Optimal. Leaf size=29 \[ -\frac {\sqrt {\frac {1}{x}+1} \sqrt {x} \sin ^{-1}(1-2 x)}{\sqrt {x+1}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {1448, 26, 53, 619, 216} \[ -\frac {\sqrt {\frac {1}{x}+1} \sqrt {x} \sin ^{-1}(1-2 x)}{\sqrt {x+1}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 26
Rule 53
Rule 216
Rule 619
Rule 1448
Rubi steps
\begin {align*} \int \frac {\sqrt {1+\frac {1}{x}}}{\sqrt {1-x^2}} \, dx &=\frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {\sqrt {1+x}}{\sqrt {x} \sqrt {1-x^2}} \, dx}{\sqrt {1+x}}\\ &=\frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {1-x} \sqrt {x}} \, dx}{\sqrt {1+x}}\\ &=\frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \int \frac {1}{\sqrt {x-x^2}} \, dx}{\sqrt {1+x}}\\ &=-\frac {\left (\sqrt {1+\frac {1}{x}} \sqrt {x}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-x^2}} \, dx,x,1-2 x\right )}{\sqrt {1+x}}\\ &=-\frac {\sqrt {1+\frac {1}{x}} \sqrt {x} \sin ^{-1}(1-2 x)}{\sqrt {1+x}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.23, size = 41, normalized size = 1.41 \[ -\tan ^{-1}\left (\frac {\sqrt {\frac {x+1}{x}} (2 x-1) \sqrt {1-x^2}}{2 \left (x^2-1\right )}\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 34, normalized size = 1.17 \[ -\arctan \left (\frac {2 \, \sqrt {-x^{2} + 1} x \sqrt {\frac {x + 1}{x}}}{2 \, x^{2} + x - 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{x} + 1}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.02, size = 40, normalized size = 1.38 \[ \frac {\sqrt {\frac {x +1}{x}}\, \sqrt {-x^{2}+1}\, x \arcsin \left (2 x -1\right )}{\left (x +1\right ) \sqrt {-\left (x -1\right ) x}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {\frac {1}{x} + 1}}{\sqrt {-x^{2} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {\sqrt {\frac {1}{x}+1}}{\sqrt {1-x^2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {1 + \frac {1}{x}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________