Optimal. Leaf size=37 \[ 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2+3 \cos (x)}}{\sqrt {2}}\right )-2 \sqrt {2+3 \cos (x)} \]
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Rubi [A]
time = 0.03, 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 = {2800, 52, 65,
213} \begin {gather*} 2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {3 \cos (x)+2}}{\sqrt {2}}\right )-2 \sqrt {3 \cos (x)+2} \end {gather*}
Antiderivative was successfully verified.
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Rule 52
Rule 65
Rule 213
Rule 2800
Rubi steps
\begin {align*} \int \sqrt {2+3 \cos (x)} \tan (x) \, dx &=-\text {Subst}\left (\int \frac {\sqrt {2+x}}{x} \, dx,x,3 \cos (x)\right )\\ &=-2 \sqrt {2+3 \cos (x)}-2 \text {Subst}\left (\int \frac {1}{x \sqrt {2+x}} \, dx,x,3 \cos (x)\right )\\ &=-2 \sqrt {2+3 \cos (x)}-4 \text {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 \begin {gather*} 2 \sqrt {2} \tanh ^{-1}\left (\sqrt {1+\frac {3 \cos (x)}{2}}\right )-2 \sqrt {2+3 \cos (x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.05, size = 31, normalized size = 0.84
method | result | size |
derivativedivides | \(2 \arctanh \left (\frac {\sqrt {2+3 \cos \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {2+3 \cos \left (x \right )}\) | \(31\) |
default | \(2 \arctanh \left (\frac {\sqrt {2+3 \cos \left (x \right )}\, \sqrt {2}}{2}\right ) \sqrt {2}-2 \sqrt {2+3 \cos \left (x \right )}\) | \(31\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 4.07, size = 47, normalized size = 1.27 \begin {gather*} -\sqrt {2} \log \left (-\frac {\sqrt {2} - \sqrt {3 \, \cos \left (x\right ) + 2}}{\sqrt {2} + \sqrt {3 \, \cos \left (x\right ) + 2}}\right ) - 2 \, \sqrt {3 \, \cos \left (x\right ) + 2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.59, size = 58, normalized size = 1.57 \begin {gather*} \frac {1}{2} \, \sqrt {2} \log \left (-\frac {9 \, \cos \left (x\right )^{2} + 4 \, {\left (3 \, \sqrt {2} \cos \left (x\right ) + 4 \, \sqrt {2}\right )} \sqrt {3 \, \cos \left (x\right ) + 2} + 48 \, \cos \left (x\right ) + 32}{\cos \left (x\right )^{2}}\right ) - 2 \, \sqrt {3 \, \cos \left (x\right ) + 2} \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 {3 \cos {\left (x \right )} + 2} \tan {\left (x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.38, size = 50, normalized size = 1.35 \begin {gather*} -\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 2 \, \sqrt {3 \, \cos \left (x\right ) + 2} \right |}}{2 \, {\left (\sqrt {2} + \sqrt {3 \, \cos \left (x\right ) + 2}\right )}}\right ) - 2 \, \sqrt {3 \, \cos \left (x\right ) + 2} \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 = -0.03 \begin {gather*} \int \mathrm {tan}\left (x\right )\,\sqrt {3\,\cos \left (x\right )+2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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