Optimal. Leaf size=43 \[ \frac {x}{\sqrt {11}}+\frac {2 \tan ^{-1}\left (\frac {4 \cos (x)-3 \sin (x)}{4 \sin (x)+3 \cos (x)+\sqrt {11}+6}\right )}{\sqrt {11}} \]
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Rubi [A] time = 0.04, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3124, 618, 204} \[ \frac {x}{\sqrt {11}}+\frac {2 \tan ^{-1}\left (\frac {4 \cos (x)-3 \sin (x)}{4 \sin (x)+3 \cos (x)+\sqrt {11}+6}\right )}{\sqrt {11}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 618
Rule 3124
Rubi steps
\begin {align*} \int \frac {1}{6+3 \cos (x)+4 \sin (x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{9+8 x+3 x^2} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {1}{-44-x^2} \, dx,x,8+6 \tan \left (\frac {x}{2}\right )\right )\right )\\ &=\frac {x}{\sqrt {11}}+\frac {2 \tan ^{-1}\left (\frac {4 \cos (x)-3 \sin (x)}{6+\sqrt {11}+3 \cos (x)+4 \sin (x)}\right )}{\sqrt {11}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 24, normalized size = 0.56 \[ \frac {2 \tan ^{-1}\left (\frac {3 \tan \left (\frac {x}{2}\right )+4}{\sqrt {11}}\right )}{\sqrt {11}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 39, normalized size = 0.91 \[ -\frac {1}{11} \, \sqrt {11} \arctan \left (-\frac {18 \, \sqrt {11} \cos \relax (x) + 24 \, \sqrt {11} \sin \relax (x) + 25 \, \sqrt {11}}{11 \, {\left (4 \, \cos \relax (x) - 3 \, \sin \relax (x)\right )}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.94, size = 49, normalized size = 1.14 \[ \frac {1}{11} \, \sqrt {11} {\left (x + 2 \, \arctan \left (-\frac {\sqrt {11} \sin \relax (x) - 4 \, \cos \relax (x) - 3 \, \sin \relax (x) - 4}{\sqrt {11} \cos \relax (x) + \sqrt {11} - 3 \, \cos \relax (x) + 4 \, \sin \relax (x) + 3}\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 20, normalized size = 0.47 \[ \frac {2 \sqrt {11}\, \arctan \left (\frac {\left (6 \tan \left (\frac {x}{2}\right )+8\right ) \sqrt {11}}{22}\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 23, normalized size = 0.53 \[ \frac {2}{11} \, \sqrt {11} \arctan \left (\frac {1}{11} \, \sqrt {11} {\left (\frac {3 \, \sin \relax (x)}{\cos \relax (x) + 1} + 4\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 21, normalized size = 0.49 \[ \frac {2\,\sqrt {11}\,\mathrm {atan}\left (\frac {3\,\sqrt {11}\,\mathrm {tan}\left (\frac {x}{2}\right )}{11}+\frac {4\,\sqrt {11}}{11}\right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.52, size = 42, normalized size = 0.98 \[ \frac {2 \sqrt {11} \left (\operatorname {atan}{\left (\frac {3 \sqrt {11} \tan {\left (\frac {x}{2} \right )}}{11} + \frac {4 \sqrt {11}}{11} \right )} + \pi \left \lfloor {\frac {\frac {x}{2} - \frac {\pi }{2}}{\pi }}\right \rfloor \right )}{11} \]
Verification of antiderivative is not currently implemented for this CAS.
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