Integrand size = 25, antiderivative size = 104 \[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {4 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a d \sqrt {\sin (c+d x)}}+\frac {2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d} \]
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Time = 0.29 (sec) , antiderivative size = 104, 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 = {3957, 2918, 2644, 30, 2649, 2721, 2719} \[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {4 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a d \sqrt {\sin (c+d x)}}+\frac {2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d} \]
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Rule 30
Rule 2644
Rule 2649
Rule 2719
Rule 2721
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) (e \sin (c+d x))^{5/2}}{-a-a \cos (c+d x)} \, dx \\ & = \frac {e^2 \int \cos (c+d x) \sqrt {e \sin (c+d x)} \, dx}{a}-\frac {e^2 \int \cos ^2(c+d x) \sqrt {e \sin (c+d x)} \, dx}{a} \\ & = -\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d}+\frac {e \text {Subst}\left (\int \sqrt {x} \, dx,x,e \sin (c+d x)\right )}{a d}-\frac {\left (2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 a} \\ & = \frac {2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d}-\frac {\left (2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a \sqrt {\sin (c+d x)}} \\ & = -\frac {4 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a d \sqrt {\sin (c+d x)}}+\frac {2 e (e \sin (c+d x))^{3/2}}{3 a d}-\frac {2 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 3.19 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.86 \[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\frac {2 e^3 \cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) \left (10 \sqrt {\csc ^2(c)} \sin ^2(c+d x)+6 \csc (c) \csc (d x-\arctan (\cot (c))) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x-\arctan (\cot (c)))\right ) \sqrt {\sin ^2(d x-\arctan (\cot (c)))}+3 \csc (c) \sec (c) (\sin (c+d x-\arctan (\cot (c)))+3 \sin (c-d x+\arctan (\cot (c))))-3 \sqrt {\csc ^2(c)} \sin (c+d x) (\sin (2 (c+d x))+4 \tan (c))\right )}{15 a d \sqrt {\csc ^2(c)} (1+\sec (c+d x)) \sqrt {e \sin (c+d x)}} \]
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Time = 5.36 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.66
method | result | size |
default | \(\frac {2 e^{3} \left (6 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-3 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+3 \cos \left (d x +c \right )^{4}-5 \cos \left (d x +c \right )^{3}-3 \cos \left (d x +c \right )^{2}+5 \cos \left (d x +c \right )\right )}{15 a \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(173\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07 \[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=-\frac {2 \, {\left (3 i \, \sqrt {2} \sqrt {-i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) - 3 i \, \sqrt {2} \sqrt {i \, e} e^{2} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (3 \, e^{2} \cos \left (d x + c\right ) - 5 \, e^{2}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right )\right )}}{15 \, a d} \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{a \sec \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{a+a \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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