Integrand size = 25, antiderivative size = 99 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}-\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \]
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Time = 0.36 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3963, 3957, 2918, 2644, 30, 2647, 2719} \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=-\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{a d \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rule 30
Rule 2644
Rule 2647
Rule 2719
Rule 2918
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sqrt {\sin (c+d x)}}{a+a \sec (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {\int \frac {\cos (c+d x) \sqrt {\sin (c+d x)}}{-a-a \cos (c+d x)} \, dx}{\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \frac {\cos (c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{a \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x)}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{a \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {2 \int \sqrt {\sin (c+d x)} \, dx}{a \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\text {Subst}\left (\int \frac {1}{x^{3/2}} \, dx,x,\sin (c+d x)\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {2 \cot (c+d x)}{a d \sqrt {e \csc (c+d x)}}-\frac {2 \csc (c+d x)}{a d \sqrt {e \csc (c+d x)}}+\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{a d \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.15 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {6 (2 i+\cot (c+d x)-\csc (c+d x))-4 \sqrt {1-e^{2 i (c+d x)}} (i+\cot (c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )}{3 a d \sqrt {e \csc (c+d x)}} \]
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Result contains complex when optimal does not.
Time = 7.46 (sec) , antiderivative size = 432, normalized size of antiderivative = 4.36
method | result | size |
default | \(-\frac {\sqrt {2}\, \left (4 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )-2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+4 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )-2 \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+\sqrt {2}\, \cos \left (d x +c \right )-\sqrt {2}\right ) \csc \left (d x +c \right )}{a d \sqrt {e \csc \left (d x +c \right )}}\) | \(432\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.88 \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {2 \, {\left (\sqrt {\frac {e}{\sin \left (d x + c\right )}} {\left (\cos \left (d x + c\right ) - 1\right )} + \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{a d e} \]
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\[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\frac {\int \frac {1}{\sqrt {e \csc {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \csc {\left (c + d x \right )}}}\, dx}{a} \]
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\[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\sqrt {e \csc \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\sqrt {e \csc \left (d x + c\right )} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {e \csc (c+d x)} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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