Integrand size = 34, antiderivative size = 69 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{c f} \]
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Time = 0.23 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.147, Rules used = {3013, 2852, 212, 2815, 2752} \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \sec (e+f x) \sqrt {a \sin (e+f x)+a}}{c f}-\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{c f} \]
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Rule 212
Rule 2752
Rule 2815
Rule 2852
Rule 3013
Rubi steps \begin{align*} \text {integral}& = \frac {\int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{c}+\int \frac {\sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx \\ & = \frac {\int \sec ^2(e+f x) (a+a \sin (e+f x))^{3/2} \, dx}{a c}-\frac {(2 a) \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{c f} \\ & = -\frac {2 \sqrt {a} \text {arctanh}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{c f}+\frac {2 \sec (e+f x) \sqrt {a+a \sin (e+f x)}}{c f} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.70 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.67 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {2 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},1-\sin (e+f x)\right ) \sec (e+f x) \sqrt {a (1+\sin (e+f x))}}{c f} \]
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Time = 0.51 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.13
method | result | size |
default | \(\frac {2 \left (1+\sin \left (f x +e \right )\right ) \left (a^{\frac {3}{2}}-\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}}{\sqrt {a}}\right ) a \sqrt {a -a \sin \left (f x +e \right )}\right )}{\sqrt {a}\, c \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(78\) |
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Leaf count of result is larger than twice the leaf count of optimal. 202 vs. \(2 (61) = 122\).
Time = 0.28 (sec) , antiderivative size = 202, normalized size of antiderivative = 2.93 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\frac {\sqrt {a} \cos \left (f x + e\right ) \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + 4 \, \sqrt {a \sin \left (f x + e\right ) + a}}{2 \, c f \cos \left (f x + e\right )} \]
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\[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=- \frac {\int \frac {\sqrt {a \sin {\left (e + f x \right )} + a}}{\sin ^{2}{\left (e + f x \right )} - \sin {\left (e + f x \right )}}\, dx}{c} \]
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\[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\int { -\frac {\sqrt {a \sin \left (f x + e\right ) + a}}{{\left (c \sin \left (f x + e\right ) - c\right )} \sin \left (f x + e\right )} \,d x } \]
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none
Time = 0.34 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=-\frac {\sqrt {2} {\left (\frac {\sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right ) \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c} + \frac {2 \, \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}{c \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}\right )} \sqrt {a}}{2 \, f} \]
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Timed out. \[ \int \frac {\csc (e+f x) \sqrt {a+a \sin (e+f x)}}{c-c \sin (e+f x)} \, dx=\int \frac {\sqrt {a+a\,\sin \left (e+f\,x\right )}}{\sin \left (e+f\,x\right )\,\left (c-c\,\sin \left (e+f\,x\right )\right )} \,d x \]
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