Optimal. Leaf size=56 \[ \frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.0456321, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2751, 2646} \[ \frac{2 \sin (c+d x) \sqrt{a \cos (c+d x)+a}}{3 d}+\frac{2 a \sin (c+d x)}{3 d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2646
Rubi steps
\begin{align*} \int \cos (c+d x) \sqrt{a+a \cos (c+d x)} \, dx &=\frac{2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3 d}+\frac{1}{3} \int \sqrt{a+a \cos (c+d x)} \, dx\\ &=\frac{2 a \sin (c+d x)}{3 d \sqrt{a+a \cos (c+d x)}}+\frac{2 \sqrt{a+a \cos (c+d x)} \sin (c+d x)}{3 d}\\ \end{align*}
Mathematica [A] time = 0.070667, size = 54, normalized size = 0.96 \[ \frac{\left (3 \sin \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{3}{2} (c+d x)\right )\right ) \sec \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{3 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.661, size = 58, normalized size = 1. \begin{align*}{\frac{2\,a\sqrt{2}}{3\,d}\cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \sin \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \left ( 2\, \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}+1 \right ){\frac{1}{\sqrt{ \left ( \cos \left ({\frac{dx}{2}}+{\frac{c}{2}} \right ) \right ) ^{2}a}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.89027, size = 49, normalized size = 0.88 \begin{align*} \frac{{\left (\sqrt{2} \sin \left (\frac{3}{2} \, d x + \frac{3}{2} \, c\right ) + 3 \, \sqrt{2} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )\right )} \sqrt{a}}{3 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.55326, size = 112, normalized size = 2. \begin{align*} \frac{2 \, \sqrt{a \cos \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right )}{3 \,{\left (d \cos \left (d x + c\right ) + d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \left (\cos{\left (c + d x \right )} + 1\right )} \cos{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \cos \left (d x + c\right ) + a} \cos \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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