\right )-{\mathrm {e}}^3+2}{x^2+\left ({\mathrm {e}}^3-3\right )\,x-{\mathrm {e}}^3+2}+x\,{\mathrm {e}}^5 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3)*(2*x - x^2) - 4*x + exp(5)*(exp(3)*(10*x - 8*x^2 + 2*x^3 - 4) - 12*x + exp(6)*(x^2 - 2*x + 1) + 13
*x^2 - 6*x^3 + x^4 + 4) + 3*x^2)/(exp(3)*(10*x - 8*x^2 + 2*x^3 - 4) - 12*x + exp(6)*(x^2 - 2*x + 1) + 13*x^2 -
6*x^3 + x^4 + 4),x)

[Out]

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

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sympy [A]  time = 0.65, size = 31, normalized size = 1.00 \begin {gather*} x e^{5} + \frac {x \left (-3 + e^{3}\right ) - e^{3} + 2}{x^{2} + x \left (-3 + e^{3}\right ) - e^{3} + 2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2-2*x+1)*exp(3)**2+(2*x**3-8*x**2+10*x-4)*exp(3)+x**4-6*x**3+13*x**2-12*x+4)*exp(5)+(-x**2+2*x
)*exp(3)+3*x**2-4*x)/((x**2-2*x+1)*exp(3)**2+(2*x**3-8*x**2+10*x-4)*exp(3)+x**4-6*x**3+13*x**2-12*x+4),x)

[Out]

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

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