Optimal. Leaf size=180 \[ \frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.838002, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{2 c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2851
Rule 2842
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{3/2} (c-c \sin (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)}} \, dx &=\frac{2 c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{5} (7 c) \int \frac{(g \cos (e+f x))^{3/2} \sqrt{c-c \sin (e+f x)}}{\sqrt{a+a \sin (e+f x)}} \, dx\\ &=\frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{5} \left (7 c^2\right ) \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx\\ &=\frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 f g \sqrt{a+a \sin (e+f x)}}+\frac{\left (7 c^2 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{5 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 f g \sqrt{a+a \sin (e+f x)}}+\frac{\left (7 c^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{5 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{14 c^2 (g \cos (e+f x))^{5/2}}{15 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{14 c^2 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{5 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{2 c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{5 f g \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 0.724199, size = 157, normalized size = 0.87 \[ -\frac{c (\sin (e+f x)-1) \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{3/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right ) \left (42 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )+\sqrt{\cos (e+f x)} (20 \cos (e+f x)-3 \sin (2 (e+f x)))\right )}{15 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{a (\sin (e+f x)+1)} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.31, size = 382, normalized size = 2.1 \begin{align*}{\frac{2}{15\,f \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+2\,\sin \left ( fx+e \right ) -2 \right ) \sin \left ( fx+e \right ) \cos \left ( fx+e \right ) } \left ( g\cos \left ( fx+e \right ) \right ) ^{{\frac{3}{2}}} \left ( 21\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ) \cos \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) -21\,i\sin \left ( fx+e \right ) \cos \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+21\,i\sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}\sin \left ( fx+e \right ){\it EllipticE} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) -21\,i\sin \left ( fx+e \right ){\it EllipticF} \left ({\frac{i \left ( -1+\cos \left ( fx+e \right ) \right ) }{\sin \left ( fx+e \right ) }},i \right ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-3\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-10\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}\sin \left ( fx+e \right ) +24\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-21\,\cos \left ( fx+e \right ) \right ) \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{a \left ( 1+\sin \left ( fx+e \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 (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (c g \cos \left (f x + e\right ) \sin \left (f x + e\right ) - c g \cos \left (f x + e\right )\right )} \sqrt{g \cos \left (f x + e\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{\sqrt{a \sin \left (f x + e\right ) + a}}, x\right ) \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 (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}}{\sqrt{a \sin \left (f x + e\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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