Optimal. Leaf size=18 \[ \frac{\log (\cos (x)+1)}{a}-\frac{\log (\cos (x))}{a} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0355734, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {2707, 36, 29, 31} \[ \frac{\log (\cos (x)+1)}{a}-\frac{\log (\cos (x))}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2707
Rule 36
Rule 29
Rule 31
Rubi steps
\begin{align*} \int \frac{\tan (x)}{a+a \cos (x)} \, dx &=-\operatorname{Subst}\left (\int \frac{1}{x (a+x)} \, dx,x,a \cos (x)\right )\\ &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x} \, dx,x,a \cos (x)\right )}{a}+\frac{\operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,a \cos (x)\right )}{a}\\ &=-\frac{\log (\cos (x))}{a}+\frac{\log (1+\cos (x))}{a}\\ \end{align*}
Mathematica [A] time = 0.0181677, size = 12, normalized size = 0.67 \[ \frac{2 \tanh ^{-1}(2 \cos (x)+1)}{a} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.059, size = 19, normalized size = 1.1 \begin{align*} -{\frac{\ln \left ( \cos \left ( x \right ) \right ) }{a}}+{\frac{\ln \left ( \cos \left ( x \right ) +1 \right ) }{a}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.17305, size = 24, normalized size = 1.33 \begin{align*} \frac{\log \left (\cos \left (x\right ) + 1\right )}{a} - \frac{\log \left (\cos \left (x\right )\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41273, size = 58, normalized size = 3.22 \begin{align*} -\frac{\log \left (-\cos \left (x\right )\right ) - \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\tan{\left (x \right )}}{\cos{\left (x \right )} + 1}\, dx}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.28405, size = 26, normalized size = 1.44 \begin{align*} \frac{\log \left (\cos \left (x\right ) + 1\right )}{a} - \frac{\log \left ({\left | \cos \left (x\right ) \right |}\right )}{a} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]