Integrand size = 25, antiderivative size = 187 \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {44 e^2 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d} \]
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Time = 0.74 (sec) , antiderivative size = 187, 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 = {3957, 2954, 2952, 2647, 2721, 2719, 2644, 14, 2649} \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {44 e^2 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d} \]
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Rule 14
Rule 2644
Rule 2647
Rule 2649
Rule 2719
Rule 2721
Rule 2952
Rule 2954
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x) (e \sin (c+d x))^{5/2}}{(-a-a \cos (c+d x))^2} \, dx \\ & = \frac {e^4 \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{3/2}} \, dx}{a^4} \\ & = \frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{3/2}}-\frac {2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{3/2}}+\frac {a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{3/2}}\right ) \, dx}{a^4} \\ & = \frac {e^4 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{a^2} \\ & = -\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {\left (2 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{a^2}-\frac {\left (6 e^2\right ) \int \cos ^2(c+d x) \sqrt {e \sin (c+d x)} \, dx}{a^2}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{x^{3/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}-\frac {\left (12 e^2\right ) \int \sqrt {e \sin (c+d x)} \, dx}{5 a^2}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^{3/2}}-\frac {\sqrt {x}}{e^2}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}-\frac {\left (2 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{a^2 \sqrt {\sin (c+d x)}} \\ & = \frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \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)}}{a^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d}-\frac {\left (12 e^2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {\sin (c+d x)}} \\ & = \frac {4 e^3}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos (c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e^3 \cos ^3(c+d x)}{a^2 d \sqrt {e \sin (c+d x)}}-\frac {44 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^2 d \sqrt {\sin (c+d x)}}+\frac {4 e (e \sin (c+d x))^{3/2}}{3 a^2 d}-\frac {12 e \cos (c+d x) (e \sin (c+d x))^{3/2}}{5 a^2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 7.28 (sec) , antiderivative size = 446, normalized size of antiderivative = 2.39 \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \csc ^2(c+d x) \sec ^2(c+d x) \left (\frac {16 \cos (d x) \sin (c)}{3 d}-\frac {16 \sec \left (\frac {c}{2}\right ) \sec (c) \left (8 \sin \left (\frac {c}{2}\right )+3 \sin \left (\frac {3 c}{2}\right )\right )}{5 d}-\frac {4 \cos (2 d x) \sin (2 c)}{5 d}+\frac {16 \sec \left (\frac {c}{2}\right ) \sec \left (\frac {c}{2}+\frac {d x}{2}\right ) \sin \left (\frac {d x}{2}\right )}{d}+\frac {16 \cos (c) \sin (d x)}{3 d}-\frac {4 \cos (2 c) \sin (2 d x)}{5 d}\right ) (e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2}-\frac {88 \cos ^4\left (\frac {c}{2}+\frac {d x}{2}\right ) \sec (c) \sec ^2(c+d x) (e \sin (c+d x))^{5/2} \left (-\frac {\cot (c) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x-\arctan (\cot (c)))\right ) \sin (d x-\arctan (\cot (c)))}{\sqrt {1-\cos (d x-\arctan (\cot (c)))} \sqrt {1+\cos (d x-\arctan (\cot (c)))} \sqrt {1+\cot ^2(c)} \sqrt {\cos (d x-\arctan (\cot (c))) \sqrt {1+\cot ^2(c)} \sin (c)}}-\frac {\frac {2 \cos (d x-\arctan (\cot (c))) \sqrt {1+\cot ^2(c)} \sin ^2(c)}{\cos ^2(c)+\sin ^2(c)}-\frac {\cot (c) \sin (d x-\arctan (\cot (c)))}{\sqrt {1+\cot ^2(c)}}}{\sqrt {\cos (d x-\arctan (\cot (c))) \sqrt {1+\cot ^2(c)} \sin (c)}}\right )}{5 d (a+a \sec (c+d x))^2 \sin ^{\frac {5}{2}}(c+d x)} \]
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Time = 7.11 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.93
method | result | size |
default | \(\frac {2 e^{3} \left (66 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-33 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+3 \cos \left (d x +c \right )^{4}-10 \cos \left (d x +c \right )^{3}-33 \cos \left (d x +c \right )^{2}+40 \cos \left (d x +c \right )\right )}{15 a^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(173\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.91 \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left ({\left (3 \, e^{2} \cos \left (d x + c\right )^{2} - 7 \, e^{2} \cos \left (d x + c\right ) - 40 \, e^{2}\right )} \sqrt {e \sin \left (d x + c\right )} \sin \left (d x + c\right ) + 33 \, {\left (i \, \sqrt {2} e^{2} \cos \left (d x + c\right ) + i \, \sqrt {2} e^{2}\right )} \sqrt {-i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 33 \, {\left (-i \, \sqrt {2} e^{2} \cos \left (d x + c\right ) - i \, \sqrt {2} e^{2}\right )} \sqrt {i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{15 \, {\left (a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{5/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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