3.755 \(\int \frac {1}{(x+\sqrt {3-2 x-x^2})^3} \, dx\)

Optimal. Leaf size=307 \[ -\frac {4 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}+\frac {2 \left (-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18\right )}{147 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}+\frac {12 \tanh ^{-1}\left (\frac {-\sqrt {3} \sqrt {-x^2-2 x+3}-\sqrt {3} x-x+3}{\sqrt {7} x}\right )}{49 \sqrt {7}} \]

[Out]

12/343*arctanh(1/7*(3-x-x*3^(1/2)-3^(1/2)*(-x^2-2*x+3)^(1/2))/x*7^(1/2))*7^(1/2)-4/21*(9-5*3^(1/2)+(21+5*3^(1/
2))*(3^(1/2)-(-x^2-2*x+3)^(1/2))/x)/(2-3^(1/2)-2*(1+3^(1/2))*(3^(1/2)-(-x^2-2*x+3)^(1/2))/x+3^(1/2)*(3^(1/2)-(
-x^2-2*x+3)^(1/2))^2/x^2)^2+2/147*(18-43*3^(1/2)-(18+49*3^(1/2))*(3^(1/2)-(-x^2-2*x+3)^(1/2))/x)/(2-3^(1/2)-2*
(1+3^(1/2))*(3^(1/2)-(-x^2-2*x+3)^(1/2))/x+3^(1/2)*(3^(1/2)-(-x^2-2*x+3)^(1/2))^2/x^2)

________________________________________________________________________________________

Rubi [A]  time = 0.25, antiderivative size = 307, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1660, 12, 618, 206} \[ -\frac {4 \left (\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-5 \sqrt {3}+9\right )}{21 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )^2}+\frac {2 \left (-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-43 \sqrt {3}+18\right )}{147 \left (\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )^2}{x^2}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {-x^2-2 x+3}\right )}{x}-\sqrt {3}+2\right )}+\frac {12 \tanh ^{-1}\left (\frac {-\sqrt {3} \sqrt {-x^2-2 x+3}-\sqrt {3} x-x+3}{\sqrt {7} x}\right )}{49 \sqrt {7}} \]

Antiderivative was successfully verified.

[In]

Int[(x + Sqrt[3 - 2*x - x^2])^(-3),x]

[Out]

(-4*(9 - 5*Sqrt[3] + ((21 + 5*Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(21*(2 - Sqrt[3] - (2*(1 + Sqrt[3]
)*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2])^2)/x^2)^2) + (2*(18 - 43*Sqrt[
3] - ((18 + 49*Sqrt[3])*(Sqrt[3] - Sqrt[3 - 2*x - x^2]))/x))/(147*(2 - Sqrt[3] - (2*(1 + Sqrt[3])*(Sqrt[3] - S
qrt[3 - 2*x - x^2]))/x + (Sqrt[3]*(Sqrt[3] - Sqrt[3 - 2*x - x^2])^2)/x^2)) + (12*ArcTanh[(3 - x - Sqrt[3]*x -
Sqrt[3]*Sqrt[3 - 2*x - x^2])/(Sqrt[7]*x)])/(49*Sqrt[7])

Rule 12

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 1660

Int[(Pq_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq, a + b*x + c*
x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + b
*x + c*x^2, x], x, 1]}, Simp[((b*f - 2*a*g + (2*c*f - b*g)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c
)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*Q - (
2*p + 3)*(2*c*f - b*g), x], x], x]] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[p, -1
]

