Optimal. Leaf size=79 \[ -\frac{\sqrt{2} (c-d) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{a} f}-\frac{2 d \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.0699759, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2751, 2649, 206} \[ -\frac{\sqrt{2} (c-d) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a \sin (e+f x)+a}}\right )}{\sqrt{a} f}-\frac{2 d \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2751
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{c+d \sin (e+f x)}{\sqrt{a+a \sin (e+f x)}} \, dx &=-\frac{2 d \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}+(c-d) \int \frac{1}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=-\frac{2 d \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}-\frac{(2 (c-d)) \operatorname{Subst}\left (\int \frac{1}{2 a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac{\sqrt{2} (c-d) \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{2} \sqrt{a+a \sin (e+f x)}}\right )}{\sqrt{a} f}-\frac{2 d \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.213114, size = 106, normalized size = 1.34 \[ \frac{2 \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (d \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )\right )+(1+i) (-1)^{3/4} (c-d) \tanh ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) (-1)^{3/4} \left (\tan \left (\frac{1}{4} (e+f x)\right )-1\right )\right )\right )}{f \sqrt{a (\sin (e+f x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.741, size = 128, normalized size = 1.6 \begin{align*} -{\frac{1+\sin \left ( fx+e \right ) }{af\cos \left ( fx+e \right ) }\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( \sqrt{a}\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{a}}}} \right ) c-\sqrt{a}\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{a}}}} \right ) d+2\,\sqrt{a-a\sin \left ( fx+e \right ) }d \right ){\frac{1}{\sqrt{a+a\sin \left ( fx+e \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{d \sin \left (f x + e\right ) + c}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.64717, size = 578, normalized size = 7.32 \begin{align*} -\frac{\frac{\sqrt{2}{\left (a c - a d +{\left (a c - a d\right )} \cos \left (f x + e\right ) +{\left (a c - a d\right )} \sin \left (f x + e\right )\right )} \log \left (-\frac{\cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) + \frac{2 \, \sqrt{2} \sqrt{a \sin \left (f x + e\right ) + a}{\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt{a}} + 3 \, \cos \left (f x + e\right ) + 2}{\cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right )}{\sqrt{a}} + 4 \,{\left (d \cos \left (f x + e\right ) - d \sin \left (f x + e\right ) + d\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{2 \,{\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a f\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{c + d \sin{\left (e + f x \right )}}{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 2.18192, size = 297, normalized size = 3.76 \begin{align*} \frac{2 \,{\left (\frac{\sqrt{2}{\left (c - d\right )} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a} + \sqrt{a}\right )}}{2 \, \sqrt{-a}}\right )}{\sqrt{-a} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )} + \frac{\frac{d \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )} - \frac{d}{\mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}}{\sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a}} - \frac{{\left (\sqrt{2} a c \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) - \sqrt{2} a d \arctan \left (\frac{\sqrt{a}}{\sqrt{-a}}\right ) - \sqrt{2} \sqrt{-a} \sqrt{a} d\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}{\sqrt{-a} a}\right )}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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