Optimal. Leaf size=244 \[ \frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.366696, antiderivative size = 244, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.296, Rules used = {2686, 2679, 2684, 2775, 203, 2833, 63, 215} \[ \frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d (a \sin (c+d x)+a \cos (c+d x)+a)}-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a \sin (c+d x)+a)^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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Rule 2686
Rule 2679
Rule 2684
Rule 2775
Rule 203
Rule 2833
Rule 63
Rule 215
Rubi steps
\begin{align*} \int \frac{(e \cos (c+d x))^{5/2}}{\sqrt{a+a \sin (c+d x)}} \, dx &=-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac{1}{4} a \int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{1}{8} \left (3 e^2\right ) \int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\left (3 e^3 \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}{8 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (3 e^3 \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}{8 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{\left (3 e^3 \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 )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (3 e^3 \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 )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{3 e^{5/2} \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)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{\left (3 e^2 \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 )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{a (e \cos (c+d x))^{7/2}}{2 d e (a+a \sin (c+d x))^{3/2}}+\frac{e (e \cos (c+d x))^{3/2}}{4 d \sqrt{a+a \sin (c+d x)}}+\frac{3 e^{5/2} \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)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{3 e^{5/2} \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)}}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.189415, size = 77, normalized size = 0.32 \[ -\frac{4 \sqrt [4]{2} (e \cos (c+d x))^{7/2} \, _2F_1\left (-\frac{1}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{7 d e (\sin (c+d x)+1)^{5/4} \sqrt{a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.149, size = 239, normalized size = 1. \begin{align*} -{\frac{1}{8\,d \left ( -1+\cos \left ( dx+c \right ) +\sin \left ( dx+c \right ) \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{2}} \left ( 3\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}\arctan \left ( 1/2\,\sqrt{2}\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}} \right ) \sqrt{2}\sin \left ( dx+c \right ) +3\,\sqrt{-2\,{\frac{\cos \left ( dx+c \right ) }{1+\cos \left ( dx+c \right ) }}}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{2}\sin \left ( dx+c \right ) }{\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 ) +4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +2\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+6\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -6\,\cos \left ( dx+c \right ) \right ) \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{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{5}{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{5}{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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