Optimal. Leaf size=194 \[ -\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)} \]
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Rubi [A] time = 0.269584, antiderivative size = 194, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.259, Rules used = {2678, 2684, 2775, 203, 2833, 63, 215} \[ -\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a \sin (c+d x)+a}}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{\sqrt{e} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{d (\sin (c+d x)+\cos (c+d x)+1)} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2684
Rule 2775
Rule 203
Rule 2833
Rule 63
Rule 215
Rubi steps
\begin{align*} \int \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)} \, dx &=-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a+a \sin (c+d x)}}+\frac{1}{2} a \int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a+a \sin (c+d x)}}+\frac{\left (a e \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \int \frac{\sqrt{1+\cos (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx}{2 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (a e \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \int \frac{\sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}} \, dx}{2 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a+a \sin (c+d x)}}+\frac{\left (a e \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{e x} \sqrt{1+x}} \, dx,x,\cos (c+d x)\right )}{2 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (a e \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+e x^2} \, dx,x,-\frac{\sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a+a \sin (c+d x)}}+\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{d (1+\cos (c+d x)+\sin (c+d x))}+\frac{\left (a \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+\frac{x^2}{e}}} \, dx,x,\sqrt{e \cos (c+d x)}\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{a (e \cos (c+d x))^{3/2}}{d e \sqrt{a+a \sin (c+d x)}}+\frac{\sqrt{e} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{d (1+\cos (c+d x)+\sin (c+d x))}+\frac{\sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{e \cos (c+d x)} \sqrt{1+\cos (c+d x)}}\right ) \sqrt{1+\cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{d (1+\cos (c+d x)+\sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.855958, size = 195, normalized size = 1.01 \[ -\frac{i \sqrt{a (\sin (c+d x)+1)} \sqrt{e \cos (c+d x)} \left (i d x e^{i (c+d x)}+e^{i (c+d x)} \sqrt{1+e^{2 i (c+d x)}}-i \sqrt{1+e^{2 i (c+d x)}}-e^{i (c+d x)} \log \left (1+\sqrt{1+e^{2 i (c+d x)}}\right )+i e^{i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{d \left (e^{i (c+d x)}+i\right ) \sqrt{1+e^{2 i (c+d x)}}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.155, size = 213, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,d \left ( 1-\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) \right ) \cos \left ( dx+c \right ) } \left ( \sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ({\frac{\sqrt{2}}{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ({\frac{\sqrt{2}\sin \left ( dx+c \right ) }{2\,\cos \left ( dx+c \right ) }\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\cos \left ( dx+c \right ) \right ) \sqrt{e\cos \left ( dx+c \right ) }\sqrt{a \left ( 1+\sin \left ( dx+c \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 \sqrt{e \cos \left (d x + c\right )} \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 [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 )} \sqrt{e \cos{\left (c + d 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 \sqrt{e \cos \left (d x + c\right )} \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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