Optimal. Leaf size=37 \[ -\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{f} \]
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Rubi [A] time = 0.0555735, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2774, 216} \[ -\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{f} \]
Antiderivative was successfully verified.
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Rule 2774
Rule 216
Rubi steps
\begin{align*} \int \frac{\sqrt{a+a \sin (e+f x)}}{\sqrt{\sin (e+f x)}} \, dx &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{x^2}{a}}} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac{2 \sqrt{a} \sin ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}\\ \end{align*}
Mathematica [C] time = 0.51352, size = 164, normalized size = 4.43 \[ \frac{(1+i) e^{\frac{1}{2} i (e+f x)} \sqrt{-i e^{-i (e+f x)} \left (-1+e^{2 i (e+f x)}\right )} \sqrt{a (\sin (e+f x)+1)} \left (\tan ^{-1}\left (\sqrt{-1+e^{2 i (e+f x)}}\right )-i \tanh ^{-1}\left (\frac{e^{i (e+f x)}}{\sqrt{-1+e^{2 i (e+f x)}}}\right )\right )}{\sqrt{2} f \sqrt{-1+e^{2 i (e+f x)}} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.141, size = 320, normalized size = 8.7 \begin{align*}{\frac{\sqrt{2}}{2\,f \left ( 1-\cos \left ( fx+e \right ) +\sin \left ( fx+e \right ) \right ) }\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{\sin \left ( fx+e \right ) } \left ( \ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) ^{-1}} \right ) +4\,\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}+1 \right ) +4\,\arctan \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}-1 \right ) +\ln \left ( -{ \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) -\sin \left ( fx+e \right ) +\cos \left ( fx+e \right ) -1 \right ) \left ( \sqrt{2}\sqrt{-{\frac{-1+\cos \left ( fx+e \right ) }{\sin \left ( fx+e \right ) }}}\sin \left ( fx+e \right ) +\sin \left ( fx+e \right ) -\cos \left ( fx+e \right ) +1 \right ) ^{-1}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.09046, size = 284, normalized size = 7.68 \begin{align*} -\frac{2 \, \sqrt{2} \sqrt{a} \left (\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )^{\frac{3}{2}} - 3 \, \sqrt{2}{\left (\sqrt{2} \arctan \left (\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} + 2 \, \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}\right ) + \sqrt{2} \arctan \left (-\frac{1}{2} \, \sqrt{2}{\left (\sqrt{2} - 2 \, \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}\right )\right )} \sqrt{a} + 6 \, \sqrt{2} \sqrt{a} \sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}} - \frac{2 \,{\left (\frac{3 \, \sqrt{2} \sqrt{a} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac{\sqrt{2} \sqrt{a} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}\right )}}{\sqrt{\frac{\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.57388, size = 934, normalized size = 25.24 \begin{align*} \left [\frac{\sqrt{-a} \log \left (\frac{128 \, a \cos \left (f x + e\right )^{5} - 128 \, a \cos \left (f x + e\right )^{4} - 416 \, a \cos \left (f x + e\right )^{3} + 128 \, a \cos \left (f x + e\right )^{2} - 8 \,{\left (16 \, \cos \left (f x + e\right )^{4} - 24 \, \cos \left (f x + e\right )^{3} - 66 \, \cos \left (f x + e\right )^{2} +{\left (16 \, \cos \left (f x + e\right )^{3} + 40 \, \cos \left (f x + e\right )^{2} - 26 \, \cos \left (f x + e\right ) - 51\right )} \sin \left (f x + e\right ) + 25 \, \cos \left (f x + e\right ) + 51\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{-a} \sqrt{\sin \left (f x + e\right )} + 289 \, a \cos \left (f x + e\right ) +{\left (128 \, a \cos \left (f x + e\right )^{4} + 256 \, a \cos \left (f x + e\right )^{3} - 160 \, a \cos \left (f x + e\right )^{2} - 288 \, a \cos \left (f x + e\right ) + a\right )} \sin \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1}\right )}{4 \, f}, \frac{\sqrt{a} \arctan \left (\frac{{\left (8 \, \cos \left (f x + e\right )^{2} + 8 \, \sin \left (f x + e\right ) - 9\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{a} \sqrt{\sin \left (f x + e\right )}}{4 \,{\left (2 \, a \cos \left (f x + e\right )^{3} + a \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 2 \, a \cos \left (f x + e\right )\right )}}\right )}{2 \, 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{\sqrt{a \left (\sin{\left (e + f x \right )} + 1\right )}}{\sqrt{\sin{\left (e + f 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 \frac{\sqrt{a \sin \left (f x + e\right ) + a}}{\sqrt{\sin \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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