Integrand size = 25, antiderivative size = 250 \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{15 a^2 d} \]
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Time = 0.64 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3963, 3957, 2954, 2952, 2647, 2716, 2719, 2644, 14} \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {4 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{15 a^2 d} \]
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Rule 14
Rule 2644
Rule 2647
Rule 2716
Rule 2719
Rule 2952
Rule 2954
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x))^2 \sin ^{\frac {3}{2}}(c+d x)} \, dx \\ & = \left (e \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 {3}{2}}(c+d x)} \, dx \\ & = \frac {\left (e \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 {11}{2}}(c+d x)} \, dx}{a^4} \\ & = \frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}\right ) \, dx}{a^4} \\ & = \frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^2}+\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^4(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^2} \\ & = -\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{9 a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{3 a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \frac {1-x^2}{x^{11/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d} \\ & = \frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^2}+\frac {\left (4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \text {Subst}\left (\int \left (\frac {1}{x^{11/2}}-\frac {1}{x^{7/2}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d} \\ & = -\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2}-\frac {\left (4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2} \\ & = -\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{15 a^2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 2.36 (sec) , antiderivative size = 247, normalized size of antiderivative = 0.99 \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{3/2} \sec (c+d x) \left (\frac {16 \sqrt {2} e^{i (c-d x)} \sqrt {\frac {i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left (3-3 e^{2 i (c+d x)}+e^{2 i d x} \left (1+e^{2 i c}\right ) \sqrt {1-e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )\right ) \sec (c+d x)}{d \left (1+e^{2 i c}\right ) \csc ^{\frac {3}{2}}(c+d x)}-\frac {2 \left (24 \cos (d x) \sec (c)+(8+13 \cos (c+d x)) \sec ^4\left (\frac {1}{2} (c+d x)\right )\right ) \tan (c+d x)}{d}\right )}{45 a^2 (1+\sec (c+d x))^2} \]
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Result contains complex when optimal does not.
Time = 9.25 (sec) , antiderivative size = 859, normalized size of antiderivative = 3.44
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 182, normalized size of antiderivative = 0.73 \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (3 \, {\left (e \cos \left (d x + c\right )^{2} + 2 \, e \cos \left (d x + c\right ) + e\right )} \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (e \cos \left (d x + c\right )^{2} + 2 \, e \cos \left (d x + c\right ) + e\right )} \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (6 \, e \cos \left (d x + c\right )^{3} + 12 \, e \cos \left (d x + c\right )^{2} + 19 \, e \cos \left (d x + c\right ) + 8 \, e\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}\right )}}{45 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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Timed out. \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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