Integrand size = 25, antiderivative size = 268 \[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {4 e^2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{231 a^2 d}+\frac {16 e^2 \cot (c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{77 a^2 d}-\frac {2 e^2 \cot ^3(c+d x) \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{11 a^2 d}-\frac {4 e^2 \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{7 a^2 d}-\frac {2 e^2 \cot (c+d x) \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{11 a^2 d}+\frac {4 e^2 \csc ^5(c+d x) \sqrt {e \csc (c+d x)}}{11 a^2 d}+\frac {4 e^2 \sqrt {e \csc (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{231 a^2 d} \]
-4/231*e^2*cot(d*x+c)*(e*csc(d*x+c))^(1/2)/a^2/d+16/77*e^2*cot(d*x+c)*csc( d*x+c)^2*(e*csc(d*x+c))^(1/2)/a^2/d-2/11*e^2*cot(d*x+c)^3*csc(d*x+c)^2*(e* csc(d*x+c))^(1/2)/a^2/d-4/7*e^2*csc(d*x+c)^3*(e*csc(d*x+c))^(1/2)/a^2/d-2/ 11*e^2*cot(d*x+c)*csc(d*x+c)^4*(e*csc(d*x+c))^(1/2)/a^2/d+4/11*e^2*csc(d*x +c)^5*(e*csc(d*x+c))^(1/2)/a^2/d-4/231*e^2*(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 = 2.11 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.43 \[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {e^3 \csc ^2\left (\frac {1}{2} (c+d x)\right ) \sec ^6\left (\frac {1}{2} (c+d x)\right ) \left (52+97 \cos (c+d x)+4 \cos (2 (c+d x))+\cos (3 (c+d x))+\csc ^4\left (\frac {1}{2} (c+d x)\right ) \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right ) \sin ^{\frac {11}{2}}(c+d x)\right )}{3696 a^2 d \sqrt {e \csc (c+d x)}} \]
-1/3696*(e^3*Csc[(c + d*x)/2]^2*Sec[(c + d*x)/2]^6*(52 + 97*Cos[c + d*x] + 4*Cos[2*(c + d*x)] + Cos[3*(c + d*x)] + Csc[(c + d*x)/2]^4*EllipticF[(-2* c + Pi - 2*d*x)/4, 2]*Sin[c + d*x]^(11/2)))/(a^2*d*Sqrt[e*Csc[c + d*x]])
Time = 0.86 (sec) , antiderivative size = 202, normalized size of antiderivative = 0.75, 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.
| \(\displaystyle \int \frac {(e \csc (c+d x))^{5/2}}{(a \sec (c+d x)+a)^2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{5/2}}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{(\sec (c+d x) a+a)^2 \sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos ^2(c+d x)}{(-\cos (c+d x) a-a)^2 \sin ^{\frac {5}{2}}(c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{5/2} \left (a \sin \left (c+d x-\frac {\pi }{2}\right )-a\right )^2}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\cos ^2(c+d x) (a-a \cos (c+d x))^2}{\sin ^{\frac {13}{2}}(c+d x)}dx}{a^4}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^2}{\cos \left (c+d x-\frac {\pi }{2}\right )^{13/2}}dx}{a^4}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \int \left (\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {13}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {13}{2}}(c+d x)}+\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {13}{2}}(c+d x)}\right )dx}{a^4}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {e^2 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)} \left (-\frac {4 a^2}{7 d \sin ^{\frac {7}{2}}(c+d x)}+\frac {4 a^2}{11 d \sin ^{\frac {11}{2}}(c+d x)}+\frac {4 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{231 d}-\frac {2 a^2 \cos ^3(c+d x)}{11 d \sin ^{\frac {11}{2}}(c+d x)}-\frac {4 a^2 \cos (c+d x)}{231 d \sin ^{\frac {3}{2}}(c+d x)}+\frac {16 a^2 \cos (c+d x)}{77 d \sin ^{\frac {7}{2}}(c+d x)}-\frac {2 a^2 \cos (c+d x)}{11 d \sin ^{\frac {11}{2}}(c+d x)}\right )}{a^4}\) |
(e^2*Sqrt[e*Csc[c + d*x]]*((4*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(231*d ) + (4*a^2)/(11*d*Sin[c + d*x]^(11/2)) - (2*a^2*Cos[c + d*x])/(11*d*Sin[c + d*x]^(11/2)) - (2*a^2*Cos[c + d*x]^3)/(11*d*Sin[c + d*x]^(11/2)) - (4*a^ 2)/(7*d*Sin[c + d*x]^(7/2)) + (16*a^2*Cos[c + d*x])/(77*d*Sin[c + d*x]^(7/ 2)) - (4*a^2*Cos[c + d*x])/(231*d*Sin[c + d*x]^(3/2)))*Sqrt[Sin[c + d*x]]) /a^4
