3.79.74 \(\int \frac {2-6 x+6 x^2-2 x^3+e^{\frac {36 x^2-36 x^3+9 x^4}{1-2 x+x^2}} (72 x-108 x^2+72 x^3-18 x^4)}{-1+3 x-3 x^2+x^3} \, dx\)

Optimal. Leaf size=26 \[ 1-e^{\left (-3+\frac {3}{-1+x}\right )^2 x^2}-2 x+\log (2) \]

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Rubi [B]  time = 0.73, antiderivative size = 57, normalized size of antiderivative = 2.19, number of steps used = 9, number of rules used = 5, integrand size = 80, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {6688, 6742, 37, 43, 6706} \begin {gather*} \frac {3 x^2}{(1-x)^2}-e^{\frac {9 (2-x)^2 x^2}{(1-x)^2}}-2 x+\frac {6}{1-x}-\frac {3}{(1-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 - 6*x + 6*x^2 - 2*x^3 + E^((36*x^2 - 36*x^3 + 9*x^4)/(1 - 2*x + x^2))*(72*x - 108*x^2 + 72*x^3 - 18*x^4
))/(-1 + 3*x - 3*x^2 + x^3),x]

[Out]

-E^((9*(2 - x)^2*x^2)/(1 - x)^2) - 3/(1 - x)^2 + 6/(1 - x) - 2*x + (3*x^2)/(1 - x)^2

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 6688

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

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-2+6 x-6 x^2+2 x^3+18 e^{\frac {9 (-2+x)^2 x^2}{(-1+x)^2}} x \left (-4+6 x-4 x^2+x^3\right )}{(1-x)^3} \, dx\\ &=\int \left (\frac {2}{(-1+x)^3}-\frac {6 x}{(-1+x)^3}+\frac {6 x^2}{(-1+x)^3}-\frac {2 x^3}{(-1+x)^3}-\frac {18 e^{\frac {9 (-2+x)^2 x^2}{(-1+x)^2}} (-2+x) x \left (2-2 x+x^2\right )}{(-1+x)^3}\right ) \, dx\\ &=-\frac {1}{(1-x)^2}-2 \int \frac {x^3}{(-1+x)^3} \, dx-6 \int \frac {x}{(-1+x)^3} \, dx+6 \int \frac {x^2}{(-1+x)^3} \, dx-18 \int \frac {e^{\frac {9 (-2+x)^2 x^2}{(-1+x)^2}} (-2+x) x \left (2-2 x+x^2\right )}{(-1+x)^3} \, dx\\ &=-e^{\frac {9 (2-x)^2 x^2}{(1-x)^2}}-\frac {1}{(1-x)^2}+\frac {3 x^2}{(1-x)^2}-2 \int \left (1+\frac {1}{(-1+x)^3}+\frac {3}{(-1+x)^2}+\frac {3}{-1+x}\right ) \, dx+6 \int \left (\frac {1}{(-1+x)^3}+\frac {2}{(-1+x)^2}+\frac {1}{-1+x}\right ) \, dx\\ &=-e^{\frac {9 (2-x)^2 x^2}{(1-x)^2}}-\frac {3}{(1-x)^2}+\frac {6}{1-x}-2 x+\frac {3 x^2}{(1-x)^2}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.28, size = 25, normalized size = 0.96 \begin {gather*} -e^{-9+\frac {9}{(-1+x)^2}-18 x+9 x^2}-2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 - 6*x + 6*x^2 - 2*x^3 + E^((36*x^2 - 36*x^3 + 9*x^4)/(1 - 2*x + x^2))*(72*x - 108*x^2 + 72*x^3 -
18*x^4))/(-1 + 3*x - 3*x^2 + x^3),x]

[Out]

-E^(-9 + 9/(-1 + x)^2 - 18*x + 9*x^2) - 2*x

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fricas [A]  time = 0.48, size = 33, normalized size = 1.27 \begin {gather*} -2 \, x - e^{\left (\frac {9 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}}{x^{2} - 2 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^4+72*x^3-108*x^2+72*x)*exp((9*x^4-36*x^3+36*x^2)/(x^2-2*x+1))-2*x^3+6*x^2-6*x+2)/(x^3-3*x^2+
3*x-1),x, algorithm="fricas")

[Out]

-2*x - e^(9*(x^4 - 4*x^3 + 4*x^2)/(x^2 - 2*x + 1))

