Optimal. Leaf size=235 \[ \frac{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}+\frac{6 a c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{a \sin (e+f x)+a}} \]
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Rubi [A] time = 1.12907, antiderivative size = 235, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ \frac{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c^2 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{5 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (c-c \sin (e+f x))^{3/2} (g \cos (e+f x))^{5/2}}{7 f g \sqrt{a \sin (e+f x)+a}}+\frac{6 a c \sqrt{c-c \sin (e+f x)} (g \cos (e+f x))^{5/2}}{35 f g \sqrt{a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 2851
Rule 2842
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int (g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx &=-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{7} (3 a) \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{6 a c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{5} (3 a 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{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt{a+a \sin (e+f x)}}+\frac{1}{5} \left (3 a 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{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt{a+a \sin (e+f x)}}+\frac{\left (3 a 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{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{6 a c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt{a+a \sin (e+f x)}}+\frac{\left (3 a 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{2 a c^2 (g \cos (e+f x))^{5/2}}{5 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{6 a 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{6 a c (g \cos (e+f x))^{5/2} \sqrt{c-c \sin (e+f x)}}{35 f g \sqrt{a+a \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (c-c \sin (e+f x))^{3/2}}{7 f g \sqrt{a+a \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 1.87459, size = 255, normalized size = 1.09 \[ \frac{c^2 g e^{-3 i (e+f x)} \left (e^{i (e+f x)}-i\right ) \left (112 e^{5 i (e+f x)} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right )+\sqrt{1+e^{2 i (e+f x)}} \left (-14 e^{i (e+f x)}+15 i e^{2 i (e+f x)}-168 e^{3 i (e+f x)}+15 i e^{4 i (e+f x)}+14 e^{5 i (e+f x)}+5 i e^{6 i (e+f x)}+5 i\right )\right ) \sqrt{a (\sin (e+f x)+1)} \sqrt{g \cos (e+f x)}}{140 f \left (e^{i (e+f x)}+i\right ) \sqrt{1+e^{2 i (e+f x)}} \sqrt{c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.346, size = 372, normalized size = 1.6 \begin{align*} -{\frac{2}{35\,f \left ( -1+\sin \left ( fx+e \right ) \right ) \sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ( -c \left ( -1+\sin \left ( fx+e \right ) \right ) \right ) ^{{\frac{3}{2}}} \left ( 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\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 ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}+5\,\sin \left ( fx+e \right ) \left ( \cos \left ( fx+e \right ) \right ) ^{4}+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}}}-21\,i\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 ) \sqrt{ \left ( \cos \left ( fx+e \right ) +1 \right ) ^{-1}}\sqrt{{\frac{\cos \left ( fx+e \right ) }{\cos \left ( fx+e \right ) +1}}}-7\, \left ( \cos \left ( fx+e \right ) \right ) ^{4}-14\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}+21\,\cos \left ( fx+e \right ) \right ) \sqrt{a \left ( 1+\sin \left ( fx+e \right ) \right ) } \left ( g\cos \left ( fx+e \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 (g \cos \left (f x + e\right )\right )^{\frac{3}{2}} \sqrt{a \sin \left (f x + e\right ) + a}{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{3}{2}}\,{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 (-{\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{a \sin \left (f x + e\right ) + a} \sqrt{-c \sin \left (f x + e\right ) + c}, 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(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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