Optimal. Leaf size=215 \[ \frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{d \left (a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right )}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{d \left (a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right )}+\frac{e (e \cos (c+d x))^{3/2}}{a d \sqrt{a \sin (c+d x)+a}} \]
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Rubi [A] time = 0.286914, antiderivative size = 215, 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 = {2679, 2684, 2775, 203, 2833, 63, 215} \[ \frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{d \left (a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right )}+\frac{3 e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{d \left (a^2 \sin (c+d x)+a^2 \cos (c+d x)+a^2\right )}+\frac{e (e \cos (c+d x))^{3/2}}{a d \sqrt{a \sin (c+d x)+a}} \]
Antiderivative was successfully verified.
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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}}{(a+a \sin (c+d x))^{3/2}} \, dx &=\frac{e (e \cos (c+d x))^{3/2}}{a d \sqrt{a+a \sin (c+d x)}}+\frac{\left (3 e^2\right ) \int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx}{2 a}\\ &=\frac{e (e \cos (c+d x))^{3/2}}{a 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}{2 a (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}{2 a (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac{e (e \cos (c+d x))^{3/2}}{a 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 )}{2 a 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 )}{a d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac{e (e \cos (c+d x))^{3/2}}{a 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)}}{d \left (a^2+a^2 \cos (c+d x)+a^2 \sin (c+d x)\right )}+\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 )}{a d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac{e (e \cos (c+d x))^{3/2}}{a 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)}}{d \left (a^2+a^2 \cos (c+d x)+a^2 \sin (c+d x)\right )}+\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)}}{d \left (a^2+a^2 \cos (c+d x)+a^2 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.183856, size = 80, normalized size = 0.37 \[ -\frac{2 \sqrt [4]{2} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{7/2} \, _2F_1\left (\frac{3}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{7 a^2 d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.113, size = 232, normalized size = 1.1 \begin{align*} -{\frac{1}{2\,d \left ( \cos \left ( dx+c \right ) \sin \left ( dx+c \right ) - \left ( \cos \left ( dx+c \right ) \right ) ^{2}-2\,\sin \left ( dx+c \right ) -\cos \left ( dx+c \right ) +2 \right ) } \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{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 ) +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 ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{3}{2}}}} \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}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{3}{2}}}\,{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(-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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