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giac [A]  time = 0.26, size = 33, normalized size = 1.27 \begin {gather*} -2 \, x - e^{\left (\frac {9 \, {\left (x^{4} - 4 \, x^{3} + 4 \, x^{2}\right )}}{x^{2} - 2 \, x + 1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^4+72*x^3-108*x^2+72*x)*exp((9*x^4-36*x^3+36*x^2)/(x^2-2*x+1))-2*x^3+6*x^2-6*x+2)/(x^3-3*x^2+
3*x-1),x, algorithm="giac")

[Out]

-2*x - e^(9*(x^4 - 4*x^3 + 4*x^2)/(x^2 - 2*x + 1))

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maple [A]  time = 0.14, size = 23, normalized size = 0.88

method	result	size
risch	\(-2 x -{\mathrm e}^{\frac {9 x^{2} \left (x -2\right )^{2}}{\left (x -1\right )^{2}}}\)	\(23\)
norman	\(\frac {6 x -2 x^{3}+2 x \,{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-x^{2} {\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-{\mathrm e}^{\frac {9 x^{4}-36 x^{3}+36 x^{2}}{x^{2}-2 x +1}}-4}{\left (x -1\right )^{2}}\)	\(111\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-18*x^4+72*x^3-108*x^2+72*x)*exp((9*x^4-36*x^3+36*x^2)/(x^2-2*x+1))-2*x^3+6*x^2-6*x+2)/(x^3-3*x^2+3*x-1)
,x,method=_RETURNVERBOSE)

[Out]

-2*x-exp(9*x^2*(x-2)^2/(x-1)^2)

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maxima [B]  time = 0.48, size = 91, normalized size = 3.50 \begin {gather*} -2 \, x + \frac {6 \, x - 5}{x^{2} - 2 \, x + 1} - \frac {3 \, {\left (4 \, x - 3\right )}}{x^{2} - 2 \, x + 1} + \frac {3 \, {\left (2 \, x - 1\right )}}{x^{2} - 2 \, x + 1} - \frac {1}{x^{2} - 2 \, x + 1} - e^{\left (9 \, x^{2} - 18 \, x + \frac {9}{x^{2} - 2 \, x + 1} - 9\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x^4+72*x^3-108*x^2+72*x)*exp((9*x^4-36*x^3+36*x^2)/(x^2-2*x+1))-2*x^3+6*x^2-6*x+2)/(x^3-3*x^2+
3*x-1),x, algorithm="maxima")

[Out]

-2*x + (6*x - 5)/(x^2 - 2*x + 1) - 3*(4*x - 3)/(x^2 - 2*x + 1) + 3*(2*x - 1)/(x^2 - 2*x + 1) - 1/(x^2 - 2*x +
1) - e^(9*x^2 - 18*x + 9/(x^2 - 2*x + 1) - 9)

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mupad [B]  time = 5.16, size = 54, normalized size = 2.08 \begin {gather*} -2\,x-{\mathrm {e}}^{\frac {9\,x^4}{x^2-2\,x+1}}\,{\mathrm {e}}^{\frac {36\,x^2}{x^2-2\,x+1}}\,{\mathrm {e}}^{-\frac {36\,x^3}{x^2-2\,x+1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((36*x^2 - 36*x^3 + 9*x^4)/(x^2 - 2*x + 1))*(72*x - 108*x^2 + 72*x^3 - 18*x^4) - 6*x + 6*x^2 - 2*x^3 +
2)/(3*x - 3*x^2 + x^3 - 1),x)

[Out]

- 2*x - exp((9*x^4)/(x^2 - 2*x + 1))*exp((36*x^2)/(x^2 - 2*x + 1))*exp(-(36*x^3)/(x^2 - 2*x + 1))

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sympy [A]  time = 0.24, size = 29, normalized size = 1.12 \begin {gather*} - 2 x - e^{\frac {9 x^{4} - 36 x^{3} + 36 x^{2}}{x^{2} - 2 x + 1}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-18*x**4+72*x**3-108*x**2+72*x)*exp((9*x**4-36*x**3+36*x**2)/(x**2-2*x+1))-2*x**3+6*x**2-6*x+2)/(x
**3-3*x**2+3*x-1),x)

[Out]

-2*x - exp((9*x**4 - 36*x**3 + 36*x**2)/(x**2 - 2*x + 1))

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