Optimal. Leaf size=37 \[ 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3 \cos (x)+2}}{\sqrt {2}}\right )-2 \sqrt {3 \cos (x)+2} \]
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Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {2721, 50, 63, 207} \[ 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3 \cos (x)+2}}{\sqrt {2}}\right )-2 \sqrt {3 \cos (x)+2} \]
Antiderivative was successfully verified.
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Rule 50
Rule 63
Rule 207
Rule 2721
Rubi steps
\begin {align*} \int \sqrt {2+3 \cos (x)} \tan (x) \, dx &=-\operatorname {Subst}\left (\int \frac {\sqrt {2+x}}{x} \, dx,x,3 \cos (x)\right )\\ &=-2 \sqrt {2+3 \cos (x)}-2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,3 \cos (x)\right )\\ &=-2 \sqrt {2+3 \cos (x)}-4 \operatorname {Subst}\left (\int \frac {1}{-2+x^2} \, dx,x,\sqrt {2+3 \cos (x)}\right )\\ &=2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2+3 \cos (x)}}{\sqrt {2}}\right )-2 \sqrt {2+3 \cos (x)}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 33, normalized size = 0.89 \[ 2 \sqrt {2} \tanh ^{-1}\left (\sqrt {\frac {3 \cos (x)}{2}+1}\right )-2 \sqrt {3 \cos (x)+2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.56, size = 58, normalized size = 1.57 \[ \frac {1}{2} \, \sqrt {2} \log \left (-\frac {9 \, \cos \relax (x)^{2} + 4 \, {\left (3 \, \sqrt {2} \cos \relax (x) + 4 \, \sqrt {2}\right )} \sqrt {3 \, \cos \relax (x) + 2} + 48 \, \cos \relax (x) + 32}{\cos \relax (x)^{2}}\right ) - 2 \, \sqrt {3 \, \cos \relax (x) + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.05, size = 50, normalized size = 1.35 \[ -\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {3 \, \cos \relax (x) + 2} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {3 \, \cos \relax (x) + 2}\right )}}\right ) - 2 \, \sqrt {3 \, \cos \relax (x) + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.03, size = 31, normalized size = 0.84 \[ 2 \sqrt {2}\, \arctanh \left (\frac {\sqrt {3 \cos \relax (x )+2}\, \sqrt {2}}{2}\right )-2 \sqrt {3 \cos \relax (x )+2} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.37, size = 47, normalized size = 1.27 \[ -\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {3 \, \cos \relax (x) + 2}}{\sqrt {2} + \sqrt {3 \, \cos \relax (x) + 2}}\right ) - 2 \, \sqrt {3 \, \cos \relax (x) + 2} \]
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 \[ \int \mathrm {tan}\relax (x)\,\sqrt {3\,\cos \relax (x)+2} \,d x \]
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 \[ \int \sqrt {3 \cos {\relax (x )} + 2} \tan {\relax (x )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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