\(\int \frac {\sqrt {1+x^2}}{(-1+x^2)^{3/2}} \, dx\) [307]

Optimal result

Integrand size = 19, antiderivative size = 73 \[ \int \frac {\sqrt {1+x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=-\frac {x \sqrt {1+x^2}}{\sqrt {-1+x^2}}+\frac {\sqrt {1-x^2} E(\arcsin (x)|-1)}{\sqrt {-1+x^2}}-\frac {\sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1+x^2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.04 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {1+x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=-\frac {x \sqrt {1+x^2}-\sqrt {1-x^2} E(\arcsin (x)|-1)+\sqrt {1-x^2} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1+x^2}} \] Input:

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

Output:

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

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(176\) vs. \(2(73)=146\).

Time = 0.28 (sec) , antiderivative size = 176, normalized size of antiderivative = 2.41, 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

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

\(\Big \downarrow \) 344

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

\(\Big \downarrow \) 835

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

\(\Big \downarrow \) 763

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

\(\Big \downarrow \) 1499

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

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] 
*EllipticF[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 [A] (verified)

Time = 2.18 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.03

method	result	size
risch	\(-\frac {x \sqrt {x^{2}+1}}{\sqrt {x^{2}-1}}-\frac {i \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right ) \sqrt {\left (x^{2}-1\right ) \left (x^{2}+1\right )}}{\sqrt {x^{4}-1}\, \sqrt {x^{2}-1}}\)	\(75\)
default	\(\frac {\left (-i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \operatorname {EllipticF}\left (i x , i\right )+i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \operatorname {EllipticE}\left (i x , i\right )-x^{3}-x \right ) \sqrt {x^{2}+1}\, \sqrt {x^{2}-1}}{x^{4}-1}\)	\(84\)
elliptic	\(\frac {\sqrt {x^{4}-1}\, \left (-\frac {x \left (x^{2}+1\right )}{\sqrt {\left (x^{2}-1\right ) \left (x^{2}+1\right )}}-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\right )}{\sqrt {x^{2}-1}\, \sqrt {x^{2}+1}}\)	\(88\)

Input:

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

Output:

-x*(x^2+1)^(1/2)/(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 [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {1+x^2}}{\left (-1+x^2\right )^{3/2}} \, dx=-\frac {\sqrt {x^{2} + 1} \sqrt {x^{2} - 1} x - {\left (-i \, x^{2} + i\right )} E(\arcsin \left (x\right )\,|\,-1) - {\left (i \, x^{2} - i\right )} F(\arcsin \left (x\right )\,|\,-1)}{x^{2} - 1} \] 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]

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

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