Integrand size = 21, antiderivative size = 74 \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1-x^2}} \, dx=\frac {\sqrt {1-x^4} E(\arcsin (x)|-1)}{\sqrt {-1-x^2} \sqrt {-1+x^2}}-\frac {2 \sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-1-x^2} \sqrt {-1+x^2}} \] Output:
(-x^4+1)^(1/2)*EllipticE(x,I)/(-x^2-1)^(1/2)/(x^2-1)^(1/2)-2*(-x^4+1)^(1/2 )*EllipticF(x,I)/(-x^2-1)^(1/2)/(x^2-1)^(1/2)
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1-x^2}} \, dx=-\frac {i \sqrt {-1+x^2} \sqrt {1+x^2} E(i \text {arcsinh}(x)|-1)}{\sqrt {-1-x^2} \sqrt {1-x^2}} \] Input:
Integrate[Sqrt[-1 + x^2]/Sqrt[-1 - x^2],x]
Output:
((-I)*Sqrt[-1 + x^2]*Sqrt[1 + x^2]*EllipticE[I*ArcSinh[x], -1])/(Sqrt[-1 - x^2]*Sqrt[1 - x^2])
Time = 0.20 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {326, 289, 329, 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.
\(\displaystyle \int \frac {\sqrt {x^2-1}}{\sqrt {-x^2-1}} \, dx\) |
\(\Big \downarrow \) 326 |
\(\displaystyle -2 \int \frac {1}{\sqrt {-x^2-1} \sqrt {x^2-1}}dx-\int \frac {\sqrt {-x^2-1}}{\sqrt {x^2-1}}dx\) |
\(\Big \downarrow \) 289 |
\(\displaystyle -\int \frac {\sqrt {-x^2-1}}{\sqrt {x^2-1}}dx-\frac {2 \sqrt {1-x^4} \int \frac {1}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1} \sqrt {x^2-1}}\) |
\(\Big \downarrow \) 329 |
\(\displaystyle \frac {\sqrt {1-x^4} \int \frac {\sqrt {x^2+1}}{\sqrt {1-x^2}}dx}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {2 \sqrt {1-x^4} \int \frac {1}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1} \sqrt {x^2-1}}\) |
\(\Big \downarrow \) 327 |
\(\displaystyle \frac {\sqrt {1-x^4} E(\arcsin (x)|-1)}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {2 \sqrt {1-x^4} \int \frac {1}{\sqrt {1-x^4}}dx}{\sqrt {-x^2-1} \sqrt {x^2-1}}\) |
\(\Big \downarrow \) 762 |
\(\displaystyle \frac {\sqrt {1-x^4} E(\arcsin (x)|-1)}{\sqrt {-x^2-1} \sqrt {x^2-1}}-\frac {2 \sqrt {1-x^4} \operatorname {EllipticF}(\arcsin (x),-1)}{\sqrt {-x^2-1} \sqrt {x^2-1}}\) |
Input:
Int[Sqrt[-1 + x^2]/Sqrt[-1 - x^2],x]
Output:
(Sqrt[1 - x^4]*EllipticE[ArcSin[x], -1])/(Sqrt[-1 - x^2]*Sqrt[-1 + x^2]) - (2*Sqrt[1 - x^4]*EllipticF[ArcSin[x], -1])/(Sqrt[-1 - x^2]*Sqrt[-1 + x^2] )
Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(p_), x_Symbol] :> Sim p[(a + b*x^2)^FracPart[p]*((c + d*x^2)^FracPart[p]/(a*c + b*d*x^4)^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] && !IntegerQ[p]
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[ a*(Sqrt[1 - b^2*(x^4/a^2)]/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2])) Int[Sqrt[1 + b*(x^2/a)]/Sqrt[1 - b*(x^2/a)], x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b *c + a*d, 0] && !(LtQ[a*c, 0] && GtQ[a*b, 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 = 0.39 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.73
method | result | size |
default | \(\frac {\sqrt {x^{2}-1}\, \sqrt {-x^{2}-1}\, \sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (-\operatorname {EllipticE}\left (x , i\right )+2 \operatorname {EllipticF}\left (x , i\right )\right )}{x^{4}-1}\) | \(54\) |
elliptic | \(\frac {\sqrt {-x^{4}+1}\, \left (-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \operatorname {EllipticF}\left (x , i\right )}{\sqrt {-x^{4}+1}}-\frac {\sqrt {-x^{2}+1}\, \sqrt {x^{2}+1}\, \left (\operatorname {EllipticF}\left (x , i\right )-\operatorname {EllipticE}\left (x , i\right )\right )}{\sqrt {-x^{4}+1}}\right )}{\sqrt {-x^{2}-1}\, \sqrt {x^{2}-1}}\) | \(97\) |
Input:
int((x^2-1)^(1/2)/(-x^2-1)^(1/2),x,method=_RETURNVERBOSE)
Output:
(x^2-1)^(1/2)*(-x^2-1)^(1/2)*(-x^2+1)^(1/2)*(x^2+1)^(1/2)*(-EllipticE(x,I) +2*EllipticF(x,I))/(x^4-1)
Time = 0.09 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.43 \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1-x^2}} \, dx=\frac {-i \, x E(\arcsin \left (\frac {1}{x}\right )\,|\,-1) - \sqrt {x^{2} - 1} \sqrt {-x^{2} - 1}}{x} \] Input:
integrate((x^2-1)^(1/2)/(-x^2-1)^(1/2),x, algorithm="fricas")
Output:
(-I*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. \[ \int \frac {\sqrt {-1+x^2}}{\sqrt {-1-x^2}} \, dx=\int \frac {\sqrt {x^2-1}}{\sqrt {-x^2-1}} \,d x \] Input:
int((x^2 - 1)^(1/2)/(- x^2 - 1)^(1/2),x)
Output:
int((x^2 - 1)^(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