Integrand size = 36, antiderivative size = 46 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \]
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Time = 0.14 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3025, 2700, 29} \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}} \]
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Rule 29
Rule 2700
Rule 3025
Rubi steps \begin{align*} \text {integral}& = \frac {\cos (e+f x) \int \csc (e+f x) \sec (e+f x) \, dx}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \text {Subst}\left (\int \frac {1}{x} \, dx,x,\tan (e+f x)\right )}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ & = \frac {\cos (e+f x) \log (\tan (e+f x))}{f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \\ \end{align*}
Time = 0.11 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.37 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {(\log (\cos (e+f x))-\log (\sin (e+f x))) \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)}}{a c f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(90\) vs. \(2(42)=84\).
Time = 1.79 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.98
method | result | size |
default | \(\frac {\cos \left (f x +e \right ) \left (\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right )-\ln \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )-1\right )-\ln \left (-\cot \left (f x +e \right )+\csc \left (f x +e \right )+1\right )\right )}{f \sqrt {a \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {-c \left (\sin \left (f x +e \right )-1\right )}}\) | \(91\) |
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none
Time = 0.33 (sec) , antiderivative size = 193, normalized size of antiderivative = 4.20 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\left [\frac {\sqrt {a c} \log \left (-\frac {4 \, {\left (2 \, a c \cos \left (f x + e\right )^{5} - 2 \, a c \cos \left (f x + e\right )^{3} + a c \cos \left (f x + e\right ) - \sqrt {a c} {\left (2 \, \cos \left (f x + e\right )^{2} - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}\right )}}{\cos \left (f x + e\right )^{5} - \cos \left (f x + e\right )^{3}}\right )}{2 \, a c f}, \frac {\sqrt {-a c} \arctan \left (\frac {\sqrt {-a c} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{2 \, a c \cos \left (f x + e\right )^{3} - a c \cos \left (f x + e\right )}\right )}{a c f}\right ] \]
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\[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{\sqrt {a \left (\sin {\left (e + f x \right )} + 1\right )} \sqrt {- c \left (\sin {\left (e + f x \right )} - 1\right )} \sin {\left (e + f x \right )}}\, dx \]
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Result contains complex when optimal does not.
Time = 0.40 (sec) , antiderivative size = 190, normalized size of antiderivative = 4.13 \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=-\frac {\left (-1\right )^{4 \, \cos \left (2 \, f x + 2 \, e\right )} \cosh \left (4 \, \pi \sin \left (2 \, f x + 2 \, e\right )\right ) \log \left (\frac {16 \, {\left (\cos \left (2 \, f x + 2 \, e\right )^{2} + \sin \left (2 \, f x + 2 \, e\right )^{2} + 2 \, \cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{a c {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}^{2}}\right ) - 2 i \, \left (-1\right )^{4 \, \cos \left (2 \, f x + 2 \, e\right )} \arctan \left (\frac {4 \, \sin \left (2 \, f x + 2 \, e\right )}{\sqrt {a} \sqrt {c} {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}}, \frac {4 \, {\left (\cos \left (2 \, f x + 2 \, e\right ) + 1\right )}}{\sqrt {a} \sqrt {c} {\left | e^{\left (2 i \, f x + 2 i \, e\right )} - 1 \right |}}\right ) \sinh \left (4 \, \pi \sin \left (2 \, f x + 2 \, e\right )\right )}{2 \, \sqrt {a} \sqrt {c} f} \]
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Exception generated. \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {\csc (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}} \, dx=\int \frac {1}{\sin \left (e+f\,x\right )\,\sqrt {a+a\,\sin \left (e+f\,x\right )}\,\sqrt {c-c\,\sin \left (e+f\,x\right )}} \,d x \]
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