Optimal. Leaf size=29 \[ x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )+\tan ^{-1}\left (\sqrt {1-2 x^2}\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {4840, 444, 63, 203} \[ x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )+\tan ^{-1}\left (\sqrt {1-2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 63
Rule 203
Rule 444
Rule 4840
Rubi steps
\begin {align*} \int \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right ) \, dx &=x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )-\int \frac {x}{\sqrt {1-2 x^2} \left (1-x^2\right )} \, dx\\ &=x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x} (1-x)} \, dx,x,x^2\right )\\ &=x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2}+\frac {x^2}{2}} \, dx,x,\sqrt {1-2 x^2}\right )\\ &=x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )+\tan ^{-1}\left (\sqrt {1-2 x^2}\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.01, size = 29, normalized size = 1.00 \[ x \sin ^{-1}\left (\frac {x}{\sqrt {1-x^2}}\right )+\tan ^{-1}\left (\sqrt {1-2 x^2}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 60, normalized size = 2.07 \[ -x \arcsin \left (\frac {\sqrt {-x^{2} + 1} x}{x^{2} - 1}\right ) + \arctan \left (\frac {x^{2} + \sqrt {-x^{2} + 1} \sqrt {\frac {2 \, x^{2} - 1}{x^{2} - 1}} - 1}{x^{2}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.33, size = 34, normalized size = 1.17 \[ x \arcsin \left (\frac {x}{\sqrt {-x^{2} + 1}}\right ) + \frac {\arctan \left (\sqrt {-2 \, x^{2} + 1}\right )}{\mathrm {sgn}\left (x^{2} - 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.12, size = 138, normalized size = 4.76 \[ x \arcsin \left (\frac {x}{\sqrt {-x^{2}+1}}\right )+\frac {\sqrt {\frac {2 x^{2}-1}{x^{2}-1}}\, \sqrt {-x^{2}+1}\, \left (-\arctan \left (\frac {2 x +1}{\sqrt {-2 x^{2}+1}}\right )+\arctan \left (\frac {2 x -1}{\sqrt {-2 x^{2}+1}}\right )+\sqrt {-2 x^{2}+1}\right )}{\sqrt {-2 x^{2}+1}\, \left (2+\sqrt {2}\right ) \left (-2+\sqrt {2}\right )}+\frac {\sqrt {\frac {2 x^{2}-1}{x^{2}-1}}\, \sqrt {-x^{2}+1}}{2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \arcsin \left (\frac {x}{\sqrt {-x^{2} + 1}}\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \mathrm {asin}\left (\frac {x}{\sqrt {1-x^2}}\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {asin}{\left (\frac {x}{\sqrt {1 - x^{2}}} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________