Optimal. Leaf size=357 \[ -\frac {14 a^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{1105 c^5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]
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Rubi [A] time = 1.81, antiderivative size = 357, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{3/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{7/2}}-\frac {14 a^2 g \sqrt {\cos (e+f x)} E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) \sqrt {g \cos (e+f x)}}{1105 c^5 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a \sin (e+f x)+a} (c-c \sin (e+f x))^{9/2}}+\frac {4 a \sqrt {a \sin (e+f x)+a} (g \cos (e+f x))^{5/2}}{17 f g (c-c \sin (e+f x))^{11/2}} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2842
Rule 2850
Rule 2852
Rubi steps
\begin {align*} \int \frac {(g \cos (e+f x))^{3/2} (a+a \sin (e+f x))^{3/2}}{(c-c \sin (e+f x))^{11/2}} \, dx &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {(7 a) \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}{17 c}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {\left (21 a^2\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^2}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {\left (7 a^2\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}{221 c^3}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {\left (7 a^2\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}{1105 c^4}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 a^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}{1105 c^5}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 a^2 g \cos (e+f x)\right ) \int \sqrt {g \cos (e+f x)} \, dx}{1105 c^5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {\left (7 a^2 g \sqrt {\cos (e+f x)} \sqrt {g \cos (e+f x)}\right ) \int \sqrt {\cos (e+f x)} \, dx}{1105 c^5 \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=\frac {4 a (g \cos (e+f x))^{5/2} \sqrt {a+a \sin (e+f x)}}{17 f g (c-c \sin (e+f x))^{11/2}}-\frac {28 a^2 (g \cos (e+f x))^{5/2}}{221 c f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{9/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{663 c^2 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{7/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^3 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{5/2}}+\frac {14 a^2 (g \cos (e+f x))^{5/2}}{1105 c^4 f g \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2}}-\frac {14 a^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 )}{1105 c^5 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 6.49, size = 532, normalized size = 1.49 \[ \frac {\sec (e+f x) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11} \left (\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {28 \sin \left (\frac {1}{2} (e+f x)\right )}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5}-\frac {160 \sin \left (\frac {1}{2} (e+f x)\right )}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7}+\frac {16 \sin \left (\frac {1}{2} (e+f x)\right )}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}+\frac {14}{1105 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {14}{663 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4}-\frac {80}{221 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6}+\frac {8}{17 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8}+\frac {14}{1105}\right )}{f (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {14 E\left (\left .\frac {1}{2} (e+f x)\right |2\right ) (a (\sin (e+f x)+1))^{3/2} (g \cos (e+f x))^{3/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{11}}{1105 f \cos ^{\frac {3}{2}}(e+f x) (c-c \sin (e+f x))^{11/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (a g \cos \left (f x + e\right ) \sin \left (f x + e\right ) + a g \cos \left (f x + e\right )\right )} \sqrt {g \cos \left (f x + e\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{c^{6} \cos \left (f x + e\right )^{6} - 18 \, c^{6} \cos \left (f x + e\right )^{4} + 48 \, c^{6} \cos \left (f x + e\right )^{2} - 32 \, c^{6} + 2 \, {\left (3 \, c^{6} \cos \left (f x + e\right )^{4} - 16 \, c^{6} \cos \left (f x + e\right )^{2} + 16 \, c^{6}\right )} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.62, size = 1298, normalized size = 3.64 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}\,{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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