Optimal. Leaf size=76 \[ \frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a x}{b^2}+\frac{\sin (c+d x)}{b d} \]
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Rubi [A] time = 0.118807, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2746, 12, 2735, 2659, 205} \[ \frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{b^2 d \sqrt{a-b} \sqrt{a+b}}-\frac{a x}{b^2}+\frac{\sin (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 2746
Rule 12
Rule 2735
Rule 2659
Rule 205
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx &=\frac{\sin (c+d x)}{b d}-\frac{\int \frac{a \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b}\\ &=\frac{\sin (c+d x)}{b d}-\frac{a \int \frac{\cos (c+d x)}{a+b \cos (c+d x)} \, dx}{b}\\ &=-\frac{a x}{b^2}+\frac{\sin (c+d x)}{b d}+\frac{a^2 \int \frac{1}{a+b \cos (c+d x)} \, dx}{b^2}\\ &=-\frac{a x}{b^2}+\frac{\sin (c+d x)}{b d}+\frac{\left (2 a^2\right ) \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^2 d}\\ &=-\frac{a x}{b^2}+\frac{2 a^2 \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{\sqrt{a-b} b^2 \sqrt{a+b} d}+\frac{\sin (c+d x)}{b d}\\ \end{align*}
Mathematica [A] time = 0.140654, size = 72, normalized size = 0.95 \[ \frac{-\frac{2 a^2 \tanh ^{-1}\left (\frac{(a-b) \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{b^2-a^2}}\right )}{\sqrt{b^2-a^2}}-a (c+d x)+b \sin (c+d x)}{b^2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.083, size = 102, normalized size = 1.3 \begin{align*} 2\,{\frac{\tan \left ( 1/2\,dx+c/2 \right ) }{bd \left ( 1+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) ^{2} \right ) }}-2\,{\frac{a\arctan \left ( \tan \left ( 1/2\,dx+c/2 \right ) \right ) }{{b}^{2}d}}+2\,{\frac{{a}^{2}}{{b}^{2}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 ) } \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 = 2.0802, size = 585, normalized size = 7.7 \begin{align*} \left [-\frac{\sqrt{-a^{2} + b^{2}} a^{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^{3} - a b^{2}\right )} d x - 2 \,{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{2 \,{\left (a^{2} b^{2} - b^{4}\right )} d}, \frac{\sqrt{a^{2} - b^{2}} a^{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^{3} - a b^{2}\right )} d x +{\left (a^{2} b - b^{3}\right )} \sin \left (d x + c\right )}{{\left (a^{2} b^{2} - b^{4}\right )} d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36573, size = 170, normalized size = 2.24 \begin{align*} -\frac{\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 )} a^{2}}{\sqrt{a^{2} - b^{2}} b^{2}} + \frac{{\left (d x + c\right )} a}{b^{2}} - \frac{2 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 1\right )} b}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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