Optimal. Leaf size=98 \[ -\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f} \]
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Rubi [A] time = 0.27, antiderivative size = 98, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2736, 2674, 2673} \[ -\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f} \]
Antiderivative was successfully verified.
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Rule 2673
Rule 2674
Rule 2736
Rubi steps
\begin {align*} \int \frac {(c-c \sin (e+f x))^{5/2}}{a+a \sin (e+f x)} \, dx &=\frac {\int \sec ^2(e+f x) (c-c \sin (e+f x))^{7/2} \, dx}{a c}\\ &=\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {8 \int \sec ^2(e+f x) (c-c \sin (e+f x))^{5/2} \, dx}{3 a}\\ &=\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}+\frac {(32 c) \int \sec ^2(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{3 a}\\ &=-\frac {64 c^2 \sec (e+f x) \sqrt {c-c \sin (e+f x)}}{3 a f}+\frac {16 c \sec (e+f x) (c-c \sin (e+f x))^{3/2}}{3 a f}+\frac {2 \sec (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a f}\\ \end {align*}
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Mathematica [A] time = 0.75, size = 102, normalized size = 1.04 \[ -\frac {c^2 \sqrt {c-c \sin (e+f x)} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) (20 \sin (e+f x)+\cos (2 (e+f x))+45)}{3 a f (\sin (e+f x)+1) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 58, normalized size = 0.59 \[ -\frac {2 \, {\left (c^{2} \cos \left (f x + e\right )^{2} + 10 \, c^{2} \sin \left (f x + e\right ) + 22 \, c^{2}\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a f \cos \left (f x + e\right )} \]
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 [A] time = 0.70, size = 59, normalized size = 0.60 \[ -\frac {2 c^{3} \left (\sin \left (f x +e \right )-1\right ) \left (\sin ^{2}\left (f x +e \right )-10 \sin \left (f x +e \right )-23\right )}{3 a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.66, size = 192, normalized size = 1.96 \[ \frac {2 \, {\left (23 \, c^{\frac {5}{2}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {40 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {65 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {20 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {23 \, c^{\frac {5}{2}} \sin \left (f x + e\right )^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}\right )}}{3 \, {\left (a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )} f {\left (\frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2}}{a+a\,\sin \left (e+f\,x\right )} \,d x \]
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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