\(\int \frac {2+x^2}{(-2+x^2) \sqrt {-2 x+2 x^2+x^3}} \, dx\) [515]

Optimal result

Integrand size = 29, antiderivative size = 40 \[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=-\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {-2 x+2 x^2+x^3}}{-2+2 x+x^2}\right ) \] Output:

-2^(1/2)*arctanh(2^(1/2)*(x^3+2*x^2-2*x)^(1/2)/(x^2+2*x-2))
 

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.55 \[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=-\frac {\sqrt {2} \sqrt {x} \sqrt {-2+2 x+x^2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {-2+2 x+x^2}}\right )}{\sqrt {x \left (-2+2 x+x^2\right )}} \] Input:

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

Output:

-((Sqrt[2]*Sqrt[x]*Sqrt[-2 + 2*x + x^2]*ArcTanh[(Sqrt[2]*Sqrt[x])/Sqrt[-2 
+ 2*x + x^2]])/Sqrt[x*(-2 + 2*x + x^2)])
 

Rubi [A] (verified)

Time = 0.44 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.60, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {2467, 25, 2035, 2537, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2+2}{\left (x^2-2\right ) \sqrt {x^3+2 x^2-2 x}} \, dx\)

\(\Big \downarrow \) 2467

\(\displaystyle \frac {\sqrt {x} \sqrt {x^2+2 x-2} \int -\frac {x^2+2}{\sqrt {x} \left (2-x^2\right ) \sqrt {x^2+2 x-2}}dx}{\sqrt {x^3+2 x^2-2 x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {\sqrt {x} \sqrt {x^2+2 x-2} \int \frac {x^2+2}{\sqrt {x} \left (2-x^2\right ) \sqrt {x^2+2 x-2}}dx}{\sqrt {x^3+2 x^2-2 x}}\)

\(\Big \downarrow \) 2035

\(\displaystyle -\frac {2 \sqrt {x} \sqrt {x^2+2 x-2} \int \frac {x^2+2}{\left (2-x^2\right ) \sqrt {x^2+2 x-2}}d\sqrt {x}}{\sqrt {x^3+2 x^2-2 x}}\)

\(\Big \downarrow \) 2537

\(\displaystyle -\frac {4 \sqrt {x} \sqrt {x^2+2 x-2} \int \frac {1}{2-4 x}d\frac {\sqrt {x}}{\sqrt {x^2+2 x-2}}}{\sqrt {x^3+2 x^2-2 x}}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {\sqrt {2} \sqrt {x} \sqrt {x^2+2 x-2} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x^2+2 x-2}}\right )}{\sqrt {x^3+2 x^2-2 x}}\)

Input:

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

Output:

-((Sqrt[2]*Sqrt[x]*Sqrt[-2 + 2*x + x^2]*ArcTanh[(Sqrt[2]*Sqrt[x])/Sqrt[-2 
+ 2*x + x^2]])/Sqrt[-2*x + 2*x^2 + x^3])
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 219

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

rule 2035

Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k   Subst 
[Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti 
onQ[m] && AlgebraicFunctionQ[Fx, x]
 

rule 2467

Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F 
racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p])   Int[x^( 
p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P 
olyQ[Px, x] &&  !IntegerQ[p] &&  !MonomialQ[Px, x] &&  !PolyQ[Fx, x]
 

rule 2537

Int[((u_)*((A_) + (B_.)*(x_)^4))/Sqrt[v_], x_Symbol] :> With[{a = Coeff[v, 
x, 0], b = Coeff[v, x, 2], c = Coeff[v, x, 4], d = Coeff[1/u, x, 0], e = Co 
eff[1/u, x, 2], f = Coeff[1/u, x, 4]}, Simp[A   Subst[Int[1/(d - (b*d - a*e 
)*x^2), x], x, x/Sqrt[v]], x] /; EqQ[a*B + A*c, 0] && EqQ[c*d - a*f, 0]] /; 
 FreeQ[{A, B}, x] && PolyQ[v, x^2, 2] && PolyQ[1/u, x^2, 2]
 

Maple [A] (verified)

Time = 0.54 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.68

method	result	size
default	\(-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+2 x -2\right )}\, \sqrt {2}}{2 x}\right )\)	\(27\)
pseudoelliptic	\(-\sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {x \left (x^{2}+2 x -2\right )}\, \sqrt {2}}{2 x}\right )\)	\(27\)
trager	\(\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x^{2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) x +4 \sqrt {x^{3}+2 x^{2}-2 x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right )}{x^{2}-2}\right )}{2}\)	\(63\)
elliptic	\(\frac {\sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}}+\frac {\sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \operatorname {EllipticF}\left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}}+\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}-\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}-\sqrt {2}\right )}+\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}-\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}-\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}+\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right )}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}+\sqrt {2}\right )}-\frac {\sqrt {2}\, \sqrt {\frac {x}{1+\sqrt {3}}+\frac {1}{1+\sqrt {3}}+\frac {\sqrt {3}}{1+\sqrt {3}}}\, \sqrt {-6 x \sqrt {3}+18-6 \sqrt {3}}\, \sqrt {-\frac {x}{1+\sqrt {3}}}\, \operatorname {EllipticPi}\left (\sqrt {\frac {x +1+\sqrt {3}}{1+\sqrt {3}}}, \frac {-1-\sqrt {3}}{-1-\sqrt {3}+\sqrt {2}}, \frac {\sqrt {6}\, \sqrt {\left (1+\sqrt {3}\right ) \sqrt {3}}}{6}\right ) \sqrt {3}}{3 \sqrt {x^{3}+2 x^{2}-2 x}\, \left (-1-\sqrt {3}+\sqrt {2}\right )}\)	\(795\)

