Optimal. Leaf size=43 \[ \frac {c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.08, antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {2738} \[ \frac {c \cos (e+f x) (a \sin (e+f x)+a)^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 2738
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^{3/2} \sqrt {c-c \sin (e+f x)} \, dx &=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{3/2}}{2 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 60, normalized size = 1.40 \[ -\frac {a \sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} (\cos (2 (e+f x))-4 \sin (e+f x))}{4 f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 61, normalized size = 1.42 \[ -\frac {{\left (a \cos \left (f x + e\right )^{2} - 2 \, a \sin \left (f x + e\right ) - a\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, f \cos \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.30, size = 63, normalized size = 1.47 \[ -\frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} \sin \left (f x +e \right ) \left (-1-\left (\cos ^{2}\left (f x +e \right )\right )+\sin \left (f x +e \right )\right )}{2 f \cos \left (f x +e \right )^{3}} \]
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 \[ \int {\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}} \sqrt {-c \sin \left (f x + e\right ) + c}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.35, size = 71, normalized size = 1.65 \[ -\frac {a\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (\cos \left (e+f\,x\right )+\cos \left (3\,e+3\,f\,x\right )-4\,\sin \left (2\,e+2\,f\,x\right )\right )}{4\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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