Optimal. Leaf size=21 \[ -\frac {1}{3} \log \left (3 \cot \left (\frac {x}{2}+\frac {\pi }{4}\right )+4\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.01, antiderivative size = 21, 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 = {3123, 31} \[ -\frac {1}{3} \log \left (3 \cot \left (\frac {x}{2}+\frac {\pi }{4}\right )+4\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3123
Rubi steps
\begin {align*} \int \frac {1}{4+3 \cos (x)+4 \sin (x)} \, dx &=-\operatorname {Subst}\left (\int \frac {1}{4+3 x} \, dx,x,\cot \left (\frac {\pi }{4}+\frac {x}{2}\right )\right )\\ &=-\frac {1}{3} \log \left (4+3 \cot \left (\frac {\pi }{4}+\frac {x}{2}\right )\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.02, size = 39, normalized size = 1.86 \[ \frac {1}{3} \log \left (\sin \left (\frac {x}{2}\right )+\cos \left (\frac {x}{2}\right )\right )-\frac {1}{3} \log \left (\sin \left (\frac {x}{2}\right )+7 \cos \left (\frac {x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.70, size = 21, normalized size = 1.00 \[ -\frac {1}{6} \, \log \left (24 \, \cos \relax (x) + 7 \, \sin \relax (x) + 25\right ) + \frac {1}{6} \, \log \left (\sin \relax (x) + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 1.09, size = 21, normalized size = 1.00 \[ -\frac {1}{3} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 7 \right |}\right ) + \frac {1}{3} \, \log \left ({\left | \tan \left (\frac {1}{2} \, x\right ) + 1 \right |}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.06, size = 20, normalized size = 0.95 \[ \frac {\ln \left (\tan \left (\frac {x}{2}\right )+1\right )}{3}-\frac {\ln \left (\tan \left (\frac {x}{2}\right )+7\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.55, size = 29, normalized size = 1.38 \[ -\frac {1}{3} \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 7\right ) + \frac {1}{3} \, \log \left (\frac {\sin \relax (x)}{\cos \relax (x) + 1} + 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.41, size = 11, normalized size = 0.52 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {x}{2}\right )}{3}+\frac {4}{3}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 0.27, size = 19, normalized size = 0.90 \[ \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 1 \right )}}{3} - \frac {\log {\left (\tan {\left (\frac {x}{2} \right )} + 7 \right )}}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________