Rubi steps

\begin {align*} \int \frac {1}{\left (x+\sqrt {3-2 x-x^2}\right )^3} \, dx &=2 \operatorname {Subst}\left (\int \frac {\sqrt {3}-2 x-2 x^3-\sqrt {3} x^4}{\left (2-\sqrt {3}+2 \left (1+\sqrt {3}\right ) x+\sqrt {3} x^2\right )^3} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}-\frac {1}{28} \operatorname {Subst}\left (\int \frac {-\frac {8}{3} \left (21+16 \sqrt {3}\right )-112 x+56 x^2}{\left (2-\sqrt {3}+2 \left (1+\sqrt {3}\right ) x+\sqrt {3} x^2\right )^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}+\frac {2 \left (18-43 \sqrt {3}-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{147 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )}+\frac {1}{784} \operatorname {Subst}\left (\int \frac {192}{2-\sqrt {3}+2 \left (1+\sqrt {3}\right ) x+\sqrt {3} x^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}+\frac {2 \left (18-43 \sqrt {3}-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{147 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )}+\frac {12}{49} \operatorname {Subst}\left (\int \frac {1}{2-\sqrt {3}+2 \left (1+\sqrt {3}\right ) x+\sqrt {3} x^2} \, dx,x,\frac {-\sqrt {3}+\sqrt {3-2 x-x^2}}{x}\right )\\ &=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}+\frac {2 \left (18-43 \sqrt {3}-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{147 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )}-\frac {24}{49} \operatorname {Subst}\left (\int \frac {1}{28-x^2} \, dx,x,\frac {2 \left (-3+x+\sqrt {3} x+\sqrt {3} \sqrt {3-2 x-x^2}\right )}{x}\right )\\ &=-\frac {4 \left (9-5 \sqrt {3}+\frac {\left (21+5 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{21 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )^2}+\frac {2 \left (18-43 \sqrt {3}-\frac {\left (18+49 \sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}\right )}{147 \left (2-\sqrt {3}-\frac {2 \left (1+\sqrt {3}\right ) \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )}{x}+\frac {\sqrt {3} \left (\sqrt {3}-\sqrt {3-2 x-x^2}\right )^2}{x^2}\right )}+\frac {12 \tanh ^{-1}\left (\frac {3-x-\sqrt {3} x-\sqrt {3} \sqrt {3-2 x-x^2}}{\sqrt {7} x}\right )}{49 \sqrt {7}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 1.11, size = 333, normalized size = 1.08 \[ \frac {\frac {7 (37-24 x)}{2 x^2+2 x-3}+\frac {98 (11 x-12)}{\left (2 x^2+2 x-3\right )^2}-6 \left (1+\sqrt {7}\right ) \sqrt {\frac {14}{4+\sqrt {7}}} \log \left (\sqrt {14 \left (4+\sqrt {7}\right )} \sqrt {-x^2-2 x+3}-\sqrt {7} x+7 x+7 \sqrt {7}+7\right )-2 \left (\sqrt {7}-1\right ) \sqrt {14 \left (4+\sqrt {7}\right )} \log \left (-\sqrt {14} \sqrt {\left (\sqrt {7}-4\right ) \left (x^2+2 x-3\right )}+\left (7+\sqrt {7}\right ) x-7 \sqrt {7}+7\right )-\frac {14 \sqrt {-x^2-2 x+3} \left (34 x^3+58 x^2-83 x-15\right )}{\left (2 x^2+2 x-3\right )^2}-12 \sqrt {7} \log \left (-2 x+\sqrt {7}-1\right )+2 \left (\sqrt {7}-1\right ) \sqrt {14 \left (4+\sqrt {7}\right )} \log \left (2 x-\sqrt {7}+1\right )+6 \left (1+\sqrt {7}\right ) \sqrt {\frac {14}{4+\sqrt {7}}} \log \left (2 x+\sqrt {7}+1\right )+12 \sqrt {7} \log \left (2 x+\sqrt {7}+1\right )}{1372} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[(x + Sqrt[3 - 2*x - x^2])^(-3),x]

[Out]

((98*(-12 + 11*x))/(-3 + 2*x + 2*x^2)^2 + (7*(37 - 24*x))/(-3 + 2*x + 2*x^2) - (14*Sqrt[3 - 2*x - x^2]*(-15 -
83*x + 58*x^2 + 34*x^3))/(-3 + 2*x + 2*x^2)^2 - 12*Sqrt[7]*Log[-1 + Sqrt[7] - 2*x] + 2*(-1 + Sqrt[7])*Sqrt[14*
(4 + Sqrt[7])]*Log[1 - Sqrt[7] + 2*x] + 12*Sqrt[7]*Log[1 + Sqrt[7] + 2*x] + 6*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[
7])]*Log[1 + Sqrt[7] + 2*x] - 6*(1 + Sqrt[7])*Sqrt[14/(4 + Sqrt[7])]*Log[7 + 7*Sqrt[7] + 7*x - Sqrt[7]*x + Sqr
t[14*(4 + Sqrt[7])]*Sqrt[3 - 2*x - x^2]] - 2*(-1 + Sqrt[7])*Sqrt[14*(4 + Sqrt[7])]*Log[7 - 7*Sqrt[7] + (7 + Sq
rt[7])*x - Sqrt[14]*Sqrt[(-4 + Sqrt[7])*(-3 + 2*x + x^2)]])/1372

________________________________________________________________________________________

fricas [A]  time = 0.43, size = 223, normalized size = 0.73 \[ -\frac {336 \, x^{3} - 6 \, \sqrt {7} {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac {x^{4} + 44 \, x^{3} - \sqrt {7} {\left (3 \, x^{3} + x^{2} - 45 \, x + 45\right )} \sqrt {-x^{2} - 2 \, x + 3} + 26 \, x^{2} - 276 \, x + 207}{4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9}\right ) - 12 \, \sqrt {7} {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )} \log \left (\frac {2 \, x^{2} + \sqrt {7} {\left (2 \, x + 1\right )} + 2 \, x + 4}{2 \, x^{2} + 2 \, x - 3}\right ) - 182 \, x^{2} + 14 \, {\left (34 \, x^{3} + 58 \, x^{2} - 83 \, x - 15\right )} \sqrt {-x^{2} - 2 \, x + 3} - 2100 \, x + 1953}{1372 \, {\left (4 \, x^{4} + 8 \, x^{3} - 8 \, x^{2} - 12 \, x + 9\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(-x^2-2*x+3)^(1/2))^3,x, algorithm="fricas")

[Out]

