Optimal. Leaf size=47 \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.0248641, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {2649, 206} \[ -\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (c+d x)}{\sqrt{a+a \sin (c+d x)}}\right )}{d}\\ &=-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{\sqrt{a} d}\\ \end{align*}
Mathematica [C] time = 0.0686954, size = 73, normalized size = 1.55 \[ \frac{(2+2 i) (-1)^{3/4} \left (\sin \left (\frac{1}{2} (c+d x)\right )+\cos \left (\frac{1}{2} (c+d x)\right )\right ) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (c+d x)\right )-1\right )\right )}{d \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.072, size = 75, normalized size = 1.6 \begin{align*} -{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) \sqrt{2}}{d\cos \left ( dx+c \right ) }\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }{\frac{1}{\sqrt{a}}}} \right ){\frac{1}{\sqrt{a}}}{\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 \frac{1}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6461, size = 463, normalized size = 9.85 \begin{align*} \left [\frac{\sqrt{2} \log \left (-\frac{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) - 2\right )} \sin \left (d x + c\right ) - \frac{2 \, \sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a}{\left (\cos \left (d x + c\right ) - \sin \left (d x + c\right ) + 1\right )}}{\sqrt{a}} + 3 \, \cos \left (d x + c\right ) + 2}{\cos \left (d x + c\right )^{2} -{\left (\cos \left (d x + c\right ) + 2\right )} \sin \left (d x + c\right ) - \cos \left (d x + c\right ) - 2}\right )}{2 \, \sqrt{a} d}, \frac{\sqrt{2} \sqrt{-\frac{1}{a}} \arctan \left (\frac{\sqrt{2} \sqrt{a \sin \left (d x + c\right ) + a} \sqrt{-\frac{1}{a}}}{\cos \left (d x + c\right )}\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 \frac{1}{\sqrt{a \sin{\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 \frac{1}{\sqrt{a \sin \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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