Optimal. Leaf size=76 \[ \frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{3/2} d} \]
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Rubi [A] time = 0.0806601, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {2679, 2649, 206} \[ \frac{2 \cos (c+d x)}{a d \sqrt{a \sin (c+d x)+a}}-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a \sin (c+d x)+a}}\right )}{a^{3/2} d} \]
Antiderivative was successfully verified.
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Rule 2679
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{\cos ^2(c+d x)}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{2 \cos (c+d x)}{a d \sqrt{a+a \sin (c+d x)}}+\frac{2 \int \frac{1}{\sqrt{a+a \sin (c+d x)}} \, dx}{a}\\ &=\frac{2 \cos (c+d x)}{a d \sqrt{a+a \sin (c+d x)}}-\frac{4 \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 )}{a d}\\ &=-\frac{2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (c+d x)}{\sqrt{2} \sqrt{a+a \sin (c+d x)}}\right )}{a^{3/2} d}+\frac{2 \cos (c+d x)}{a d \sqrt{a+a \sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.145741, size = 84, normalized size = 1.11 \[ \frac{2 \cos ^3(c+d x) \left (\sqrt{1-\sin (c+d x)}-\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )\right )}{d (1-\sin (c+d x))^{3/2} (a (\sin (c+d x)+1))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.134, size = 94, normalized size = 1.2 \begin{align*} 2\,{\frac{ \left ( 1+\sin \left ( dx+c \right ) \right ) \sqrt{-a \left ( \sin \left ( dx+c \right ) -1 \right ) }}{{a}^{2}\cos \left ( dx+c \right ) \sqrt{a+a\sin \left ( dx+c \right ) }d} \left ( \sqrt{a-a\sin \left ( dx+c \right ) }-\sqrt{a}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{a-a\sin \left ( dx+c \right ) }\sqrt{2}}{\sqrt{a}}} \right ) \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{\cos \left (d x + c\right )^{2}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.26083, size = 536, normalized size = 7.05 \begin{align*} \frac{\frac{\sqrt{2}{\left (a \cos \left (d x + c\right ) + a \sin \left (d x + c\right ) + a\right )} \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 )}{\sqrt{a}} + 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 )}}{a^{2} d \cos \left (d x + c\right ) + a^{2} d \sin \left (d x + c\right ) + a^{2} 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 \frac{\cos ^{2}{\left (c + d x \right )}}{\left (a \left (\sin{\left (c + d x \right )} + 1\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.90494, size = 257, normalized size = 3.38 \begin{align*} -\frac{2 \,{\left (\frac{\frac{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}{a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )} - \frac{1}{a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}}{\sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a}} + \frac{\sqrt{2}{\left (2 \, \sqrt{a} \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) + \sqrt{-a}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}{\sqrt{-a} a^{\frac{3}{2}}} - \frac{2 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{a} \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \sqrt{a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + a} + \sqrt{a}\right )}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1\right )}\right )}}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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