Optimal. Leaf size=26 \[ \frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}} \]
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Rubi [A] time = 0.0131807, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.071, Rules used = {2646} \[ \frac{2 a \sin (c+d x)}{d \sqrt{a \cos (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2646
Rubi steps
\begin{align*} \int \sqrt{a+a \cos (c+d x)} \, dx &=\frac{2 a \sin (c+d x)}{d \sqrt{a+a \cos (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.0326455, size = 29, normalized size = 1.12 \[ \frac{2 \tan \left (\frac{1}{2} (c+d x)\right ) \sqrt{a (\cos (c+d x)+1)}}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.496, size = 43, normalized size = 1.7 \begin{align*} 2\,{\frac{a\cos \left ( 1/2\,dx+c/2 \right ) \sin \left ( 1/2\,dx+c/2 \right ) \sqrt{2}}{\sqrt{ \left ( \cos \left ( 1/2\,dx+c/2 \right ) \right ) ^{2}a}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.67626, size = 27, normalized size = 1.04 \begin{align*} \frac{2 \, \sqrt{2} \sqrt{a} \sin \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.50136, size = 84, normalized size = 3.23 \begin{align*} \frac{2 \, \sqrt{a \cos \left (d x + c\right ) + a} \sin \left (d x + c\right )}{d \cos \left (d x + c\right ) + d} \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 \cos{\left (c + d x \right )} + a}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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