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 = 0.040102, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, 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 3124
Rule 618
Rule 204
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*}
Mathematica [A] time = 0.026355, 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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Maple [A] time = 0.03, size = 20, normalized size = 0.5 \begin{align*}{\frac{2\,\sqrt{11}}{11}\arctan \left ({\frac{\sqrt{11}}{22} \left ( 6\,\tan \left ( x/2 \right ) +8 \right ) } \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.42151, size = 31, normalized size = 0.72 \begin{align*} \frac{2}{11} \, \sqrt{11} \arctan \left (\frac{1}{11} \, \sqrt{11}{\left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} + 4\right )}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.89124, size = 146, normalized size = 3.4 \begin{align*} -\frac{1}{11} \, \sqrt{11} \arctan \left (-\frac{18 \, \sqrt{11} \cos \left (x\right ) + 24 \, \sqrt{11} \sin \left (x\right ) + 25 \, \sqrt{11}}{11 \,{\left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right )\right )}}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.516389, size = 42, normalized size = 0.98 \begin{align*} \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} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.09596, size = 66, normalized size = 1.53 \begin{align*} \frac{1}{11} \, \sqrt{11}{\left (x + 2 \, \arctan \left (-\frac{\sqrt{11} \sin \left (x\right ) - 4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) - 4}{\sqrt{11} \cos \left (x\right ) + \sqrt{11} - 3 \, \cos \left (x\right ) + 4 \, \sin \left (x\right ) + 3}\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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