Integrand size = 25, antiderivative size = 162 \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {52 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {4 e^3 \sqrt {e \sin (c+d x)}}{a^2 d}+\frac {26 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a^2 d}+\frac {4 e (e \sin (c+d x))^{5/2}}{5 a^2 d} \]
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Time = 0.68 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3957, 2954, 2952, 2649, 2721, 2720, 2644, 14} \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {52 e^4 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {4 e^3 \sqrt {e \sin (c+d x)}}{a^2 d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a^2 d}+\frac {26 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}+\frac {4 e (e \sin (c+d x))^{5/2}}{5 a^2 d} \]
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Rule 14
Rule 2644
Rule 2649
Rule 2720
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))^{7/2}}{(-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}{\sqrt {e \sin (c+d x)}} \, dx}{a^4} \\ & = \frac {e^4 \int \left (\frac {a^2 \cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}}-\frac {2 a^2 \cos ^3(c+d x)}{\sqrt {e \sin (c+d x)}}+\frac {a^2 \cos ^4(c+d x)}{\sqrt {e \sin (c+d x)}}\right ) \, dx}{a^4} \\ & = \frac {e^4 \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{a^2}+\frac {e^4 \int \frac {\cos ^4(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{a^2}-\frac {\left (2 e^4\right ) \int \frac {\cos ^3(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{a^2} \\ & = \frac {2 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a^2 d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a^2 d}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \frac {1-\frac {x^2}{e^2}}{\sqrt {x}} \, dx,x,e \sin (c+d x)\right )}{a^2 d}+\frac {\left (2 e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a^2}+\frac {\left (6 e^4\right ) \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{7 a^2} \\ & = \frac {26 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a^2 d}-\frac {\left (2 e^3\right ) \text {Subst}\left (\int \left (\frac {1}{\sqrt {x}}-\frac {x^{3/2}}{e^2}\right ) \, dx,x,e \sin (c+d x)\right )}{a^2 d}+\frac {\left (4 e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{7 a^2}+\frac {\left (2 e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a^2 \sqrt {e \sin (c+d x)}} \\ & = \frac {4 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a^2 d \sqrt {e \sin (c+d x)}}-\frac {4 e^3 \sqrt {e \sin (c+d x)}}{a^2 d}+\frac {26 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a^2 d}+\frac {4 e (e \sin (c+d x))^{5/2}}{5 a^2 d}+\frac {\left (4 e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{7 a^2 \sqrt {e \sin (c+d x)}} \\ & = \frac {52 e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d \sqrt {e \sin (c+d x)}}-\frac {4 e^3 \sqrt {e \sin (c+d x)}}{a^2 d}+\frac {26 e^3 \cos (c+d x) \sqrt {e \sin (c+d x)}}{21 a^2 d}+\frac {2 e^3 \cos ^3(c+d x) \sqrt {e \sin (c+d x)}}{7 a^2 d}+\frac {4 e (e \sin (c+d x))^{5/2}}{5 a^2 d} \\ \end{align*}
Time = 2.54 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.58 \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {e^3 \left (520 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )+(756-305 \cos (c+d x)+84 \cos (2 (c+d x))-15 \cos (3 (c+d x))) \sqrt {\sin (c+d x)}\right ) \sqrt {e \sin (c+d x)}}{210 a^2 d \sqrt {\sin (c+d x)}} \]
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Time = 6.87 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {2 e^{4} \left (-15 \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+65 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+42 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )-65 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )+168 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{105 a^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}\) | \(145\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.77 \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\frac {2 \, {\left (65 \, \sqrt {2} \sqrt {-i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 65 \, \sqrt {2} \sqrt {i \, e} e^{3} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (15 \, e^{3} \cos \left (d x + c\right )^{3} - 42 \, e^{3} \cos \left (d x + c\right )^{2} + 65 \, e^{3} \cos \left (d x + c\right ) - 168 \, e^{3}\right )} \sqrt {e \sin \left (d x + c\right )}\right )}}{105 \, a^{2} d} \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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