Optimal. Leaf size=29 \[ -\frac {\sqrt {1+\frac {1}{x}} \sqrt {x} \sin ^{-1}(1-2 x)}{\sqrt {1+x}} \]
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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 = {1462, 26, 55,
633, 222} \begin {gather*} -\frac {\sqrt {\frac {1}{x}+1} \sqrt {x} \text {ArcSin}(1-2 x)}{\sqrt {x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 26
Rule 55
Rule 222
Rule 633
Rule 1462
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 ) \text {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*}
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Mathematica [A]
time = 1.57, size = 41, normalized size = 1.41 \begin {gather*} -\tan ^{-1}\left (\frac {\sqrt {\frac {1+x}{x}} (-1+2 x) \sqrt {1-x^2}}{2 \left (-1+x^2\right )}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.48, size = 40, normalized size = 1.38
method | result | size |
default | \(\frac {\sqrt {\frac {1+x}{x}}\, x \sqrt {-x^{2}+1}\, \arcsin \left (2 x -1\right )}{\left (1+x \right ) \sqrt {-x \left (-1+x \right )}}\) | \(40\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 0.41, size = 34, normalized size = 1.17 \begin {gather*} -\arctan \left (\frac {2 \, \sqrt {-x^{2} + 1} x \sqrt {\frac {x + 1}{x}}}{2 \, x^{2} + x - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 + \frac {1}{x}}}{\sqrt {- \left (x - 1\right ) \left (x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {\frac {1}{x}+1}}{\sqrt {1-x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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