Optimal. Leaf size=93 \[ \frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d} \]
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Rubi [A] time = 0.339329, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.185, Rules used = {2874, 2976, 2981, 2773, 206} \[ \frac{2 \cos (c+d x) \sqrt{a \sin (c+d x)+a}}{3 d}+\frac{2 a \cos (c+d x)}{3 d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a \sin (c+d x)+a}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 2874
Rule 2976
Rule 2981
Rule 2773
Rule 206
Rubi steps
\begin{align*} \int \cos (c+d x) \cot (c+d x) \sqrt{a+a \sin (c+d x)} \, dx &=\frac{\int \csc (c+d x) (a-a \sin (c+d x)) (a+a \sin (c+d x))^{3/2} \, dx}{a^2}\\ &=\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\frac{2 \int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \left (\frac{3 a^2}{2}-\frac{1}{2} a^2 \sin (c+d x)\right ) \, dx}{3 a^2}\\ &=\frac{2 a \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}+\int \csc (c+d x) \sqrt{a+a \sin (c+d x)} \, dx\\ &=\frac{2 a \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}+\frac{2 a \cos (c+d x)}{3 d \sqrt{a+a \sin (c+d x)}}+\frac{2 \cos (c+d x) \sqrt{a+a \sin (c+d x)}}{3 d}\\ \end{align*}
Mathematica [A] time = 0.213736, size = 143, normalized size = 1.54 \[ \frac{\sqrt{a (\sin (c+d x)+1)} \left (-3 \sin \left (\frac{1}{2} (c+d x)\right )+\sin \left (\frac{3}{2} (c+d x)\right )+3 \cos \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{3}{2} (c+d x)\right )-3 \log \left (-\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )+1\right )+3 \log \left (\sin \left (\frac{1}{2} (c+d x)\right )-\cos \left (\frac{1}{2} (c+d x)\right )+1\right )\right )}{3 d \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.801, size = 103, normalized size = 1.1 \begin{align*} -{\frac{2+2\,\sin \left ( dx+c \right ) }{3\,ad\cos \left ( dx+c \right ) }\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) } \left ( 3\,{a}^{3/2}{\it Artanh} \left ({\frac{\sqrt{a-a\sin \left ( dx+c \right ) }}{\sqrt{a}}} \right ) + \left ( a-a\sin \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}-3\,a\sqrt{a-a\sin \left ( dx+c \right ) } \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} \csc \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.68882, size = 693, normalized size = 7.45 \begin{align*} \frac{3 \, \sqrt{a}{\left (\cos \left (d x + c\right ) + \sin \left (d x + c\right ) + 1\right )} \log \left (\frac{a \cos \left (d x + c\right )^{3} - 7 \, a \cos \left (d x + c\right )^{2} - 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) + 3\right )} \sin \left (d x + c\right ) - 2 \, \cos \left (d x + c\right ) - 3\right )} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{a} - 9 \, a \cos \left (d x + c\right ) +{\left (a \cos \left (d x + c\right )^{2} + 8 \, a \cos \left (d x + c\right ) - a\right )} \sin \left (d x + c\right ) - a}{\cos \left (d x + c\right )^{3} + \cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right )^{2} - 1\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 1}\right ) + 4 \,{\left (\cos \left (d x + c\right )^{2} +{\left (\cos \left (d x + c\right ) - 1\right )} \sin \left (d x + c\right ) + 2 \, \cos \left (d x + c\right ) + 1\right )} \sqrt{a \sin \left (d x + c\right ) + a}}{6 \,{\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 )} \csc{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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