Optimal. Leaf size=15 \[ \frac {1}{4} \log \left (4 \tan \left (\frac {x}{2}\right )+3\right ) \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3124, 31} \[ \frac {1}{4} \log \left (4 \tan \left (\frac {x}{2}\right )+3\right ) \]
Antiderivative was successfully verified.
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Rule 31
Rule 3124
Rubi steps
\begin {align*} \int \frac {1}{3+3 \cos (x)+4 \sin (x)} \, dx &=2 \operatorname {Subst}\left (\int \frac {1}{6+8 x} \, dx,x,\tan \left (\frac {x}{2}\right )\right )\\ &=\frac {1}{4} \log \left (3+4 \tan \left (\frac {x}{2}\right )\right )\\ \end {align*}
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Mathematica [B] time = 0.02, size = 34, normalized size = 2.27 \[ \frac {1}{4} \log \left (4 \sin \left (\frac {x}{2}\right )+3 \cos \left (\frac {x}{2}\right )\right )-\frac {1}{4} \log \left (\cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
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fricas [B] time = 0.76, size = 23, normalized size = 1.53 \[ -\frac {1}{8} \, \log \left (\frac {1}{2} \, \cos \relax (x) + \frac {1}{2}\right ) + \frac {1}{8} \, \log \left (-\frac {7}{2} \, \cos \relax (x) + 12 \, \sin \relax (x) + \frac {25}{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.13, size = 12, normalized size = 0.80 \[ \frac {1}{4} \, \log \left ({\left | 4 \, \tan \left (\frac {1}{2} \, x\right ) + 3 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 12, normalized size = 0.80 \[ \frac {\ln \left (4 \tan \left (\frac {x}{2}\right )+3\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 15, normalized size = 1.00 \[ \frac {1}{4} \, \log \left (\frac {4 \, \sin \relax (x)}{\cos \relax (x) + 1} + 3\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 11, normalized size = 0.73 \[ \frac {\ln \left (4\,\mathrm {tan}\left (\frac {x}{2}\right )+3\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.26, size = 10, normalized size = 0.67 \[ \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + \frac {3}{4} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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