Integrand size = 25, antiderivative size = 224 \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d e^2 \sqrt {\sin (c+d x)}} \]
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Time = 0.81 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3957, 2954, 2952, 2647, 2716, 2721, 2719, 2644, 14} \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d e^2 \sqrt {\sin (c+d x)}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}} \]
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Rule 14
Rule 2644
Rule 2647
Rule 2716
Rule 2719
Rule 2721
Rule 2952
Rule 2954
Rule 3957
Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx \\ & = \frac {e^4 \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{(e \sin (c+d x))^{11/2}} \, dx}{a^4} \\ & = \frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{11/2}}-\frac {2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{11/2}}+\frac {a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{11/2}}\right ) \, dx}{a^4} \\ & = \frac {e^4 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{(e \sin (c+d x))^{11/2}} \, dx}{a^2} \\ & = -\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {\left (2 e^2\right ) \int \frac {1}{(e \sin (c+d x))^{7/2}} \, dx}{9 a^2}-\frac {\left (2 e^2\right ) \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{3 a^2}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{x^{11/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{15 a^2}+\frac {4 \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{15 a^2}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{x^{11/2}}-\frac {1}{e^2 x^{7/2}}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d} \\ & = \frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}+\frac {2 \int \sqrt {e \sin (c+d x)} \, dx}{15 a^2 e^2}-\frac {4 \int \sqrt {e \sin (c+d x)} \, dx}{15 a^2 e^2} \\ & = \frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}+\frac {\left (2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2 e^2 \sqrt {\sin (c+d x)}}-\frac {\left (4 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2 e^2 \sqrt {\sin (c+d x)}} \\ & = \frac {4 e^3}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos (c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {2 e^3 \cos ^3(c+d x)}{9 a^2 d (e \sin (c+d x))^{9/2}}-\frac {4 e}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{45 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 \cos (c+d x)}{15 a^2 d e \sqrt {e \sin (c+d x)}}-\frac {4 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{15 a^2 d e^2 \sqrt {\sin (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.99 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.73 \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\frac {\sec ^4\left (\frac {1}{2} (c+d x)\right ) (\cos (c+d x)+i \sin (c+d x)) \left (-31-40 \cos (c+d x)-19 \cos (2 (c+d x))+e^{-2 i (c+d x)} \left (1+e^{i (c+d x)}\right )^4 \sqrt {1-e^{2 i (c+d x)}} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},e^{2 i (c+d x)}\right )+16 i \sin (c+d x)+13 i \sin (2 (c+d x))\right )}{180 a^2 d e \sqrt {e \sin (c+d x)}} \]
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Time = 6.28 (sec) , antiderivative size = 213, normalized size of antiderivative = 0.95
method | result | size |
default | \(\frac {\frac {4 e^{3} \left (9 \cos \left (d x +c \right )^{2}-4\right )}{45 a^{2} \left (e \sin \left (d x +c \right )\right )^{\frac {9}{2}}}+\frac {\frac {4 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {11}{2}} \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}-\frac {2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {11}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )}{15}+\frac {4 \sin \left (d x +c \right )^{7}}{15}-\frac {38 \sin \left (d x +c \right )^{5}}{45}+\frac {46 \sin \left (d x +c \right )^{3}}{45}-\frac {4 \sin \left (d x +c \right )}{9}}{e \,a^{2} \sin \left (d x +c \right )^{5} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(213\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.99 \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=-\frac {2 \, {\left (3 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + {\left (6 \, \cos \left (d x + c\right )^{3} + 12 \, \cos \left (d x + c\right )^{2} + 19 \, \cos \left (d x + c\right ) + 8\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{45 \, {\left (a^{2} d e^{2} \cos \left (d x + c\right )^{2} + 2 \, a^{2} d e^{2} \cos \left (d x + c\right ) + a^{2} d e^{2}\right )} \sin \left (d x + c\right )} \]
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Timed out. \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \sec (c+d x))^2 (e \sin (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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