-1/1372*(336*x^3 - 6*sqrt(7)*(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)*log((x^4 + 44*x^3 - sqrt(7)*(3*x^3 + x^2 - 45*
x + 45)*sqrt(-x^2 - 2*x + 3) + 26*x^2 - 276*x + 207)/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)) - 12*sqrt(7)*(4*x^4 +
8*x^3 - 8*x^2 - 12*x + 9)*log((2*x^2 + sqrt(7)*(2*x + 1) + 2*x + 4)/(2*x^2 + 2*x - 3)) - 182*x^2 + 14*(34*x^3
+ 58*x^2 - 83*x - 15)*sqrt(-x^2 - 2*x + 3) - 2100*x + 1953)/(4*x^4 + 8*x^3 - 8*x^2 - 12*x + 9)

________________________________________________________________________________________

giac [A]  time = 0.56, size = 452, normalized size = 1.47 \[ -\frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | 4 \, x - 2 \, \sqrt {7} + 2 \right |}}{{\left | 4 \, x + 2 \, \sqrt {7} + 2 \right |}}\right ) + \frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {6 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + 4 \right |}}\right ) - \frac {3}{343} \, \sqrt {7} \log \left (\frac {{\left | -2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}{{\left | 2 \, \sqrt {7} + \frac {2 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} - 4 \right |}}\right ) - \frac {48 \, x^{3} - 26 \, x^{2} - 300 \, x + 279}{196 \, {\left (2 \, x^{2} + 2 \, x - 3\right )}^{2}} + \frac {4 \, {\left (\frac {231 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {3286 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} - \frac {4441 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {18906 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - \frac {12487 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{5}}{{\left (x + 1\right )}^{5}} + \frac {946 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{6}}{{\left (x + 1\right )}^{6}} + \frac {1977 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{7}}{{\left (x + 1\right )}^{7}} - 414\right )}}{441 \, {\left (\frac {8 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}}{x + 1} + \frac {26 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{2}}{{\left (x + 1\right )}^{2}} + \frac {8 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{3}}{{\left (x + 1\right )}^{3}} - \frac {3 \, {\left (\sqrt {-x^{2} - 2 \, x + 3} - 2\right )}^{4}}{{\left (x + 1\right )}^{4}} - 3\right )}^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(-x^2-2*x+3)^(1/2))^3,x, algorithm="giac")

[Out]

-3/343*sqrt(7)*log(abs(4*x - 2*sqrt(7) + 2)/abs(4*x + 2*sqrt(7) + 2)) + 3/343*sqrt(7)*log(abs(-2*sqrt(7) + 6*(
sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)/abs(2*sqrt(7) + 6*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) + 4)) - 3/343*sqrt
(7)*log(abs(-2*sqrt(7) + 2*(sqrt(-x^2 - 2*x + 3) - 2)/(x + 1) - 4)/abs(2*sqrt(7) + 2*(sqrt(-x^2 - 2*x + 3) - 2
)/(x + 1) - 4)) - 1/196*(48*x^3 - 26*x^2 - 300*x + 279)/(2*x^2 + 2*x - 3)^2 + 4/441*(231*(sqrt(-x^2 - 2*x + 3)
- 2)/(x + 1) + 3286*(sqrt(-x^2 - 2*x + 3) - 2)^2/(x + 1)^2 - 4441*(sqrt(-x^2 - 2*x + 3) - 2)^3/(x + 1)^3 - 18
906*(sqrt(-x^2 - 2*x + 3) - 2)^4/(x + 1)^4 - 12487*(sqrt(-x^2 - 2*x + 3) - 2)^5/(x + 1)^5 + 946*(sqrt(-x^2 - 2
*x + 3) - 2)^6/(x + 1)^6 + 1977*(sqrt(-x^2 - 2*x + 3) - 2)^7/(x + 1)^7 - 414)/(8*(sqrt(-x^2 - 2*x + 3) - 2)/(x
+ 1) + 26*(sqrt(-x^2 - 2*x + 3) - 2)^2/(x + 1)^2 + 8*(sqrt(-x^2 - 2*x + 3) - 2)^3/(x + 1)^3 - 3*(sqrt(-x^2 -
2*x + 3) - 2)^4/(x + 1)^4 - 3)^2

________________________________________________________________________________________

maple [B]  time = 0.07, size = 5984, normalized size = 19.49 \[ \text {output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x+(-x^2-2*x+3)^(1/2))^3,x)

[Out]

result too large to display

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (x + \sqrt {-x^{2} - 2 \, x + 3}\right )}^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(-x^2-2*x+3)^(1/2))^3,x, algorithm="maxima")

[Out]

integrate((x + sqrt(-x^2 - 2*x + 3))^(-3), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\left (x+\sqrt {-x^2-2\,x+3}\right )}^3} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x + (3 - x^2 - 2*x)^(1/2))^3,x)

[Out]

int(1/(x + (3 - x^2 - 2*x)^(1/2))^3, x)

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (x + \sqrt {- x^{2} - 2 x + 3}\right )^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x+(-x**2-2*x+3)**(1/2))**3,x)

[Out]

Integral((x + sqrt(-x**2 - 2*x + 3))**(-3), x)

________________________________________________________________________________________