Optimal. Leaf size=31 \[ \frac {\sqrt {1+x^2} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{\sqrt {-1-x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {432, 430}
\begin {gather*} \frac {\sqrt {x^2+1} F\left (\left .\text {ArcSin}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{\sqrt {-x^2-1}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 430
Rule 432
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1-x^2} \sqrt {2-x^2}} \, dx &=\frac {\sqrt {1+x^2} \int \frac {1}{\sqrt {2-x^2} \sqrt {1+x^2}} \, dx}{\sqrt {-1-x^2}}\\ &=\frac {\sqrt {1+x^2} F\left (\left .\sin ^{-1}\left (\frac {x}{\sqrt {2}}\right )\right |-2\right )}{\sqrt {-1-x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.21, size = 39, normalized size = 1.26 \begin {gather*} -\frac {i \sqrt {1+x^2} F\left (i \sinh ^{-1}(x)|-\frac {1}{2}\right )}{\sqrt {2} \sqrt {-1-x^2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 34, normalized size = 1.10
method | result | size |
default | \(\frac {i \EllipticF \left (i x , \frac {i \sqrt {2}}{2}\right ) \sqrt {2}\, \sqrt {-x^{2}-1}}{2 \sqrt {x^{2}+1}}\) | \(34\) |
elliptic | \(-\frac {i \sqrt {\left (x^{2}-2\right ) \left (x^{2}+1\right )}\, \sqrt {x^{2}+1}\, \sqrt {-2 x^{2}+4}\, \EllipticF \left (i x , \frac {i \sqrt {2}}{2}\right )}{2 \sqrt {-x^{2}-1}\, \sqrt {-x^{2}+2}\, \sqrt {x^{4}-x^{2}-2}}\) | \(74\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.26, size = 16, normalized size = 0.52 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \sqrt {-2} {\rm ellipticF}\left (\frac {1}{2} \, \sqrt {2} x, -2\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {2 - x^{2}} \sqrt {- x^{2} - 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________