Optimal. Leaf size=46 \[ \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \log (\sin (e+f x))}{f} \]
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Rubi [A] time = 0.17, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2948, 3475} \[ \frac {\sec (e+f x) \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)} \log (\sin (e+f x))}{f} \]
Antiderivative was successfully verified.
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Rule 2948
Rule 3475
Rubi steps
\begin {align*} \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)} \, dx &=\left (\sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}\right ) \int \cot (e+f x) \, dx\\ &=\frac {\log (\sin (e+f x)) \sec (e+f x) \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}{f}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 62, normalized size = 1.35 \[ \frac {\sec (e+f x) \sqrt {a (\sin (e+f x)+1)} \sqrt {c-c \sin (e+f x)} \left (\log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )+\log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )\right )}{f} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.64, size = 202, normalized size = 4.39 \[ \left [\frac {\sqrt {a c} \log \left (\frac {4 \, {\left (256 \, a c \cos \left (f x + e\right )^{5} - 512 \, a c \cos \left (f x + e\right )^{3} + 337 \, a c \cos \left (f x + e\right ) + {\left (256 \, \cos \left (f x + e\right )^{4} - 512 \, \cos \left (f x + e\right )^{2} + 175\right )} \sqrt {a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{\cos \left (f x + e\right )^{3} - \cos \left (f x + e\right )}\right )}{2 \, f}, -\frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} {\left (16 \, \cos \left (f x + e\right )^{2} - 7\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{16 \, a c \cos \left (f x + e\right )^{3} - 25 \, a c \cos \left (f x + e\right )}\right )}{f}\right ] \]
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 = 0.66, size = 74, normalized size = 1.61 \[ -\frac {\left (\ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-\ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )\right ) \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+\sin \left (f x +e \right )\right )}}{f \cos \left (f x +e \right )} \]
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 \[ \int \frac {\sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{\sin \left (f x + e\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )} \,d x \]
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 \[ \int \frac {\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )}}{\sin {\left (e + f x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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