Optimal. Leaf size=21 \[ \frac{1}{6} \log (1-\sin (x))-\frac{1}{6} \log (\sin (x)+5) \]
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Rubi [A] time = 0.0276304, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3258, 616, 31} \[ \frac{1}{6} \log (1-\sin (x))-\frac{1}{6} \log (\sin (x)+5) \]
Antiderivative was successfully verified.
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Rule 3258
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{\cos (x)}{-5+4 \sin (x)+\sin ^2(x)} \, dx &=\operatorname{Subst}\left (\int \frac{1}{-5+4 x+x^2} \, dx,x,\sin (x)\right )\\ &=\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\sin (x)\right )-\frac{1}{6} \operatorname{Subst}\left (\int \frac{1}{5+x} \, dx,x,\sin (x)\right )\\ &=\frac{1}{6} \log (1-\sin (x))-\frac{1}{6} \log (5+\sin (x))\\ \end{align*}
Mathematica [A] time = 0.0471494, size = 30, normalized size = 1.43 \[ \frac{1}{6} \left (2 \log \left (\cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\log (\sin (x)+5)\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.062, size = 16, normalized size = 0.8 \begin{align*}{\frac{\ln \left ( -1+\sin \left ( x \right ) \right ) }{6}}-{\frac{\ln \left ( 5+\sin \left ( x \right ) \right ) }{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.956759, size = 20, normalized size = 0.95 \begin{align*} -\frac{1}{6} \, \log \left (\sin \left (x\right ) + 5\right ) + \frac{1}{6} \, \log \left (\sin \left (x\right ) - 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.38426, size = 61, normalized size = 2.9 \begin{align*} -\frac{1}{6} \, \log \left (\sin \left (x\right ) + 5\right ) + \frac{1}{6} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.232809, size = 15, normalized size = 0.71 \begin{align*} \frac{\log{\left (\sin{\left (x \right )} - 1 \right )}}{6} - \frac{\log{\left (\sin{\left (x \right )} + 5 \right )}}{6} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12437, size = 23, normalized size = 1.1 \begin{align*} -\frac{1}{6} \, \log \left (\sin \left (x\right ) + 5\right ) + \frac{1}{6} \, \log \left (-\sin \left (x\right ) + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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