Integrand size = 23, antiderivative size = 53 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=-\frac {\sqrt {1+x^2} E(\arcsin (x)|-1)}{\sqrt {-1-x^2}}+\frac {2 \sqrt {1+x^2} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1-x^2}} \] Output:
-(x^2+1)^(1/2)*EllipticE(x,I)/(-x^2-1)^(1/2)+2*(x^2+1)^(1/2)*EllipticF(x,I )/(-x^2-1)^(1/2)
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=-\frac {i \sqrt {1+x^2} E(i \text {arcsinh}(x)|-1)}{\sqrt {1-x^2} \sqrt {\frac {1+x^2}{-1+x^2}}} \] Input:
Integrate[Sqrt[1 - x^2]/Sqrt[-1 - x^2],x]
Output:
((-I)*Sqrt[1 + x^2]*EllipticE[I*ArcSinh[x], -1])/(Sqrt[1 - x^2]*Sqrt[(1 + x^2)/(-1 + x^2)])
Time = 0.19 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {326, 288, 330, 327, 762}
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.
\(\displaystyle \int \frac {\sqrt {1-x^2}}{\sqrt {-x^2-1}} \, dx\) |
\(\Big \downarrow \) 326 |
\(\displaystyle 2 \int \frac {1}{\sqrt {-x^2-1} \sqrt {1-x^2}}dx+\int \frac {\sqrt {-x^2-1}}{\sqrt {1-x^2}}dx\) |
\(\Big \downarrow \) 288 |
\(\displaystyle \int \frac {\sqrt {-x^2-1}}{\sqrt {1-x^2}}dx+\frac {2 \sqrt {x^2+1} \int \frac {1}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1}}\) |
\(\Big \downarrow \) 330 |
\(\displaystyle \frac {\sqrt {-x^2-1} \int \frac {\sqrt {x^2+1}}{\sqrt {1-x^2}}dx}{\sqrt {x^2+1}}+\frac {2 \sqrt {x^2+1} \int \frac {1}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {2 \sqrt {x^2+1} \int \frac {1}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1}}+\frac {\sqrt {-x^2-1} E(\arcsin (x)|-1)}{\sqrt {x^2+1}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {2 \sqrt {x^2+1} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-x^2-1}}+\frac {\sqrt {-x^2-1} E(\arcsin (x)|-1)}{\sqrt {x^2+1}}\) |
Input:
Int[Sqrt[1 - x^2]/Sqrt[-1 - x^2],x]
Output:
(Sqrt[-1 - x^2]*EllipticE[ArcSin[x], -1])/Sqrt[1 + x^2] + (2*Sqrt[1 + x^2] *EllipticF[ArcSin[x], -1])/Sqrt[-1 - x^2]
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(c + d*x^2)^FracPart[p]/((-1)^IntPart[p]*(-c - d*x^2)^FracPart[p]) Int[ ((-a)*c - b*d*x^4)^p, x], x] /; FreeQ[{a, b, c, d, p}, x] && EqQ[b*c + a*d, 0] && GtQ[a, 0] && LtQ[c, 0]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ b/d Int[Sqrt[c + d*x^2]/Sqrt[a + b*x^2], x], x] - Simp[(b*c - a*d)/d In t[1/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && NegQ[b/a]
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]
Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[ Sqrt[a + b*x^2]/Sqrt[1 + (b/a)*x^2] Int[Sqrt[1 + (b/a)*x^2]/Sqrt[c + d*x^ 2], x], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && !GtQ[a, 0]
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]
Time = 0.43 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.51
method | result | size |
default | \(\frac {i \operatorname {EllipticE}\left (i x , i\right ) \sqrt {-x^{2}-1}}{\sqrt {x^{2}+1}}\) | \(27\) |
elliptic | \(\frac {\sqrt {x^{4}-1}\, \left (-\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \operatorname {EllipticF}\left (i x , i\right )}{\sqrt {x^{4}-1}}+\frac {i \sqrt {x^{2}+1}\, \sqrt {-x^{2}+1}\, \left (\operatorname {EllipticF}\left (i x , i\right )-\operatorname {EllipticE}\left (i x , i\right )\right )}{\sqrt {x^{4}-1}}\right )}{\sqrt {-x^{2}-1}\, \sqrt {-x^{2}+1}}\) | \(104\) |
Input:
int((-x^2+1)^(1/2)/(-x^2-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
I*EllipticE(I*x,I)/(x^2+1)^(1/2)*(-x^2-1)^(1/2)
Time = 0.08 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.62 \[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=-\frac {x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) + \sqrt {-x^{2} + 1} \sqrt {-x^{2} - 1}}{x} \] Input:
integrate((-x^2+1)^(1/2)/(-x^2-1)^(1/2),x, algorithm="fricas")
Output:
-(x*elliptic_e(arcsin(1/x), -1) + sqrt(-x^2 + 1)*sqrt(-x^2 - 1))/x
\[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=\int \frac {\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}{\sqrt {- x^{2} - 1}}\, dx \] Input:
integrate((-x**2+1)**(1/2)/(-x**2-1)**(1/2),x)
Output:
Integral(sqrt(-(x - 1)*(x + 1))/sqrt(-x**2 - 1), x)
\[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=\int { \frac {\sqrt {-x^{2} + 1}}{\sqrt {-x^{2} - 1}} \,d x } \] Input:
integrate((-x^2+1)^(1/2)/(-x^2-1)^(1/2),x, algorithm="maxima")
Output:
integrate(sqrt(-x^2 + 1)/sqrt(-x^2 - 1), x)
\[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=\int { \frac {\sqrt {-x^{2} + 1}}{\sqrt {-x^{2} - 1}} \,d x } \] Input:
integrate((-x^2+1)^(1/2)/(-x^2-1)^(1/2),x, algorithm="giac")
Output:
integrate(sqrt(-x^2 + 1)/sqrt(-x^2 - 1), x)
Timed out. \[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=\int \frac {\sqrt {1-x^2}}{\sqrt {-x^2-1}} \,d x \] Input:
int((1 - x^2)^(1/2)/(- x^2 - 1)^(1/2),x)
Output:
int((1 - x^2)^(1/2)/(- x^2 - 1)^(1/2), x)
\[ \int \frac {\sqrt {1-x^2}}{\sqrt {-1-x^2}} \, dx=-\left (\int \frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}}{x^{2}+1}d x \right ) i \] Input:
int((-x^2+1)^(1/2)/(-x^2-1)^(1/2),x)
Output:
- int((sqrt( - x**2 + 1)*sqrt(x**2 + 1))/(x**2 + 1),x)*i