Optimal. Leaf size=45 \[ -\sqrt {2} \log \left (x+\sqrt {-1+x^2}-\sqrt {2} \sqrt {x^2+x \sqrt {-1+x^2}}\right ) \]
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Rubi [A]
time = 0.12, antiderivative size = 18, normalized size of antiderivative = 0.40, number of steps
used = 2, number of rules used = 2, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2155, 221}
\begin {gather*} \sqrt {2} \sinh ^{-1}\left (\sqrt {x^2-1}+x\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 2155
Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+x \sqrt {-1+x^2}}}{x \sqrt {-1+x^2}} \, dx &=\sqrt {2} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x+\sqrt {-1+x^2}\right )\\ &=\sqrt {2} \sinh ^{-1}\left (x+\sqrt {-1+x^2}\right )\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 70, normalized size = 1.56 \begin {gather*} \frac {\sqrt {2} \sqrt {x \left (x+\sqrt {-1+x^2}\right )} \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {x}}{\sqrt {x+\sqrt {-1+x^2}}}\right )}{\sqrt {x} \sqrt {x+\sqrt {-1+x^2}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+x \sqrt {x^{2}-1}}}{x \sqrt {x^{2}-1}}\, dx\]
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 = 0.56, size = 57, normalized size = 1.27 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-4 \, x^{2} - 2 \, \sqrt {x^{2} + \sqrt {x^{2} - 1} x} {\left (\sqrt {2} x + \sqrt {2} \sqrt {x^{2} - 1}\right )} - 4 \, \sqrt {x^{2} - 1} 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 {x \left (x + \sqrt {x^{2} - 1}\right )}}{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 = -0.02 \begin {gather*} \int \frac {\sqrt {x\,\sqrt {x^2-1}+x^2}}{x\,\sqrt {x^2-1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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