\(\int \frac {\sqrt {-1-x^2}}{(-1+x^2)^{3/2}} \, dx\) [304]

Optimal result

Integrand size = 21, antiderivative size = 97 \[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=-\frac {x \sqrt {-1-x^2}}{\sqrt {-1+x^2}}-\frac {\sqrt {1-x^4} E(\arcsin (x)|-1)}{\sqrt {-1-x^2} \sqrt {-1+x^2}}+\frac {\sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1-x^2} \sqrt {-1+x^2}} \] Output:

-x*(-x^2-1)^(1/2)/(x^2-1)^(1/2)-(-x^4+1)^(1/2)*EllipticE(x,I)/(-x^2-1)^(1/ 
2)/(x^2-1)^(1/2)+(-x^4+1)^(1/2)*EllipticF(x,I)/(-x^2-1)^(1/2)/(x^2-1)^(1/2 
)
 

Mathematica [A] (verified)

Time = 1.06 (sec) , antiderivative size = 59, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=\frac {x+x^3-\sqrt {1-x^4} E(\arcsin (x)|-1)+\sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1-x^2} \sqrt {-1+x^2}} \] Input:

Integrate[Sqrt[-1 - x^2]/(-1 + x^2)^(3/2),x]
 

Output:

(x + x^3 - Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1] + Sqrt[1 - x^4]*Elliptic 
F[ArcSin[x], -1])/(Sqrt[-1 - x^2]*Sqrt[-1 + x^2])
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {314, 25, 344, 836, 762, 1388, 327}

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.

\(\displaystyle \int \frac {\sqrt {-x^2-1}}{\left (x^2-1\right )^{3/2}} \, dx\)

\(\Big \downarrow \) 314

\(\displaystyle \int -\frac {x^2}{\sqrt {-x^2-1} \sqrt {x^2-1}}dx-\frac {x \sqrt {-x^2-1}}{\sqrt {x^2-1}}\)

\(\Big \downarrow \) 25

\(\displaystyle -\int \frac {x^2}{\sqrt {-x^2-1} \sqrt {x^2-1}}dx-\frac {\sqrt {-x^2-1} x}{\sqrt {x^2-1}}\)

\(\Big \downarrow \) 344

\(\displaystyle -\frac {\sqrt {1-x^4} \int \frac {x^2}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {\sqrt {-x^2-1} x}{\sqrt {x^2-1}}\)

\(\Big \downarrow \) 836

\(\displaystyle -\frac {\sqrt {1-x^4} \left (\int \frac {x^2+1}{\sqrt {1-x^4}}dx-\int \frac {1}{\sqrt {1-x^4}}dx\right )}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {\sqrt {-x^2-1} x}{\sqrt {x^2-1}}\)

\(\Big \downarrow \) 762

\(\displaystyle -\frac {\sqrt {1-x^4} \left (\int \frac {x^2+1}{\sqrt {1-x^4}}dx-\operatorname {EllipticF}(\arcsin (x),-1)\right )}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {\sqrt {-x^2-1} x}{\sqrt {x^2-1}}\)

\(\Big \downarrow \) 1388

\(\displaystyle -\frac {\sqrt {1-x^4} \left (\int \frac {\sqrt {x^2+1}}{\sqrt {1-x^2}}dx-\operatorname {EllipticF}(\arcsin (x),-1)\right )}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {\sqrt {-x^2-1} x}{\sqrt {x^2-1}}\)

\(\Big \downarrow \) 327

\(\displaystyle -\frac {\sqrt {1-x^4} (E(\arcsin (x)|-1)-\operatorname {EllipticF}(\arcsin (x),-1))}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {\sqrt {-x^2-1} x}{\sqrt {x^2-1}}\)

Input:

Int[Sqrt[-1 - x^2]/(-1 + x^2)^(3/2),x]
 

Output:

-((x*Sqrt[-1 - x^2])/Sqrt[-1 + x^2]) - (Sqrt[1 - x^4]*(EllipticE[ArcSin[x] 
, -1] - EllipticF[ArcSin[x], -1]))/(Sqrt[-1 - x^2]*Sqrt[-1 + x^2])
 

