Optimal. Leaf size=46 \[ \frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\sin ^{-1}(x),-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {x^2-1}} \]
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Rubi [A] time = 0.04, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {421, 419} \[ \frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {x^2-1}} \]
Antiderivative was successfully verified.
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Rule 419
Rule 421
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {-1+x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx &=\frac {\sqrt {1-x^2} \int \frac {1}{\sqrt {1-x^2} \sqrt {7-4 \sqrt {3}+x^2}} \, dx}{\sqrt {-1+x^2}}\\ &=\frac {\sqrt {1-x^2} F\left (\sin ^{-1}(x)|-7-4 \sqrt {3}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {-1+x^2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 48, normalized size = 1.04 \[ \frac {\sqrt {1-x^2} \operatorname {EllipticF}\left (\sin ^{-1}(x),\frac {1}{4 \sqrt {3}-7}\right )}{\sqrt {7-4 \sqrt {3}} \sqrt {x^2-1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.62, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (x^{2} + 4 \, \sqrt {3} + 7\right )} \sqrt {x^{2} - 4 \, \sqrt {3} + 7} \sqrt {x^{2} - 1}}{x^{6} + 13 \, x^{4} - 13 \, x^{2} - 1}, x\right ) \]
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 \[ \int \frac {1}{\sqrt {x^{2} - 4 \, \sqrt {3} + 7} \sqrt {x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.14, size = 117, normalized size = 2.54 \[ -\frac {i \sqrt {-x^{2}+1}\, \sqrt {-\left (-x^{2}+4 \sqrt {3}-7\right ) \left (-4 \sqrt {3}+7\right )}\, \left (-2+\sqrt {3}\right ) \sqrt {x^{2}-1}\, \sqrt {x^{2}+7-4 \sqrt {3}}\, \EllipticF \left (\frac {i x}{-2+\sqrt {3}}, 2 i-i \sqrt {3}\right )}{\left (4 \sqrt {3}-7\right ) \left (-x^{4}+4 \sqrt {3}\, x^{2}-6 x^{2}-4 \sqrt {3}+7\right )} \]
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 \[ \int \frac {1}{\sqrt {x^{2} - 4 \, \sqrt {3} + 7} \sqrt {x^{2} - 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\sqrt {x^2-1}\,\sqrt {x^2-4\,\sqrt {3}+7}} \,d x \]
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 \[ \int \frac {1}{\sqrt {\left (x - 1\right ) \left (x + 1\right )} \sqrt {x^{2} - 4 \sqrt {3} + 7}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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