Optimal. Leaf size=43 \[ \frac{1}{4} \log \left (3 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )-3 \sin \left (\frac{x}{2}\right )\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0187309, antiderivative size = 43, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.375, Rules used = {2660, 616, 31} \[ \frac{1}{4} \log \left (3 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )-3 \sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2660
Rule 616
Rule 31
Rubi steps
\begin{align*} \int \frac{1}{3-5 \sin (x)} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{3-10 x+3 x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-9+3 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-\frac{3}{4} \operatorname{Subst}\left (\int \frac{1}{-1+3 x} \, dx,x,\tan \left (\frac{x}{2}\right )\right )\\ &=-\frac{1}{4} \log \left (1-3 \tan \left (\frac{x}{2}\right )\right )+\frac{1}{4} \log \left (3-\tan \left (\frac{x}{2}\right )\right )\\ \end{align*}
Mathematica [A] time = 0.0113205, size = 43, normalized size = 1. \[ \frac{1}{4} \log \left (3 \cos \left (\frac{x}{2}\right )-\sin \left (\frac{x}{2}\right )\right )-\frac{1}{4} \log \left (\cos \left (\frac{x}{2}\right )-3 \sin \left (\frac{x}{2}\right )\right ) \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 22, normalized size = 0.5 \begin{align*} -{\frac{1}{4}\ln \left ( 3\,\tan \left ( x/2 \right ) -1 \right ) }+{\frac{1}{4}\ln \left ( \tan \left ({\frac{x}{2}} \right ) -3 \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.927992, size = 41, normalized size = 0.95 \begin{align*} -\frac{1}{4} \, \log \left (\frac{3 \, \sin \left (x\right )}{\cos \left (x\right ) + 1} - 1\right ) + \frac{1}{4} \, \log \left (\frac{\sin \left (x\right )}{\cos \left (x\right ) + 1} - 3\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.30637, size = 95, normalized size = 2.21 \begin{align*} \frac{1}{8} \, \log \left (4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) + 5\right ) - \frac{1}{8} \, \log \left (-4 \, \cos \left (x\right ) - 3 \, \sin \left (x\right ) + 5\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 0.20946, size = 20, normalized size = 0.47 \begin{align*} \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - 3 \right )}}{4} - \frac{\log{\left (\tan{\left (\frac{x}{2} \right )} - \frac{1}{3} \right )}}{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.0834, size = 31, normalized size = 0.72 \begin{align*} -\frac{1}{4} \, \log \left ({\left | 3 \, \tan \left (\frac{1}{2} \, x\right ) - 1 \right |}\right ) + \frac{1}{4} \, \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) - 3 \right |}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]