3.3.100.3.1 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 = 9.06 (sec) , antiderivative size = 382, normalized size of antiderivative = 1.43
| method | result | size |
| default | \(\frac {\sqrt {2}\, {\left (\frac {e \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )+\sin \left (d x +c \right )\right )}{1-\cos \left (d x +c \right )}\right )}^{\frac {5}{2}} \left (1-\cos \left (d x +c \right )\right )^{2} \left (21 \left (1-\cos \left (d x +c \right )\right )^{8} \csc \left (d x +c \right )^{8}+16 i \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {2}\, \sqrt {-i \left (i+\cot \left (d x +c \right )-\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )-6 \left (1-\cos \left (d x +c \right )\right )^{6} \csc \left (d x +c \right )^{6}-136 \left (1-\cos \left (d x +c \right )\right )^{4} \csc \left (d x +c \right )^{4}-186 \left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}-77\right ) \csc \left (d x +c \right )^{2}}{1848 a^{2} d \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right )^{2} \sqrt {\left (1-\cos \left (d x +c \right )\right ) \left (\left (1-\cos \left (d x +c \right )\right )^{2} \csc \left (d x +c \right )^{2}+1\right ) \csc \left (d x +c \right )}\, \sqrt {\left (1-\cos \left (d x +c \right )\right )^{3} \csc \left (d x +c \right )^{3}+\csc \left (d x +c \right )-\cot \left (d x +c \right )}}\) | \(382\) |
1/1848/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)))^(5/2)*(1-cos(d*x+c))^2*(21*(1-cos(d*x+c))^8*csc(d*x+c)^8+16*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))*(-cot(d*x+c)+csc(d*x+c))-6*(1-cos(d*x+c))^6*csc(d* x+c)^6-136*(1-cos(d*x+c))^4*csc(d*x+c)^4-186*(1-cos(d*x+c))^2*csc(d*x+c)^2 -77)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1)^2/((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)^2
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.82 \[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=-\frac {2 \, {\left ({\left (i \, e^{2} \cos \left (d x + c\right )^{2} + 2 i \, e^{2} \cos \left (d x + c\right ) + i \, e^{2}\right )} \sqrt {2 i \, e} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + {\left (-i \, e^{2} \cos \left (d x + c\right )^{2} - 2 i \, e^{2} \cos \left (d x + c\right ) - i \, e^{2}\right )} \sqrt {-2 i \, e} \sin \left (d x + c\right ) {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + {\left (2 \, e^{2} \cos \left (d x + c\right )^{3} + 4 \, e^{2} \cos \left (d x + c\right )^{2} + 47 \, e^{2} \cos \left (d x + c\right ) + 24 \, e^{2}\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}}\right )}}{231 \, {\left (a^{2} d \cos \left (d x + c\right )^{2} + 2 \, a^{2} d \cos \left (d x + c\right ) + a^{2} d\right )} \sin \left (d x + c\right )} \]
-2/231*((I*e^2*cos(d*x + c)^2 + 2*I*e^2*cos(d*x + c) + I*e^2)*sqrt(2*I*e)* sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) + I*sin(d*x + c)) + (- I*e^2*cos(d*x + c)^2 - 2*I*e^2*cos(d*x + c) - I*e^2)*sqrt(-2*I*e)*sin(d*x + c)*weierstrassPInverse(4, 0, cos(d*x + c) - I*sin(d*x + c)) + (2*e^2*cos (d*x + c)^3 + 4*e^2*cos(d*x + c)^2 + 47*e^2*cos(d*x + c) + 24*e^2)*sqrt(e/ sin(d*x + c)))/((a^2*d*cos(d*x + c)^2 + 2*a^2*d*cos(d*x + c) + a^2*d)*sin( d*x + c))
Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\int { \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {5}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {(e \csc (c+d x))^{5/2}}{(a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{5/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]