Optimal. Leaf size=27 \[ \sqrt {\frac {1}{1-x^2}} \sqrt {1-x^2} \sin ^{-1}(x) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 27, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {6688, 6720, 216} \[ \sqrt {\frac {1}{1-x^2}} \sqrt {1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 216
Rule 6688
Rule 6720
Rubi steps
\begin {align*} \int \sqrt {\frac {1+x^2}{1-x^4}} \, dx &=\int \sqrt {\frac {1}{1-x^2}} \, dx\\ &=\left (\sqrt {\frac {1}{1-x^2}} \sqrt {1-x^2}\right ) \int \frac {1}{\sqrt {1-x^2}} \, dx\\ &=\sqrt {\frac {1}{1-x^2}} \sqrt {1-x^2} \sin ^{-1}(x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 27, normalized size = 1.00 \[ \sqrt {\frac {1}{1-x^2}} \sqrt {1-x^2} \sin ^{-1}(x) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.41, size = 26, normalized size = 0.96 \[ 2 \, \arctan \left (\frac {{\left (x^{2} - 1\right )} \sqrt {-\frac {1}{x^{2} - 1}} + 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.34, size = 10, normalized size = 0.37 \[ -\arcsin \relax (x) \mathrm {sgn}\left (x^{2} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.01, size = 30, normalized size = 1.11 \[ \sqrt {-\frac {1}{x^{2}-1}}\, \sqrt {x^{2}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {-\frac {x^{2} + 1}{x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \sqrt {-\frac {x^2+1}{x^4-1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \sqrt {\frac {x^{2} + 1}{1 - x^{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________