Integrand size = 25, antiderivative size = 120 \[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=-\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{5 a d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 \sin (c+d x)}{3 a d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \sin (c+d x)}{5 a d e^2 \sqrt {e \csc (c+d x)}} \]
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Time = 0.37 (sec) , antiderivative size = 120, 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, 2649, 2719} \[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\frac {2 \sin (c+d x)}{3 a d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 \sin (c+d x) \cos (c+d x)}{5 a d e^2 \sqrt {e \csc (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{5 a d e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}} \]
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Rule 30
Rule 2644
Rule 2649
Rule 2719
Rule 2918
Rule 3957
Rule 3963
Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\sin ^{\frac {5}{2}}(c+d x)}{a+a \sec (c+d x)} \, dx}{e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {\int \frac {\cos (c+d x) \sin ^{\frac {5}{2}}(c+d x)}{-a-a \cos (c+d x)} \, dx}{e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = \frac {\int \cos (c+d x) \sqrt {\sin (c+d x)} \, dx}{a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}-\frac {\int \cos ^2(c+d x) \sqrt {\sin (c+d x)} \, dx}{a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {2 \cos (c+d x) \sin (c+d x)}{5 a d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 \int \sqrt {\sin (c+d x)} \, dx}{5 a e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {\text {Subst}\left (\int \sqrt {x} \, dx,x,\sin (c+d x)\right )}{a d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}} \\ & = -\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right )}{5 a d e^2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {2 \sin (c+d x)}{3 a d e^2 \sqrt {e \csc (c+d x)}}-\frac {2 \cos (c+d x) \sin (c+d x)}{5 a d e^2 \sqrt {e \csc (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.54 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.83 \[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\frac {8 \sqrt {1-e^{2 i (c+d x)}} (i+\cot (c+d x)) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )+20 \sin (c+d x)-6 (4 i+\sin (2 (c+d x)))}{30 a d e^2 \sqrt {e \csc (c+d x)}} \]
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Result contains complex when optimal does not.
Time = 8.59 (sec) , antiderivative size = 462, normalized size of antiderivative = 3.85
method | result | size |
default | \(\frac {\sqrt {2}\, \left (12 \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \cos \left (d x +c \right )-6 \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+12 \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticE}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}-6 \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {i \left (-i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+3 \cos \left (d x +c \right )^{3} \sqrt {2}-5 \sqrt {2}\, \cos \left (d x +c \right )^{2}+3 \sqrt {2}\, \cos \left (d x +c \right )-\sqrt {2}\right ) \csc \left (d x +c \right )}{15 a d \sqrt {e \csc \left (d x +c \right )}\, e^{2}}\) | \(462\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.92 \[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\frac {2 \, {\left ({\left (3 \, \cos \left (d x + c\right )^{3} - 5 \, \cos \left (d x + c\right )^{2} - 3 \, \cos \left (d x + c\right ) + 5\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} - 3 \, \sqrt {2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 \, \sqrt {-2 i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{15 \, a d e^{3}} \]
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Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{5/2} (a+a \sec (c+d x))} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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