Optimal. Leaf size=63 \[ -\frac{8 a^2 \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.110775, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2674, 2673} \[ -\frac{8 a^2 \cos ^3(c+d x)}{15 d (a \sin (c+d x)+a)^{3/2}}-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2674
Rule 2673
Rubi steps
\begin{align*} \int \cos ^2(c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{5} (4 a) \int \frac{\cos ^2(c+d x)}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{8 a^2 \cos ^3(c+d x)}{15 d (a+a \sin (c+d x))^{3/2}}-\frac{2 a \cos ^3(c+d x)}{5 d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.168565, size = 79, normalized size = 1.25 \[ -\frac{2 (3 \sin (c+d x)+7) \sqrt{a (\sin (c+d x)+1)} \left (\cos \left (\frac{1}{2} (c+d x)\right )-\sin \left (\frac{1}{2} (c+d x)\right )\right )^3}{15 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.102, size = 55, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 2+2\,\sin \left ( dx+c \right ) \right ) a \left ( \sin \left ( dx+c \right ) -1 \right ) ^{2} \left ( 3\,\sin \left ( dx+c \right ) +7 \right ) }{15\,d\cos \left ( dx+c \right ) }{\frac{1}{\sqrt{a+a\sin \left ( dx+c \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.69323, size = 246, normalized size = 3.9 \begin{align*} -\frac{2 \,{\left (3 \, \cos \left (d x + c\right )^{3} - \cos \left (d x + c\right )^{2} -{\left (3 \, \cos \left (d x + c\right )^{2} + 4 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right ) + 4 \, \cos \left (d x + c\right ) + 8\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{15 \,{\left (d \cos \left (d x + c\right ) + d \sin \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 (\sin{\left (c + d x \right )} + 1\right )} \cos ^{2}{\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 \sin \left (d x + c\right ) + a} \cos \left (d x + c\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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