Input:

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

Output:

-2^(1/2)*arctanh(1/2*(x*(x^2+2*x-2))^(1/2)/x*2^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.60 \[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=\frac {1}{4} \, \sqrt {2} \log \left (\frac {x^{4} + 16 \, x^{3} - 4 \, \sqrt {2} \sqrt {x^{3} + 2 \, x^{2} - 2 \, x} {\left (x^{2} + 4 \, x - 2\right )} + 28 \, x^{2} - 32 \, x + 4}{x^{4} - 4 \, x^{2} + 4}\right ) \] Input:

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

Output:

1/4*sqrt(2)*log((x^4 + 16*x^3 - 4*sqrt(2)*sqrt(x^3 + 2*x^2 - 2*x)*(x^2 + 4 
*x - 2) + 28*x^2 - 32*x + 4)/(x^4 - 4*x^2 + 4))
 

Sympy [F]

\[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=\int \frac {x^{2} + 2}{\sqrt {x \left (x^{2} + 2 x - 2\right )} \left (x^{2} - 2\right )}\, dx \] Input:

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

Output:

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

Maxima [F]

\[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=\int { \frac {x^{2} + 2}{\sqrt {x^{3} + 2 \, x^{2} - 2 \, x} {\left (x^{2} - 2\right )}} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=\int { \frac {x^{2} + 2}{\sqrt {x^{3} + 2 \, x^{2} - 2 \, x} {\left (x^{2} - 2\right )}} \,d x } \] Input:

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

Output:

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

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 227, normalized size of antiderivative = 5.68 \[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=-\frac {2\,\sqrt {x}\,\sqrt {\frac {1}{\sqrt {3}+1}}\,\Pi \left (\sqrt {2}\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right );\mathrm {asin}\left (\sqrt {\frac {x}{\sqrt {3}-1}}\right )\middle |-\frac {\sqrt {3}-1}{\sqrt {3}+1}\right )\,\sqrt {x+\sqrt {3}+1}\,\sqrt {\sqrt {3}-x-1}+2\,\sqrt {x}\,\sqrt {\frac {1}{\sqrt {3}+1}}\,\Pi \left (-\sqrt {2}\,\left (\frac {\sqrt {3}}{2}-\frac {1}{2}\right );\mathrm {asin}\left (\sqrt {\frac {x}{\sqrt {3}-1}}\right )\middle |-\frac {\sqrt {3}-1}{\sqrt {3}+1}\right )\,\sqrt {x+\sqrt {3}+1}\,\sqrt {\sqrt {3}-x-1}-2\,\sqrt {x}\,\sqrt {\frac {1}{\sqrt {3}+1}}\,\mathrm {F}\left (\mathrm {asin}\left (\sqrt {\frac {x}{\sqrt {3}-1}}\right )\middle |-\frac {\sqrt {3}-1}{\sqrt {3}+1}\right )\,\sqrt {x+\sqrt {3}+1}\,\sqrt {\sqrt {3}-x-1}}{\sqrt {x^3+2\,x^2-\left (\sqrt {3}-1\right )\,\left (\sqrt {3}+1\right )\,x}} \] Input:

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

Output:

-(2*x^(1/2)*(1/(3^(1/2) + 1))^(1/2)*ellipticPi(2^(1/2)*(3^(1/2)/2 - 1/2), 
asin((x/(3^(1/2) - 1))^(1/2)), -(3^(1/2) - 1)/(3^(1/2) + 1))*(x + 3^(1/2) 
+ 1)^(1/2)*(3^(1/2) - x - 1)^(1/2) + 2*x^(1/2)*(1/(3^(1/2) + 1))^(1/2)*ell 
ipticPi(-2^(1/2)*(3^(1/2)/2 - 1/2), asin((x/(3^(1/2) - 1))^(1/2)), -(3^(1/ 
2) - 1)/(3^(1/2) + 1))*(x + 3^(1/2) + 1)^(1/2)*(3^(1/2) - x - 1)^(1/2) - 2 
*x^(1/2)*(1/(3^(1/2) + 1))^(1/2)*ellipticF(asin((x/(3^(1/2) - 1))^(1/2)), 
-(3^(1/2) - 1)/(3^(1/2) + 1))*(x + 3^(1/2) + 1)^(1/2)*(3^(1/2) - x - 1)^(1 
/2))/(2*x^2 + x^3 - x*(3^(1/2) - 1)*(3^(1/2) + 1))^(1/2)
 

Reduce [F]

\[ \int \frac {2+x^2}{\left (-2+x^2\right ) \sqrt {-2 x+2 x^2+x^3}} \, dx=\int \frac {\sqrt {x}\, \sqrt {x^{2}+2 x -2}\, x}{x^{4}+2 x^{3}-4 x^{2}-4 x +4}d x +2 \left (\int \frac {\sqrt {x}\, \sqrt {x^{2}+2 x -2}}{x^{5}+2 x^{4}-4 x^{3}-4 x^{2}+4 x}d x \right ) \] Input:

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

Output:

int((sqrt(x)*sqrt(x**2 + 2*x - 2)*x)/(x**4 + 2*x**3 - 4*x**2 - 4*x + 4),x) 
 + 2*int((sqrt(x)*sqrt(x**2 + 2*x - 2))/(x**5 + 2*x**4 - 4*x**3 - 4*x**2 + 
 4*x),x)