Optimal. Leaf size=100 \[ -\frac{a B \log (a+b \cos (x))}{a^2-b^2}+\frac{2 A \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}+\frac{B \log (\cos (x)+1)}{2 (a-b)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.156366, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4401, 2659, 205, 2721, 801} \[ -\frac{a B \log (a+b \cos (x))}{a^2-b^2}+\frac{2 A \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}+\frac{B \log (\cos (x)+1)}{2 (a-b)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4401
Rule 2659
Rule 205
Rule 2721
Rule 801
Rubi steps
\begin{align*} \int \frac{A+B \cot (x)}{a+b \cos (x)} \, dx &=\int \left (\frac{A}{a+b \cos (x)}+\frac{B \cot (x)}{a+b \cos (x)}\right ) \, dx\\ &=A \int \frac{1}{a+b \cos (x)} \, dx+B \int \frac{\cot (x)}{a+b \cos (x)} \, dx\\ &=(2 A) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{x}{2}\right )\right )-B \operatorname{Subst}\left (\int \frac{x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cos (x)\right )\\ &=\frac{2 A \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}-B \operatorname{Subst}\left (\int \left (\frac{1}{2 (a+b) (b-x)}+\frac{a}{(a-b) (a+b) (a+x)}-\frac{1}{2 (a-b) (b+x)}\right ) \, dx,x,b \cos (x)\right )\\ &=\frac{2 A \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} \sqrt{a+b}}+\frac{B \log (1-\cos (x))}{2 (a+b)}+\frac{B \log (1+\cos (x))}{2 (a-b)}-\frac{a B \log (a+b \cos (x))}{a^2-b^2}\\ \end{align*}
Mathematica [A] time = 0.251473, size = 134, normalized size = 1.34 \[ \frac{\sin (x) (A+B \cot (x)) \left (B \sqrt{b^2-a^2} \left ((a-b) \log \left (\sin \left (\frac{x}{2}\right )\right )+(a+b) \log \left (\cos \left (\frac{x}{2}\right )\right )-a \log (a+b \cos (x))\right )-2 A \left (a^2-b^2\right ) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )\right )}{(a-b) (a+b) \sqrt{b^2-a^2} (A \sin (x)+B \cos (x))} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.029, size = 135, normalized size = 1.4 \begin{align*}{\frac{B}{a+b}\ln \left ( \tan \left ({\frac{x}{2}} \right ) \right ) }-{\frac{Ba}{ \left ( a+b \right ) \left ( a-b \right ) }\ln \left ( \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}a- \left ( \tan \left ({\frac{x}{2}} \right ) \right ) ^{2}b+a+b \right ) }+2\,{\frac{Aa}{ \left ( a+b \right ) \sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }+2\,{\frac{Ab}{ \left ( a+b \right ) \sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}\arctan \left ({\frac{ \left ( a-b \right ) \tan \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 39.9767, size = 683, normalized size = 6.83 \begin{align*} \left [-\frac{B a \log \left (b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}\right ) + \sqrt{-a^{2} + b^{2}} A \log \left (\frac{2 \, a b \cos \left (x\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (x\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (x\right ) + b\right )} \sin \left (x\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}}\right ) -{\left (B a + B b\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (B a - B b\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{2 \,{\left (a^{2} - b^{2}\right )}}, -\frac{B a \log \left (b^{2} \cos \left (x\right )^{2} + 2 \, a b \cos \left (x\right ) + a^{2}\right ) - 2 \, \sqrt{a^{2} - b^{2}} A \arctan \left (-\frac{a \cos \left (x\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (x\right )}\right ) -{\left (B a + B b\right )} \log \left (\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right ) -{\left (B a - B b\right )} \log \left (-\frac{1}{2} \, \cos \left (x\right ) + \frac{1}{2}\right )}{2 \,{\left (a^{2} - b^{2}\right )}}\right ] \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*} \int \frac{A + B \cot{\left (x \right )}}{a + b \cos{\left (x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.16589, size = 157, normalized size = 1.57 \begin{align*} -\frac{B a \log \left (-a \tan \left (\frac{1}{2} \, x\right )^{2} + b \tan \left (\frac{1}{2} \, x\right )^{2} - a - b\right )}{a^{2} - b^{2}} - \frac{2 \,{\left (\pi \left \lfloor \frac{x}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, x\right ) - b \tan \left (\frac{1}{2} \, x\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )} A}{\sqrt{a^{2} - b^{2}}} + \frac{B \log \left ({\left | \tan \left (\frac{1}{2} \, x\right ) \right |}\right )}{a + b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]