Optimal. Leaf size=414 \[ \frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{663 c^6 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}} \]
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Rubi [A] time = 2.17137, antiderivative size = 414, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}+\frac{220 a^3 \sqrt{a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{22 a^4 g \sqrt{\cos (e+f x)} E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) \sqrt{g \cos (e+f x)}}{663 c^6 f \sqrt{a \sin (e+f x)+a} \sqrt{c-c \sin (e+f x)}}-\frac{20 a^2 (a \sin (e+f x)+a)^{3/2} (g \cos (e+f x))^{5/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{4 a (a \sin (e+f x)+a)^{5/2} (g \cos (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}} \]
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} (a+a \sin (e+f x))^{7/2}}{(c-c \sin (e+f x))^{13/2}} \, dx &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{(5 a) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{5/2}}{(c-c \sin (e+f x))^{11/2}} \, dx}{7 c}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{\left (55 a^2\right ) \int \frac{(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{9/2}} \, dx}{119 c^2}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{\left (55 a^3\right ) \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}{221 c^3}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{\left (55 a^4\right ) \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}{663 c^4}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{\left (11 a^4\right ) \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}{663 c^5}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\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}{663 c^6}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac{\left (11 a^4 g \cos (e+f x)\right ) \int \sqrt{g \cos (e+f x)} \, dx}{663 c^6 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\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}{663 c^6 \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ &=\frac{4 a (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{5/2}}{21 f g (c-c \sin (e+f x))^{13/2}}-\frac{20 a^2 (g \cos (e+f x))^{5/2} (a+a \sin (e+f x))^{3/2}}{119 c f g (c-c \sin (e+f x))^{11/2}}+\frac{220 a^3 (g \cos (e+f x))^{5/2} \sqrt{a+a \sin (e+f x)}}{1547 c^2 f g (c-c \sin (e+f x))^{9/2}}-\frac{220 a^4 (g \cos (e+f x))^{5/2}}{1989 c^3 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^4 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac{22 a^4 (g \cos (e+f x))^{5/2}}{663 c^5 f g \sqrt{a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\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 )}{663 c^6 f \sqrt{a+a \sin (e+f x)} \sqrt{c-c \sin (e+f x)}}\\ \end{align*}
Mathematica [A] time = 6.81841, size = 600, normalized size = 1.45 \[ \frac{\sec (e+f x) (a (\sin (e+f x)+1))^{7/2} (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 )^{13} \left (\frac{44 \sin \left (\frac{1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )}+\frac{44 \sin \left (\frac{1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3}-\frac{2432 \sin \left (\frac{1}{2} (e+f x)\right )}{1989 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5}+\frac{928 \sin \left (\frac{1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{704 \sin \left (\frac{1}{2} (e+f x)\right )}{119 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^9}+\frac{64 \sin \left (\frac{1}{2} (e+f x)\right )}{21 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{11}}+\frac{22}{663 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2}-\frac{1216}{1989 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4}+\frac{464}{221 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6}-\frac{352}{119 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8}+\frac{32}{21 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{10}}+\frac{22}{663}\right )}{f (c-c \sin (e+f x))^{13/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^7}-\frac{22 E\left (\left .\frac{1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{7/2} (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 )^{13}}{663 f \cos ^{\frac{3}{2}}(e+f x) (c-c \sin (e+f x))^{13/2} \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 = 0.417, size = 1484, normalized size = 3.6 \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}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{13}{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{{\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}}{7 \, c^{7} \cos \left (f x + e\right )^{6} - 56 \, c^{7} \cos \left (f x + e\right )^{4} + 112 \, c^{7} \cos \left (f x + e\right )^{2} - 64 \, c^{7} -{\left (c^{7} \cos \left (f x + e\right )^{6} - 24 \, c^{7} \cos \left (f x + e\right )^{4} + 80 \, c^{7} \cos \left (f x + e\right )^{2} - 64 \, c^{7}\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}}{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac{7}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{13}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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