Optimal. Leaf size=24 \[ -\frac {\sqrt {x^4-1} \sin ^{-1}(x)}{\sqrt {1-x^4}} \]
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Rubi [A] time = 0.01, antiderivative size = 40, normalized size of antiderivative = 1.67, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {1152, 217, 206} \[ \frac {\sqrt {x^2-1} \sqrt {x^2+1} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2-1}}\right )}{\sqrt {x^4-1}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 217
Rule 1152
Rubi steps
\begin {align*} \int \frac {\sqrt {1+x^2}}{\sqrt {-1+x^4}} \, dx &=\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \int \frac {1}{\sqrt {-1+x^2}} \, dx}{\sqrt {-1+x^4}}\\ &=\frac {\left (\sqrt {-1+x^2} \sqrt {1+x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {-1+x^2}}\right )}{\sqrt {-1+x^4}}\\ &=\frac {\sqrt {-1+x^2} \sqrt {1+x^2} \tanh ^{-1}\left (\frac {x}{\sqrt {-1+x^2}}\right )}{\sqrt {-1+x^4}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 34, normalized size = 1.42 \[ \log \left (x^3+\sqrt {x^2+1} \sqrt {x^4-1}+x\right )-\log \left (x^2+1\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.88, size = 65, normalized size = 2.71 \[ \frac {1}{2} \, \log \left (\frac {x^{3} + \sqrt {x^{4} - 1} \sqrt {x^{2} + 1} + x}{x^{3} + x}\right ) - \frac {1}{2} \, \log \left (-\frac {x^{3} - \sqrt {x^{4} - 1} \sqrt {x^{2} + 1} + x}{x^{3} + x}\right ) \]
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 \[ \int \frac {\sqrt {x^{2} + 1}}{\sqrt {x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 1.38 \[ \frac {\sqrt {x^{4}-1}\, \ln \left (x +\sqrt {x^{2}-1}\right )}{\sqrt {x^{2}+1}\, \sqrt {x^{2}-1}} \]
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 \[ \int \frac {\sqrt {x^{2} + 1}}{\sqrt {x^{4} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\sqrt {x^2+1}}{\sqrt {x^4-1}} \,d x \]
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 \[ \int \frac {\sqrt {x^{2} + 1}}{\sqrt {\left (x - 1\right ) \left (x + 1\right ) \left (x^{2} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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