Optimal. Leaf size=261 \[ \frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d \left (a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right )}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d \left (a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right )}+\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a \sin (c+d x)+a)^{3/2}} \]
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Rubi [A] time = 0.46874, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 9, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {2680, 2686, 2679, 2684, 2775, 203, 2833, 63, 215} \[ \frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a \sin (c+d x)+a}}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \tan ^{-1}\left (\frac{\sqrt{e} \sin (c+d x)}{\sqrt{\cos (c+d x)+1} \sqrt{e \cos (c+d x)}}\right )}{4 d \left (a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right )}+\frac{21 e^{9/2} \sqrt{\cos (c+d x)+1} \sqrt{a \sin (c+d x)+a} \sinh ^{-1}\left (\frac{\sqrt{e \cos (c+d x)}}{\sqrt{e}}\right )}{4 d \left (a^3 \sin (c+d x)+a^3 \cos (c+d x)+a^3\right )}+\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a \sin (c+d x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2680
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))^{9/2}}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac{4 e (e \cos (c+d x))^{7/2}}{a d (a+a \sin (c+d x))^{3/2}}+\frac{\left (7 e^2\right ) \int \frac{(e \cos (c+d x))^{5/2}}{\sqrt{a+a \sin (c+d x)}} \, dx}{a^2}\\ &=\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a+a \sin (c+d x))^{3/2}}+\frac{\left (7 e^2\right ) \int \frac{(e \cos (c+d x))^{5/2}}{(a+a \sin (c+d x))^{3/2}} \, dx}{4 a}\\ &=\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a+a \sin (c+d x))^{3/2}}+\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\left (21 e^4\right ) \int \frac{\sqrt{e \cos (c+d x)}}{\sqrt{a+a \sin (c+d x)}} \, dx}{8 a^2}\\ &=\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a+a \sin (c+d x))^{3/2}}+\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\left (21 e^5 \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^2 (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (21 e^5 \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^2 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a+a \sin (c+d x))^{3/2}}+\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{\left (21 e^5 \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 a^2 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac{\left (21 e^5 \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 a^2 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a+a \sin (c+d x))^{3/2}}+\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{21 e^{9/2} \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 \left (a^3+a^3 \cos (c+d x)+a^3 \sin (c+d x)\right )}+\frac{\left (21 e^4 \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 a^2 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac{e (e \cos (c+d x))^{7/2}}{2 a d (a+a \sin (c+d x))^{3/2}}+\frac{7 e^3 (e \cos (c+d x))^{3/2}}{4 a^2 d \sqrt{a+a \sin (c+d x)}}+\frac{21 e^{9/2} \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 \left (a^3+a^3 \cos (c+d x)+a^3 \sin (c+d x)\right )}+\frac{21 e^{9/2} \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 \left (a^3+a^3 \cos (c+d x)+a^3 \sin (c+d x)\right )}\\ \end{align*}
Mathematica [C] time = 0.16791, size = 80, normalized size = 0.31 \[ -\frac{2 \sqrt [4]{2} \sqrt{a (\sin (c+d x)+1)} (e \cos (c+d x))^{11/2} \, _2F_1\left (\frac{3}{4},\frac{11}{4};\frac{15}{4};\frac{1}{2} (1-\sin (c+d x))\right )}{11 a^3 d e (\sin (c+d x)+1)^{13/4}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.146, size = 282, normalized size = 1.1 \begin{align*}{\frac{1}{8\,d \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{2}\sin \left ( dx+c \right ) + \left ( \cos \left ( dx+c \right ) \right ) ^{3}+2\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}-4\,\sin \left ( dx+c \right ) -2\,\cos \left ( dx+c \right ) +4 \right ) } \left ( e\cos \left ( dx+c \right ) \right ) ^{{\frac{9}{2}}} \left ( 21\,\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 ) +21\,\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 ) ^{2}\sin \left ( dx+c \right ) -4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}-22\,\cos \left ( dx+c \right ) \sin \left ( dx+c \right ) -18\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}+22\,\cos \left ( dx+c \right ) \right ) \left ( a \left ( 1+\sin \left ( dx+c \right ) \right ) \right ) ^{-{\frac{5}{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{9}{2}}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac{5}{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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