Integrand size = 25, antiderivative size = 201 \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {16 \cot (c+d x) \sqrt {e \csc (c+d x)}}{21 a^2 d}-\frac {2 \cot ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {4 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a^2 d}-\frac {2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}+\frac {20 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^2 d} \]
16/21*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/a^2/d-2/7*cot(d*x+c)^3*(e*csc(d*x+c) )^(1/2)/a^2/d-4/3*csc(d*x+c)*(e*csc(d*x+c))^(1/2)/a^2/d-2/7*cot(d*x+c)*csc (d*x+c)^2*(e*csc(d*x+c))^(1/2)/a^2/d+4/7*csc(d*x+c)^3*(e*csc(d*x+c))^(1/2) /a^2/d-20/21*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/sin(1/2*c+1/4*Pi+1/2*d*x) *EllipticF(cos(1/2*c+1/4*Pi+1/2*d*x),2^(1/2))*(e*csc(d*x+c))^(1/2)*sin(d*x +c)^(1/2)/a^2/d
Time = 1.44 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.41 \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 \csc ^3(c+d x) \sqrt {e \csc (c+d x)} \left (2 (8+11 \cos (c+d x)) \sin ^4\left (\frac {1}{2} (c+d x)\right )+5 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {7}{2}}(c+d x)\right )}{21 a^2 d} \]
(-4*Csc[c + d*x]^3*Sqrt[e*Csc[c + d*x]]*(2*(8 + 11*Cos[c + d*x])*Sin[(c + d*x)/2]^4 + 5*EllipticF[(-2*c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(7/2)))/(21 *a^2*d)
Time = 0.80 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.86, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3042, 4366, 3042, 4360, 3042, 3354, 3042, 3352, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sqrt {e \csc (c+d x)}}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sqrt {e \sec \left (c+d x-\frac {\pi }{2}\right )}}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{(\sec (c+d x) a+a)^2 \sqrt {\sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos ^2(c+d x)}{(-\cos (c+d x) a-a)^2 \sqrt {\sin (c+d x)}}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2}{\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )} \left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos ^2(c+d x) (a-a \cos (c+d x))^2}{\sin ^{\frac {9}{2}}(c+d x)}dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{9/2}}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \left (\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}+\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {9}{2}}(c+d x)}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {4 a^2}{3 d \sin ^{\frac {3}{2}}(c+d x)}+\frac {4 a^2}{7 d \sin ^{\frac {7}{2}}(c+d x)}+\frac {20 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{21 d}-\frac {2 a^2 \cos ^3(c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}+\frac {16 a^2 \cos (c+d x)}{21 d \sin ^{\frac {3}{2}}(c+d x)}-\frac {2 a^2 \cos (c+d x)}{7 d \sin ^{\frac {7}{2}}(c+d x)}\right )}{a^4}\) |
(Sqrt[e*Csc[c + d*x]]*((20*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(21*d) + (4*a^2)/(7*d*Sin[c + d*x]^(7/2)) - (2*a^2*Cos[c + d*x])/(7*d*Sin[c + d*x]^ (7/2)) - (2*a^2*Cos[c + d*x]^3)/(7*d*Sin[c + d*x]^(7/2)) - (4*a^2)/(3*d*Si n[c + d*x]^(3/2)) + (16*a^2*Cos[c + d*x])/(21*d*Sin[c + d*x]^(3/2)))*Sqrt[ Sin[c + d*x]])/a^4
3.4.2.3.1 Defintions of rubi rules used
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Int[ExpandTrig [(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x] /; F reeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(a/g)^(2* m) Int[(g*Cos[e + f*x])^(2*m + p)*((d*Sin[e + f*x])^n/(a - b*Sin[e + f*x] )^m), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && ILtQ[m, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.), x_Symbol] :> Int[(g*Cos[e + f*x])^p*((b + a*Sin[e + f*x])^m/Si n[e + f*x]^m), x] /; FreeQ[{a, b, e, f, g, p}, x] && IntegerQ[m]
Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_.)*((g_.)*sec[(e_.) + (f_.)*( x_)])^(p_), x_Symbol] :> Simp[g^IntPart[p]*(g*Sec[e + f*x])^FracPart[p]*Cos [e + f*x]^FracPart[p] Int[(a + b*Csc[e + f*x])^m/Cos[e + f*x]^p, x], x] / ; FreeQ[{a, b, e, f, g, m, p}, x] && !IntegerQ[p]
Result contains complex when optimal does not.
Time = 8.06 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.54
| method | result | size |
| default | \(\frac {\sqrt {2}\, \sqrt {\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}}\, \left (1-\cos \left (d x +c \right )\right ) \left (20 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+3 \left (1-\cos \left (d x +c \right )\right )^{5} \csc \left (d x +c \right )^{5}-16 \left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}-19 \csc \left (d x +c \right )+19 \cot \left (d x +c \right )\right ) \csc \left (d x +c \right )}{42 a^{2} d \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\) | \(309\) |
1/42/a^2/d*2^(1/2)*(e/(1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)+sin(d*x+ c)))^(1/2)*(1-cos(d*x+c))*(20*I*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2)*2^(1/ 2)*(-I*(I+cot(d*x+c)-csc(d*x+c)))^(1/2)*(I*(-cot(d*x+c)+csc(d*x+c)))^(1/2) *EllipticF((-I*(I-cot(d*x+c)+csc(d*x+c)))^(1/2),1/2*2^(1/2))+3*(1-cos(d*x+ c))^5*csc(d*x+c)^5-16*(1-cos(d*x+c))^3*csc(d*x+c)^3-19*csc(d*x+c)+19*cot(d *x+c))/((1-cos(d*x+c))*((1-cos(d*x+c))^2*csc(d*x+c)^2+1)*csc(d*x+c))^(1/2) /((1-cos(d*x+c))^3*csc(d*x+c)^3+csc(d*x+c)-cot(d*x+c))^(1/2)*csc(d*x+c)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left (\sqrt {\frac {e}{\sin \left (d x + c\right )}} {\left (11 \, \cos \left (d x + c\right ) + 8\right )} \sin \left (d x + c\right ) + 5 \, {\left (i \, \cos \left (d x + c\right )^{2} + 2 i \, \cos \left (d x + c\right ) + i\right )} \sqrt {2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 5 \, {\left (-i \, \cos \left (d x + c\right )^{2} - 2 i \, \cos \left (d x + c\right ) - i\right )} \sqrt {-2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{21 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )}} \]
-2/21*(sqrt(e/sin(d*x + c))*(11*cos(d*x + c) + 8)*sin(d*x + c) + 5*(I*cos( d*x + c)^2 + 2*I*cos(d*x + c) + I)*sqrt(2*I*e)*weierstrassPInverse(4, 0, c os(d*x + c) + I*sin(d*x + c)) + 5*(-I*cos(d*x + c)^2 - 2*I*cos(d*x + c) - I)*sqrt(-2*I*e)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)))/ (a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)
\[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\frac {\int \frac {\sqrt {e \csc {\left (c + d x \right )}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Timed out. \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\sqrt {e \csc \left (d x + c\right )}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {e \csc (c+d x)}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]