Optimal. Leaf size=18 \[ -\frac{1}{5} \tanh ^{-1}\left (\frac{1}{5} (4 \sin (x)+3 \cos (x))\right ) \]
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Rubi [A] time = 0.0120571, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {3074, 206} \[ -\frac{1}{5} \tanh ^{-1}\left (\frac{1}{5} (4 \sin (x)+3 \cos (x))\right ) \]
Antiderivative was successfully verified.
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Rule 3074
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{-4 \cos (x)+3 \sin (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{25-x^2} \, dx,x,3 \cos (x)+4 \sin (x)\right )\\ &=-\frac{1}{5} \tanh ^{-1}\left (\frac{1}{5} (3 \cos (x)+4 \sin (x))\right )\\ \end{align*}
Mathematica [B] time = 0.0166594, size = 41, normalized size = 2.28 \[ \frac{1}{5} \log \left (\cos \left (\frac{x}{2}\right )-2 \sin \left (\frac{x}{2}\right )\right )-\frac{1}{5} \log \left (\sin \left (\frac{x}{2}\right )+2 \cos \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.031, size = 22, normalized size = 1.2 \begin{align*} -{\frac{1}{5}\ln \left ( \tan \left ({\frac{x}{2}} \right ) +2 \right ) }+{\frac{1}{5}\ln \left ( 2\,\tan \left ( x/2 \right ) -1 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.936192, size = 41, normalized size = 2.28 \begin{align*} \frac{1}{5} \, \log \left (\frac{2 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) - \frac{1}{5} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} + 2\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92764, size = 109, normalized size = 6.06 \begin{align*} -\frac{1}{10} \, \log \left (\frac{3}{2} \, \cos \left (x\right ) + 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) + \frac{1}{10} \, \log \left (-\frac{3}{2} \, \cos \left (x\right ) - 2 \, \sin \left (x\right ) + \frac{5}{2}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.250552, size = 20, normalized size = 1.11 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{2} \right )}}{5} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} + 2 \right )}}{5} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.07951, size = 31, normalized size = 1.72 \begin{align*} \frac{1}{5} \, \log \left ({\left | 2 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) - \frac{1}{5} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) + 2 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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