Optimal. Leaf size=42 \[ -\frac {\sqrt {1-\frac {1}{x^4}} x^2 \operatorname {EllipticF}\left (\csc ^{-1}(x),-1\right )}{\sqrt {2-2 x^2} \sqrt {-x^2-1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 65, normalized size of antiderivative = 1.55, 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 = {253, 222} \[ \frac {\sqrt {x^2-1} \sqrt {x^2+1} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2-1}}\right )|\frac {1}{2}\right )}{2 \sqrt {-x^2-1} \sqrt {1-x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 222
Rule 253
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {2-2 x^2} \sqrt {-1-x^2}} \, dx &=\frac {\sqrt {-2+2 x^4} \int \frac {1}{\sqrt {-2+2 x^4}} \, dx}{\sqrt {2-2 x^2} \sqrt {-1-x^2}}\\ &=\frac {\sqrt {-1+x^2} \sqrt {1+x^2} F\left (\sin ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {-1+x^2}}\right )|\frac {1}{2}\right )}{2 \sqrt {-1-x^2} \sqrt {1-x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.02, size = 48, normalized size = 1.14 \[ \frac {x \sqrt {1-x^4} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};x^4\right )}{\sqrt {2-2 x^2} \sqrt {-x^2-1}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-x^{2} - 1} \sqrt {-2 \, x^{2} + 2}}{2 \, {\left (x^{4} - 1\right )}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-2 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 30, normalized size = 0.71 \[ \frac {i \sqrt {2}\, \sqrt {-x^{2}-1}\, \EllipticF \left (i x , i\right )}{2 \sqrt {x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-x^{2} - 1} \sqrt {-2 \, x^{2} + 2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {-x^2-1}\,\sqrt {2-2\,x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 5.45, size = 73, normalized size = 1.74 \[ \frac {\sqrt {2} {G_{6, 6}^{5, 3}\left (\begin {matrix} \frac {1}{2}, 1, 1 & \frac {3}{4}, \frac {3}{4}, \frac {5}{4} \\\frac {1}{4}, \frac {1}{2}, \frac {3}{4}, 1, \frac {5}{4} & 0 \end {matrix} \middle | {\frac {1}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} - \frac {\sqrt {2} {G_{6, 6}^{3, 5}\left (\begin {matrix} - \frac {1}{4}, 0, \frac {1}{4}, \frac {1}{2}, \frac {3}{4} & 1 \\0, \frac {1}{2}, 0 & - \frac {1}{4}, \frac {1}{4}, \frac {1}{4} \end {matrix} \middle | {\frac {e^{- 2 i \pi }}{x^{4}}} \right )}}{16 \pi ^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________