Integrand size = 25, antiderivative size = 201 \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {20 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d} \]
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Time = 0.56 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3963, 3957, 2954, 2952, 2647, 2716, 2720, 2644, 14} \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}+\frac {20 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right ) \sqrt {e \csc (c+d x)}}{21 a^2 d} \]
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Rule 14
Rule 2644
Rule 2647
Rule 2716
Rule 2720
Rule 2952
Rule 2954
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x))^2 \sqrt {\sin (c+d x)}} \, dx \\ & = \left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 \sqrt {\sin (c+d x)}} \, dx \\ & = \frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^4} \\ & = \frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}\right ) \, dx}{a^4} \\ & = \frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^2}+\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^4(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^2}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)} \, dx}{a^2} \\ & = -\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{7 a^2}-\frac {\left (6 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{7 a^2}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{x^{9/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d} \\ & = \frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a^2}+\frac {\left (4 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{7 a^2}-\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {1}{x^{9/2}}-\frac {1}{x^{5/2}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d} \\ & = \frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {20 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d} \\ \end{align*}
Time = 1.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)} \left (2 (8+11 \cos (c+d x)) \sin ^4\left (\frac {1}{2} (c+d x)\right )+5 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {7}{2}}(c+d x)\right )}{21 a^2 d} \]
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Result contains complex when optimal does not.
Time = 8.06 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.54
method | result | size |
default | \(\frac {\sqrt {2}\, \sqrt {\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}}\, \left (1-\cos \left (d x +c \right )\right ) \left (20 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+3 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-16 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-19 \csc \left (d x +c \right )+19 \cot \left (d x +c \right )\right ) \csc \left (d x +c \right )}{42 a^{2} d \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\) | \(309\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (\sqrt {\frac {e}{\sin \left (d x + c\right )}} {\left (11 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right ) + 5 \, {\left (i \, \cos \left (d x + c\right )^{2} + 2 i \, \cos \left (d x + c\right ) + i\right )} \sqrt {2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (-i \, \cos \left (d x + c\right )^{2} - 2 i \, \cos \left (d x + c\right ) - i\right )} \sqrt {-2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{21 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\sqrt {e \csc {\left (c + d x \right )}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Timed out. \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\sqrt {e \csc \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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