Defintions of rubi rules used

rule 25

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

rule 314

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] + Simp[1/(2*a* 
(p + 1))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1)*Simp[c*(2*p + 3) + d 
*(2*(p + q + 1) + 1)*x^2, x], x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - 
 a*d, 0] && LtQ[p, -1] && LtQ[0, q, 1] && IntBinomialQ[a, b, c, d, 2, p, q, 
 x]
 

rule 327

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ 
(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*EllipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d) 
)], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0]
 

rule 344

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_ 
), x_Symbol] :> Simp[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c 
+ b*d*x^4)^FracPart[p])   Int[(e*x)^m*(a*c + b*d*x^4)^p, x], x] /; FreeQ[{a 
, b, c, d, e, m, p}, x] && EqQ[b*c + a*d, 0] &&  !IntegerQ[p]
 

rule 762

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> Simp[(1/(Sqrt[a]*Rt[-b/a, 4]) 
)*EllipticF[ArcSin[Rt[-b/a, 4]*x], -1], x] /; FreeQ[{a, b}, x] && NegQ[b/a] 
 && GtQ[a, 0]
 

rule 836

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[-q^(-1)   Int[1/Sqrt[a + b*x^4], x], x] + Simp[1/q   Int[(1 + q*x^2)/S 
qrt[a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && NegQ[b/a]
 

rule 1388

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), 
x_Symbol] :> Int[u*(d + e*x^n)^(p + q)*(a/d + (c/e)*x^n)^p, x] /; FreeQ[{a, 
 c, d, e, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[c*d^2 + a*e^2, 0] && (Integer 
Q[p] || (GtQ[a, 0] && GtQ[d, 0]))
 

Maple [A] (verified)

Time = 1.81 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.79

method	result	size
default	\(\frac {\left (-\operatorname {EllipticF}\left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}+\operatorname {EllipticE}\left (x , i\right ) \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}-x^{3}-x \right ) \sqrt {-x^{2}-1}\, \sqrt {x^{2}-1}}{x^{4}-1}\)	\(77\)
elliptic	\(\frac {\sqrt {-x^{4}+1}\, \left (-\frac {\left (-x^{2}-1\right ) x}{\sqrt {\left (-x^{2}-1\right ) \left (x^{2}-1\right )}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , i\right )-\operatorname {EllipticE}\left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\right )}{\sqrt {-x^{2}-1}\, \sqrt {x^{2}-1}}\)	\(90\)
risch	\(\frac {x \left (x^{2}+1\right ) \sqrt {\left (-x^{2}-1\right ) \left (x^{2}-1\right )}}{\sqrt {-\left (x^{2}-1\right ) \left (x^{2}+1\right )}\, \sqrt {-x^{2}-1}\, \sqrt {x^{2}-1}}+\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , i\right )-\operatorname {EllipticE}\left (x , i\right )\right ) \sqrt {\left (-x^{2}-1\right ) \left (x^{2}-1\right )}}{\sqrt {-x^{4}+1}\, \sqrt {-x^{2}-1}\, \sqrt {x^{2}-1}}\)	\(122\)

Input:

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

Output:

(-EllipticF(x,I)*(-x^2+1)^(1/2)*(x^2+1)^(1/2)+EllipticE(x,I)*(-x^2+1)^(1/2 
)*(x^2+1)^(1/2)-x^3-x)*(-x^2-1)^(1/2)*(x^2-1)^(1/2)/(x^4-1)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.51 \[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=-\frac {\sqrt {x^{2} - 1} \sqrt {-x^{2} - 1} x + {\left (x^{2} - 1\right )} E(\arcsin \left (x\right )\,|\,-1) - {\left (x^{2} - 1\right )} F(\arcsin \left (x\right )\,|\,-1)}{x^{2} - 1} \] Input:

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

Output:

-(sqrt(x^2 - 1)*sqrt(-x^2 - 1)*x + (x^2 - 1)*elliptic_e(arcsin(x), -1) - ( 
x^2 - 1)*elliptic_f(arcsin(x), -1))/(x^2 - 1)
 

Sympy [F]

\[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {- x^{2} - 1}}{\left (\left (x - 1\right ) \left (x + 1\right )\right )^{\frac {3}{2}}}\, dx \] Input:

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

Output:

Integral(sqrt(-x**2 - 1)/((x - 1)*(x + 1))**(3/2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-x^{2} - 1}}{{\left (x^{2} - 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integrate(sqrt(-x^2 - 1)/(x^2 - 1)^(3/2), x)
 

Giac [F]

\[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {-x^{2} - 1}}{{\left (x^{2} - 1\right )}^{\frac {3}{2}}} \,d x } \] Input:

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

Output:

integrate(sqrt(-x^2 - 1)/(x^2 - 1)^(3/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {-x^2-1}}{{\left (x^2-1\right )}^{3/2}} \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int \frac {\sqrt {-1-x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=\left (\int \frac {\sqrt {x^{2}+1}\, \sqrt {x^{2}-1}}{x^{4}-2 x^{2}+1}d x \right ) i \] Input:

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

Output:

int((sqrt(x**2 + 1)*sqrt(x**2 - 1))/(x**4 - 2*x**2 + 1),x)*i