Optimal. Leaf size=67 \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b d \sqrt{a-b} \sqrt{a+b}}+\frac{B x}{b} \]
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Rubi [A] time = 0.0777636, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2735, 2659, 205} \[ \frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b d \sqrt{a-b} \sqrt{a+b}}+\frac{B x}{b} \]
Antiderivative was successfully verified.
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Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{A+B \cos (c+d x)}{a+b \cos (c+d x)} \, dx &=\frac{B x}{b}-\frac{(-A b+a B) \int \frac{1}{a+b \cos (c+d x)} \, dx}{b}\\ &=\frac{B x}{b}+\frac{(2 (A b-a B)) \operatorname{Subst}\left (\int \frac{1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac{1}{2} (c+d x)\right )\right )}{b d}\\ &=\frac{B x}{b}+\frac{2 (A b-a B) \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b \sqrt{a+b} d}\\ \end{align*}
Mathematica [A] time = 0.11225, size = 68, normalized size = 1.01 \[ \frac{\frac{2 (a B-A b) \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}+B (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.098, size = 113, normalized size = 1.7 \begin{align*} 2\,{\frac{\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) B}{bd}}+2\,{\frac{A}{d\sqrt{ \left ( a-b \right ) \left ( a+b \right ) }}\arctan \left ({\frac{\tan \left ( 1/2\,dx+c/2 \right ) \left ( a-b \right ) }{\sqrt{ \left ( a-b \right ) \left ( a+b \right ) }}} \right ) }-2\,{\frac{aB}{bd\sqrt{ \left ( a-b \right ) \left ( a+b \right ) }}\arctan \left ({\frac{\tan \left ( 1/2\,dx+c/2 \right ) \left ( a-b \right ) }{\sqrt{ \left ( a-b \right ) \left ( a+b \right ) }}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A] time = 1.8237, size = 524, normalized size = 7.82 \begin{align*} \left [\frac{2 \,{\left (B a^{2} - B b^{2}\right )} d x +{\left (B a - A b\right )} \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right )}{2 \,{\left (a^{2} b - b^{3}\right )} d}, \frac{{\left (B a^{2} - B b^{2}\right )} d x -{\left (B a - A b\right )} \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (d x + c\right )}\right )}{{\left (a^{2} b - b^{3}\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 127.075, size = 524, normalized size = 7.82 \begin{align*} \begin{cases} \frac{\tilde{\infty } x \left (A + B \cos{\left (c \right )}\right )}{\cos{\left (c \right )}} & \text{for}\: a = 0 \wedge b = 0 \wedge d = 0 \\\frac{A \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{b d} + \frac{B x}{b} - \frac{B \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}}{b d} & \text{for}\: a = b \\\frac{A}{b d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}} + \frac{B x}{b} + \frac{B}{b d \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )}} & \text{for}\: a = - b \\\frac{A x + \frac{B \sin{\left (c + d x \right )}}{d}}{a} & \text{for}\: b = 0 \\\frac{x \left (A + B \cos{\left (c \right )}\right )}{a + b \cos{\left (c \right )}} & \text{for}\: d = 0 \\\frac{A b \log{\left (- \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} + \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} \right )}}{a b d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} - b^{2} d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}} - \frac{A b \log{\left (\sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} + \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} \right )}}{a b d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} - b^{2} d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}} + \frac{B a d x \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}}{a b d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} - b^{2} d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}} - \frac{B a \log{\left (- \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} + \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} \right )}}{a b d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} - b^{2} d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}} + \frac{B a \log{\left (\sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} + \tan{\left (\frac{c}{2} + \frac{d x}{2} \right )} \right )}}{a b d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} - b^{2} d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}} - \frac{B b d x \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}}{a b d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}} - b^{2} d \sqrt{- \frac{a}{a - b} - \frac{b}{a - b}}} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.58013, size = 136, normalized size = 2.03 \begin{align*} \frac{\frac{{\left (d x + c\right )} B}{b} + \frac{2 \,{\left (\pi \left \lfloor \frac{d x + c}{2 \, \pi } + \frac{1}{2} \right \rfloor \mathrm{sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - b \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{\sqrt{a^{2} - b^{2}}}\right )\right )}{\left (B a - A b\right )}}{\sqrt{a^{2} - b^{2}} b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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