Optimal. Leaf size=63 \[ \frac {1}{4} \left (3 x+\sqrt {-1+x^2}\right ) \sqrt {1-x^2+x \sqrt {-1+x^2}}+\frac {3 \sin ^{-1}\left (x-\sqrt {-1+x^2}\right )}{4 \sqrt {2}} \]
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Rubi [F]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx &=\int \sqrt {1-x^2+x \sqrt {-1+x^2}} \, dx\\ \end {align*}
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Mathematica [A]
time = 0.53, size = 110, normalized size = 1.75 \begin {gather*} \frac {1}{8} \left (\frac {2 \left (-1+x^2\right ) \left (3 x+\sqrt {-1+x^2}\right )}{\sqrt {1-x^2+x \sqrt {-1+x^2}} \left (-1+x^2+x \sqrt {-1+x^2}\right )}-3 \sqrt {2} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {-1+x^2}}{\sqrt {1-x^2+x \sqrt {-1+x^2}}}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \sqrt {1-x^{2}+x \sqrt {x^{2}-1}}\, dx\]
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 = 0.89, size = 68, normalized size = 1.08 \begin {gather*} \frac {1}{4} \, \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1} {\left (3 \, x + \sqrt {x^{2} - 1}\right )} + \frac {3}{8} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt {-x^{2} + \sqrt {x^{2} - 1} x + 1}}{2 \, \sqrt {x^{2} - 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 \sqrt {- x^{2} + x \sqrt {x^{2} - 1} + 1}\, 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 = -0.02 \begin {gather*} \int \sqrt {x\,\sqrt {x^2-1}-x^2+1} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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