Integrand size = 25, antiderivative size = 101 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac {2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a d \sqrt {e \sin (c+d x)}} \]
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Time = 0.28 (sec) , antiderivative size = 101, 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, 2647, 2721, 2720} \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=-\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac {2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac {4 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 a d \sqrt {e \sin (c+d x)}} \]
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Rule 30
Rule 2644
Rule 2647
Rule 2720
Rule 2721
Rule 2918
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx \\ & = \frac {e^2 \int \frac {\cos (c+d x)}{(e \sin (c+d x))^{5/2}} \, dx}{a}-\frac {e^2 \int \frac {\cos ^2(c+d x)}{(e \sin (c+d x))^{5/2}} \, dx}{a} \\ & = \frac {2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac {2 \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a}+\frac {e \text {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,e \sin (c+d x)\right )}{a d} \\ & = -\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac {2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac {\left (2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a \sqrt {e \sin (c+d x)}} \\ & = -\frac {2 e}{3 a d (e \sin (c+d x))^{3/2}}+\frac {2 e \cos (c+d x)}{3 a d (e \sin (c+d x))^{3/2}}+\frac {4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 a d \sqrt {e \sin (c+d x)}} \\ \end{align*}
Time = 1.13 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\frac {2 \cot \left (\frac {1}{2} (c+d x)\right ) \left (-1+\cos (c+d x)-2 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {3}{2}}(c+d x)\right )}{3 a d (1+\cos (c+d x)) \sqrt {e \sin (c+d x)}} \]
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Time = 4.43 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.20
method | result | size |
default | \(\frac {-\frac {2 e}{3 a \left (e \sin \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2 \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sin \left (d x +c \right )^{\frac {5}{2}} \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+\sin \left (d x +c \right )^{3}-\sin \left (d x +c \right )\right )}{3 a \sin \left (d x +c \right )^{2} \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(121\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.07 \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\frac {2 \, {\left ({\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {-i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (\sqrt {2} \cos \left (d x + c\right ) + \sqrt {2}\right )} \sqrt {i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - \sqrt {e \sin \left (d x + c\right )}\right )}}{3 \, {\left (a d e \cos \left (d x + c\right ) + a d e\right )}} \]
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\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\frac {\int \frac {1}{\sqrt {e \sin {\left (c + d x \right )}} \sec {\left (c + d x \right )} + \sqrt {e \sin {\left (c + d x \right )}}}\, dx}{a} \]
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\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}} \,d x } \]
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\[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\int { \frac {1}{{\left (a \sec \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(a+a \sec (c+d x)) \sqrt {e \sin (c+d x)}} \, dx=\int \frac {\cos \left (c+d\,x\right )}{a\,\sqrt {e\,\sin \left (c+d\,x\right )}\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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