\(\int \frac {\sqrt {-1+x^2}}{(1+x^2)^{3/2}} \, dx\) [311]

Optimal result

Integrand size = 19, antiderivative size = 42 \[ \int \frac {\sqrt {-1+x^2}}{\left (1+x^2\right )^{3/2}} \, dx=\frac {\sqrt {-1+x^2} E(\arctan (x)|2)}{\sqrt {\frac {1-x^2}{1+x^2}} \sqrt {1+x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.40 \[ \int \frac {\sqrt {-1+x^2}}{\left (1+x^2\right )^{3/2}} \, dx=\frac {-x+x^3-\sqrt {1-x^4} E(\arcsin (x)|-1)+\sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1+x^2} \sqrt {1+x^2}} \] 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]*Ellipti 
cF[ArcSin[x], -1])/(Sqrt[-1 + x^2]*Sqrt[1 + x^2])
 

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(42)=84\).

Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 4.19, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {314, 344, 835, 763, 1499}

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

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 835

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

\(\Big \downarrow \) 763

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

\(\Big \downarrow \) 1499

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

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

Defintions of rubi rules used

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 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 763

Int[1/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[(-a)*b, 2]}, Sim 
p[Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2)/q]/(Sqrt[2]*Sqrt[-a]*Sqrt[a + b*x^4])) 
*EllipticF[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; IntegerQ[q]] /; F 
reeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
 

rule 835

Int[(x_)^2/Sqrt[(a_) + (b_.)*(x_)^4], x_Symbol] :> With[{q = Rt[-b/a, 2]}, 
Simp[1/q   Int[1/Sqrt[a + b*x^4], x], x] - Simp[1/q   Int[(1 - q*x^2)/Sqrt[ 
a + b*x^4], x], x]] /; FreeQ[{a, b}, x] && LtQ[a, 0] && GtQ[b, 0]
 

rule 1499

Int[((d_) + (e_.)*(x_)^2)/Sqrt[(a_) + (c_.)*(x_)^4], x_Symbol] :> With[{q = 
 Rt[(-a)*c, 2]}, Simp[e*x*((q + c*x^2)/(c*Sqrt[a + c*x^4])), x] - Simp[Sqrt 
[2]*e*q*Sqrt[-a + q*x^2]*(Sqrt[(a + q*x^2)/q]/(Sqrt[-a]*c*Sqrt[a + c*x^4])) 
*EllipticE[ArcSin[x/Sqrt[(a + q*x^2)/(2*q)]], 1/2], x] /; EqQ[c*d + e*q, 0] 
 && IntegerQ[q]] /; FreeQ[{a, c, d, e}, x] && LtQ[a, 0] && GtQ[c, 0]
 

Maple [C] (verified)

Result contains complex when optimal does not.

Time = 0.94 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.76

method	result	size
risch	\(\frac {\sqrt {x^{2}-1}\, x}{\sqrt {x^{2}+1}}+\frac {i \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right ) \sqrt {\left (x^{2}-1\right ) \left (x^{2}+1\right )}}{\sqrt {x^{4}-1}\, \sqrt {x^{2}-1}}\)	\(74\)
default	\(\frac {\sqrt {x^{2}-1}\, \sqrt {x^{2}+1}\, \left (i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \operatorname {EllipticF}\left (i x , i\right )-i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \operatorname {EllipticE}\left (i x , i\right )+x^{3}-x \right )}{x^{4}-1}\)	\(82\)
elliptic	\(\frac {\sqrt {x^{4}-1}\, \left (\frac {\left (x^{2}-1\right ) x}{\sqrt {\left (x^{2}-1\right ) \left (x^{2}+1\right )}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\right )}{\sqrt {x^{2}-1}\, \sqrt {x^{2}+1}}\)	\(87\)

Input:

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

Output:

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {-1+x^2}}{\left (1+x^2\right )^{3/2}} \, dx=\frac {\sqrt {x^{2} + 1} \sqrt {x^{2} - 1} x + {\left (i \, x^{2} + i\right )} E(\arcsin \left (x\right )\,|\,-1) + {\left (-i \, x^{2} - i\right )} F(\arcsin \left (x\right )\,|\,-1)}{x^{2} + 1} \] 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 + (I*x^2 + I)*elliptic_e(arcsin(x), -1) + ( 
-I*x^2 - I)*elliptic_f(arcsin(x), -1))/(x^2 + 1)
 

Sympy [F]

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

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

Output:

Integral(sqrt((x - 1)*(x + 1))/(x**2 + 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]

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