Optimal. Leaf size=292 \[ -\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{65 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}} \]
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Rubi [A] time = 1.45982, antiderivative size = 292, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.119, Rules used = {2850, 2852, 2842, 2640, 2639} \[ -\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac{2 a g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{65 c^4 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2850
Rule 2852
Rule 2842
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{(g \cos (e+f x))^{3/2} \sqrt{a+a \sin (e+f x)}}{(c-c \sin (e+f x))^{9/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{(3 a) \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}} \, dx}{13 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{a \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}} \, dx}{13 c^2}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{a \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}} \, dx}{65 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{a \int \frac{(g \cos (e+f x))^{3/2}}{\sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}} \, dx}{65 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{(a g \cos (e+f x)) \int \sqrt{g \cos (e+f x)} \, dx}{65 c^4 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{\left (a g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{65 c^4 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2}}{13 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{39 c f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^2 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}-\frac{2 a (g \cos (e+f x))^{5/2}}{65 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}+\frac{2 a g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{65 c^4 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [C] time = 2.86199, size = 291, normalized size = 1. \[ \frac{4 e^{3 i (e+f x)} \left (g e^{-i (e+f x)} \left (1+e^{2 i (e+f x)}\right )\right )^{3/2} \left (\sqrt{1+e^{2 i (e+f x)}} \left (149 i e^{i (e+f x)}+44 e^{2 i (e+f x)}-64 i e^{3 i (e+f x)}+21 e^{4 i (e+f x)}+3 i e^{5 i (e+f x)}-1\right )-i \left (e^{i (e+f x)}-i\right )^7 \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{7}{4};-e^{2 i (e+f x)}\right )\right ) \sqrt{a (\sin (e+f x)+1)}}{195 c^4 f \left (1-i e^{i (e+f x)}\right ) \left (e^{i (e+f x)}-i\right )^6 \left (1+e^{2 i (e+f x)}\right )^{3/2} \sqrt{i c e^{-i (e+f x)} \left (e^{i (e+f x)}-i\right )^2}} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.385, size = 1126, normalized size = 3.9 \begin{align*} \text{result too large to display} \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}} \sqrt{a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{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 (\frac{\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} g \cos \left (f x + e\right )}{5 \, c^{5} \cos \left (f x + e\right )^{4} - 20 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5} -{\left (c^{5} \cos \left (f x + e\right )^{4} - 12 \, c^{5} \cos \left (f x + e\right )^{2} + 16 \, c^{5}\right )} \sin \left (f x + e\right )}, 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}} \sqrt{a \sin \left (f x + e\right ) + a}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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