Optimal. Leaf size=319 \[ -\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a \sin (c+d x)+a)^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{64 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{7/2}}{4 d e} \]
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Rubi [A] time = 0.564244, antiderivative size = 319, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2678, 2686, 2679, 2684, 2775, 203, 2833, 63, 215} \[ -\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a \sin (c+d x)+a}}-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a \sin (c+d x)+a)^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a \sin (c+d x)+a}}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{64 d (\sin (c+d x)+\cos (c+d x)+1)}+\frac{45 a e^{5/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{64 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac{a \sqrt{a \sin (c+d x)+a} (e \cos (c+d x))^{7/2}}{4 d e} \]
Antiderivative was successfully verified.
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Rule 2678
Rule 2686
Rule 2679
Rule 2684
Rule 2775
Rule 203
Rule 2833
Rule 63
Rule 215
Rubi steps
\begin{align*} \int (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} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{1}{8} (9 a) \int (e \cos (c+d x))^{5/2} \sqrt{a+a \sin (c+d x)} \, dx\\ &=-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{1}{16} \left (15 a^2\right ) \int \frac{(e \cos (c+d x))^{5/2}}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a+a \sin (c+d x))^{3/2}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{1}{64} \left (15 a^3\right ) \int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{3/2}} \, dx\\ &=-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a+a \sin (c+d x))^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{1}{128} \left (45 a^2 e^2\right ) \int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx\\ &=-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a+a \sin (c+d x))^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{\left (45 a^2 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}{128 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (45 a^2 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}{128 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a+a \sin (c+d x))^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{\left (45 a^2 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 )}{128 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (45 a^2 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 )}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a+a \sin (c+d x))^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{45 a^2 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)}}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{\left (45 a^2 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 )}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac{15 a^3 (e \cos (c+d x))^{7/2}}{32 d e (a+a \sin (c+d x))^{3/2}}+\frac{15 a^2 e (e \cos (c+d x))^{3/2}}{64 d \sqrt{a+a \sin (c+d x)}}-\frac{3 a^2 (e \cos (c+d x))^{7/2}}{8 d e \sqrt{a+a \sin (c+d x)}}-\frac{a (e \cos (c+d x))^{7/2} \sqrt{a+a \sin (c+d x)}}{4 d e}+\frac{45 a^2 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)}}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac{45 a^2 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)}}{64 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end{align*}
Mathematica [C] time = 0.166952, size = 78, normalized size = 0.24 \[ -\frac{16 \sqrt [4]{2} a \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{7/2} \, _2F_1\left (-\frac{9}{4},\frac{7}{4};\frac{11}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{7 d e (\sin (c+d x)+1)^{9/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.201, size = 314, normalized size = 1. \begin{align*}{\frac{1}{128\,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 ( \cos \left ( dx+c \right ) \right ) ^{3}} \left ( 32\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}-32\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}+48\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}\sin \left ( dx+c \right ) +45\,\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 ) +45\,\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 ) +80\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}-60\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) +12\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+90\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) +30\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-90\,\cos \left ( dx+c \right ) \right ) \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}} \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 \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \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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