Optimal. Leaf size=288 \[ -\frac{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}} \]
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Rubi [A] time = 1.43713, antiderivative size = 288, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 42, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2851, 2842, 2640, 2639} \[ -\frac{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{21 f g \sqrt{c-c \sin (e+f x)}}+\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{3 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{2 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}} \]
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} (a+a \sin (e+f x))^{7/2}}{\sqrt{c-c \sin (e+f x)}} \, dx &=-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{3} (5 a) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{21} \left (55 a^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{\sqrt{c-c \sin (e+f x)}} \, dx\\ &=-\frac{22 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{3} \left (11 a^3\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{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{1}{3} \left (11 a^4\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{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{3 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}+\frac{\left (11 a^4 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)}\right ) \int \sqrt{\cos (e+f x)} \, dx}{3 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=-\frac{22 a^4 (g \cos (e+f x))^{5/2}}{9 f g \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}+\frac{22 a^4 g \sqrt{\cos (e+f x)} \sqrt{g \cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right )}{3 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}-\frac{22 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{10 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{21 f g \sqrt{c-c \sin (e+f x)}}-\frac{2 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{9 f g \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 3.94527, size = 181, normalized size = 0.63 \[ -\frac{a^3 (\sin (e+f x)+1)^3 \sqrt{a (\sin (e+f x)+1)} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (\sqrt{\cos (e+f x)} (350 \sin (2 (e+f x))-7 \sin (4 (e+f x))+1128 \cos (e+f x)-72 \cos (3 (e+f x)))-1848 E\left (\left .\frac{1}{2} (e+f x)\right |2\right )\right )}{252 f \cos ^{\frac{3}{2}}(e+f x) \sqrt{c-c \sin (e+f x)} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.352, size = 436, normalized size = 1.5 \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}}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{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 (3 \, a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right ) +{\left (a^{3} g \cos \left (f x + e\right )^{3} - 4 \, a^{3} g \cos \left (f x + e\right )\right )} \sin \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}}{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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (g \cos \left (f x + e\right )\right )^{\frac{3}{2}}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}{\sqrt{-c \sin \left (f x + e\right ) + c}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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