Optimal. Leaf size=92 \[ -\frac {a (A+B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a+a \sin (e+f x)}} \]
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Rubi [A]
time = 0.21, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.050, Rules used = {3050, 2817}
\begin {gather*} \frac {a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a \sin (e+f x)+a}}-\frac {a (A+B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
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Rule 2817
Rule 3050
Rubi steps
\begin {align*} \int \sqrt {a+a \sin (e+f x)} (A+B \sin (e+f x)) \sqrt {c-c \sin (e+f x)} \, dx &=(A+B) \int \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx-\frac {B \int \sqrt {a+a \sin (e+f x)} (c-c \sin (e+f x))^{3/2} \, dx}{c}\\ &=-\frac {a (A+B) \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{f \sqrt {a+a \sin (e+f x)}}+\frac {a B \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{2 c f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 63, normalized size = 0.68 \begin {gather*} \frac {\sec (e+f x) \sqrt {a (1+\sin (e+f x))} (-B \cos (2 (e+f x))+4 A \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{4 f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.37, size = 57, normalized size = 0.62
method | result | size |
default | \(\frac {\left (B \sin \left (f x +e \right )+2 A \right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right ) \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{2 f \cos \left (f x +e \right )}\) | \(57\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 66, normalized size = 0.72 \begin {gather*} -\frac {{\left (B \cos \left (f x + e\right )^{2} - 2 \, A \sin \left (f x + e\right ) - B\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 )} \end {gather*}
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 \begin {gather*} \int \sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )} \left (A + B \sin {\left (e + f x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.51, size = 148, normalized size = 1.61 \begin {gather*} -\frac {2 \, {\left (B \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + A \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - B \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.94, size = 75, normalized size = 0.82 \begin {gather*} -\frac {\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (B\,\cos \left (e+f\,x\right )+B\,\cos \left (3\,e+3\,f\,x\right )-4\,A\,\sin \left (2\,e+2\,f\,x\right )\right )}{4\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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