Optimal. Leaf size=200 \[ \frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a d} \]
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Rubi [A] time = 0.278785, antiderivative size = 200, 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 = {2685, 2677, 2775, 203, 2833, 63, 215} \[ \frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{e^{3/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{e \sqrt{a \sin (c+d x)+a} \sqrt{e \cos (c+d x)}}{a d} \]
Antiderivative was successfully verified.
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Rule 2685
Rule 2677
Rule 2775
Rule 203
Rule 2833
Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{3/2}}{\sqrt{a+a \sin (c+d x)}} \, dx &=\frac{e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a d}+\frac{e^2 \int \frac{\sqrt{a+a \sin (c+d x)}}{\sqrt{e \cos (c+d x)}} \, dx}{2 a}\\ &=\frac{e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a d}+\frac{\left (e^2 \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 (e^2 \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{e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a d}-\frac{\left (e^2 \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 (e^2 \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{e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a d}+\frac{e^{3/2} \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 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (e \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{e \sqrt{e \cos (c+d x)} \sqrt{a+a \sin (c+d x)}}{a d}-\frac{e^{3/2} \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 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{e^{3/2} \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 (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.116269, size = 77, normalized size = 0.38 \[ -\frac{2\ 2^{3/4} (e \cos (c+d x))^{5/2} \, _2F_1\left (\frac{1}{4},\frac{5}{4};\frac{9}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{5 d e (\sin (c+d x)+1)^{3/4} \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.125, size = 212, 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 ) \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\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 \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{\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(-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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (e \cos \left (d x + c\right )\right )^{\frac{3}{2}}}{\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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