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 = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.00, 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 206
Rule 3074
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*}
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Mathematica [B] time = 0.02, 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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fricas [B] time = 0.44, size = 27, normalized size = 1.50 \[ -\frac {1}{10} \, \log \left (\frac {3}{2} \, \cos \relax (x) + 2 \, \sin \relax (x) + \frac {5}{2}\right ) + \frac {1}{10} \, \log \left (-\frac {3}{2} \, \cos \relax (x) - 2 \, \sin \relax (x) + \frac {5}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.96, size = 23, normalized size = 1.28 \[ \frac {1}{5} \, \log \left ({\left | 2 \, \tan \left (\frac {1}{2} \, x\right ) - 1 \right |}\right ) - \frac {1}{5} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 2 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 22, normalized size = 1.22 \[ -\frac {\ln \left (\tan \left (\frac {x}{2}\right )+2\right )}{5}+\frac {\ln \left (2 \tan \left (\frac {x}{2}\right )-1\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.55, size = 30, normalized size = 1.67 \[ \frac {1}{5} \, \log \left (\frac {2 \, \sin \relax (x)}{\cos \relax (x) + 1} - 1\right ) - \frac {1}{5} \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 2\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.49, size = 11, normalized size = 0.61 \[ -\frac {2\,\mathrm {atanh}\left (\frac {4\,\mathrm {tan}\left (\frac {x}{2}\right )}{5}+\frac {3}{5}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.29, size = 20, normalized size = 1.11 \[ \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} - \frac {1}{2} \right )}}{5} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 2 \right )}}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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