Integrand size = 25, antiderivative size = 188 \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 e^3}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 e}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {16 e \cos (c+d x)}{5 a^2 d \sqrt {e \sin (c+d x)}}+\frac {28 E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}} \]
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Time = 0.70 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 15, 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} \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {4 e^3}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 e}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {16 e \cos (c+d x)}{5 a^2 d \sqrt {e \sin (c+d x)}}+\frac {28 E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{5 a^2 d \sqrt {\sin (c+d x)}} \]
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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) \sqrt {e \sin (c+d x)}}{(-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))^{7/2}} \, dx}{a^4} \\ & = \frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{(e \sin (c+d x))^{7/2}}-\frac {2 a^2 \cos ^3(c+d x)}{(e \sin (c+d x))^{7/2}}+\frac {a^2 \cos ^4(c+d x)}{(e \sin (c+d x))^{7/2}}\right ) \, dx}{a^4} \\ & = \frac {e^4 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{(e \sin (c+d x))^{7/2}} \, dx}{a^2} \\ & = -\frac {2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {\left (2 e^2\right ) \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx}{5 a^2}-\frac {\left (6 e^2\right ) \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{3/2}} \, dx}{5 a^2}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{x^{7/2}} \, dx,x,e \sin (c+d x)\right )}{a^2 d} \\ & = -\frac {2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}+\frac {16 e \cos (c+d x)}{5 a^2 d \sqrt {e \sin (c+d x)}}+\frac {2 \int \sqrt {e \sin (c+d x)} \, dx}{5 a^2}+\frac {12 \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^{7/2}}-\frac {1}{e^2 x^{3/2}}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d} \\ & = \frac {4 e^3}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 e}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {16 e \cos (c+d x)}{5 a^2 d \sqrt {e \sin (c+d x)}}+\frac {\left (2 \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a^2 \sqrt {\sin (c+d x)}}+\frac {\left (12 \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}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos (c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {2 e^3 \cos ^3(c+d x)}{5 a^2 d (e \sin (c+d x))^{5/2}}-\frac {4 e}{a^2 d \sqrt {e \sin (c+d x)}}+\frac {16 e \cos (c+d x)}{5 a^2 d \sqrt {e \sin (c+d x)}}+\frac {28 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)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 6.27 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec ^2(c+d x) \sqrt {e \sin (c+d x)} \left (\sec (c) \sec ^3\left (\frac {1}{2} (c+d x)\right ) \left (49 \sin \left (\frac {1}{2} (c-d x)\right )+35 \sin \left (\frac {1}{2} (3 c+d x)\right )-23 \sin \left (\frac {1}{2} (c+3 d x)\right )+5 \sin \left (\frac {1}{2} (5 c+3 d x)\right )\right )-56 \sqrt {\csc ^2(c)} \csc (c+d x) \csc (d x-\arctan (\cot (c))) \, _2F_1\left (-\frac {1}{2},-\frac {1}{4};\frac {3}{4};\cos ^2(d x-\arctan (\cot (c)))\right ) \sin (c) \sqrt {\sin ^2(d x-\arctan (\cot (c)))}-28 \sqrt {\csc ^2(c)} \csc (c+d x) (\sin (c+d x-\arctan (\cot (c)))+3 \sin (c-d x+\arctan (\cot (c)))) \tan (c)\right )}{5 a^2 d (1+\sec (c+d x))^2} \]
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Time = 5.92 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.09
method | result | size |
default | \(\frac {-\frac {2 e \left (-\frac {2 e^{2}}{5 \left (e \sin \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {2}{\sqrt {e \sin \left (d x +c \right )}}\right )}{a^{2}}-\frac {2 e \left (14 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-7 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {7}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+9 \sin \left (d x +c \right )^{5}-11 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{5 a^{2} \sin \left (d x +c \right )^{3} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(205\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (\sqrt {e \sin \left (d x + c\right )} {\left (9 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right ) + 7 \, {\left (-i \, \sqrt {2} \cos \left (d x + c\right )^{2} - 2 i \, \sqrt {2} \cos \left (d x + c\right ) - i \, \sqrt {2}\right )} \sqrt {-i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 7 \, {\left (i \, \sqrt {2} \cos \left (d x + c\right )^{2} + 2 i \, \sqrt {2} \cos \left (d x + c\right ) + i \, \sqrt {2}\right )} \sqrt {i \, e} {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )\right )}}{5 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
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\[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\sqrt {e \sin {\left (c + d x \right )}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
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\[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\sqrt {e \sin \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {\sqrt {e \sin (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\sqrt {e\,\sin \left (c+d\,x\right )}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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