Integrand size = 25, antiderivative size = 172 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=-\frac {4}{a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {26 \cos (c+d x)}{21 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {2 \cos ^3(c+d x)}{7 a^2 d e^3 \sqrt {e \csc (c+d x)}}+\frac {52 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right )}{21 a^2 d e^3 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}}+\frac {4 \sin ^2(c+d x)}{5 a^2 d e^3 \sqrt {e \csc (c+d x)}} \]
-4/a^2/d/e^3/(e*csc(d*x+c))^(1/2)+26/21*cos(d*x+c)/a^2/d/e^3/(e*csc(d*x+c) )^(1/2)+2/7*cos(d*x+c)^3/a^2/d/e^3/(e*csc(d*x+c))^(1/2)+4/5*sin(d*x+c)^2/a ^2/d/e^3/(e*csc(d*x+c))^(1/2)-52/21*(sin(1/2*c+1/4*Pi+1/2*d*x)^2)^(1/2)/si n(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))/a^2/d /e^3/(e*csc(d*x+c))^(1/2)/sin(d*x+c)^(1/2)
Time = 3.01 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.55 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\frac {\sqrt {e \csc (c+d x)} \left (-520 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )+(-756+305 \cos (c+d x)-84 \cos (2 (c+d x))+15 \cos (3 (c+d x))) \sqrt {\sin (c+d x)}\right ) \sqrt {\sin (c+d x)}}{210 a^2 d e^4} \]
(Sqrt[e*Csc[c + d*x]]*(-520*EllipticF[(-2*c + Pi - 2*d*x)/4, 2] + (-756 + 305*Cos[c + d*x] - 84*Cos[2*(c + d*x)] + 15*Cos[3*(c + d*x)])*Sqrt[Sin[c + d*x]])*Sqrt[Sin[c + d*x]])/(210*a^2*d*e^4)
Time = 0.81 (sec) , antiderivative size = 148, 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.
| \(\displaystyle \int \frac {1}{(a \sec (c+d x)+a)^2 (e \csc (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2 \left (e \sec \left (c+d x-\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4366 |
| \(\displaystyle \frac {\int \frac {\sin ^{\frac {7}{2}}(c+d x)}{(\sec (c+d x) a+a)^2}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^{7/2}}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^2}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) \sin ^{\frac {7}{2}}(c+d x)}{(-\cos (c+d x) a-a)^2}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\left (-\cos \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2} \sin \left (c+d x+\frac {\pi }{2}\right )^2}{\left (-\sin \left (c+d x+\frac {\pi }{2}\right ) a-a\right )^2}dx}{e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle \frac {\int \frac {\cos ^2(c+d x) (a-a \cos (c+d x))^2}{\sqrt {\sin (c+d x)}}dx}{a^4 e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {\sin \left (c+d x-\frac {\pi }{2}\right )^2 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^2}{\sqrt {\cos \left (c+d x-\frac {\pi }{2}\right )}}dx}{a^4 e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle \frac {\int \left (\frac {a^2 \cos ^4(c+d x)}{\sqrt {\sin (c+d x)}}-\frac {2 a^2 \cos ^3(c+d x)}{\sqrt {\sin (c+d x)}}+\frac {a^2 \cos ^2(c+d x)}{\sqrt {\sin (c+d x)}}\right )dx}{a^4 e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {4 a^2 \sin ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {4 a^2 \sqrt {\sin (c+d x)}}{d}+\frac {52 a^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{21 d}+\frac {2 a^2 \sqrt {\sin (c+d x)} \cos ^3(c+d x)}{7 d}+\frac {26 a^2 \sqrt {\sin (c+d x)} \cos (c+d x)}{21 d}}{a^4 e^3 \sqrt {\sin (c+d x)} \sqrt {e \csc (c+d x)}}\) |
((52*a^2*EllipticF[(c - Pi/2 + d*x)/2, 2])/(21*d) - (4*a^2*Sqrt[Sin[c + d* x]])/d + (26*a^2*Cos[c + d*x]*Sqrt[Sin[c + d*x]])/(21*d) + (2*a^2*Cos[c + d*x]^3*Sqrt[Sin[c + d*x]])/(7*d) + (4*a^2*Sin[c + d*x]^(5/2))/(5*d))/(a^4* e^3*Sqrt[e*Csc[c + d*x]]*Sqrt[Sin[c + d*x]])
3.4.6.3.1 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 = 9.86 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.84
| method | result | size |
| default | \(\frac {\sqrt {2}\, \left (130 i \sqrt {i \left (-i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right ) \cos \left (d x +c \right )+130 i \sqrt {i \left (-i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {i \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}\, \operatorname {EllipticF}\left (\sqrt {-i \left (i-\cot \left (d x +c \right )+\csc \left (d x +c \right )\right )}, \frac {\sqrt {2}}{2}\right )+15 \sqrt {2}\, \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )-42 \cos \left (d x +c \right )^{2} \sqrt {2}\, \sin \left (d x +c \right )+65 \cos \left (d x +c \right ) \sin \left (d x +c \right ) \sqrt {2}-168 \sqrt {2}\, \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )^{3}}{105 a^{2} d \,e^{3} \sqrt {e \csc \left (d x +c \right )}\, \left (\cos \left (d x +c \right )-1\right )^{2} \left (\cos \left (d x +c \right )+1\right )^{2}}\) | \(316\) |
1/105/a^2/d*2^(1/2)*(130*I*(-I*(I-cot(d*x+c)+csc(d*x+c)))^(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))*cos(d*x+c)+130*I*(-I*(I-co t(d*x+c)+csc(d*x+c)))^(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))+15*2^(1/2)*cos(d*x+c)^3*sin(d*x+c)-42*cos(d*x+c)^2*2^(1/2)*sin (d*x+c)+65*cos(d*x+c)*sin(d*x+c)*2^(1/2)-168*2^(1/2)*sin(d*x+c))/e^3/(e*cs c(d*x+c))^(1/2)/(cos(d*x+c)-1)^2/(cos(d*x+c)+1)^2*sin(d*x+c)^3
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\frac {2 \, {\left ({\left (15 \, \cos \left (d x + c\right )^{3} - 42 \, \cos \left (d x + c\right )^{2} + 65 \, \cos \left (d x + c\right ) - 168\right )} \sqrt {\frac {e}{\sin \left (d x + c\right )}} \sin \left (d x + c\right ) - 65 i \, \sqrt {2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 65 i \, \sqrt {-2 i \, e} {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right )}}{105 \, a^{2} d e^{4}} \]
2/105*((15*cos(d*x + c)^3 - 42*cos(d*x + c)^2 + 65*cos(d*x + c) - 168)*sqr t(e/sin(d*x + c))*sin(d*x + c) - 65*I*sqrt(2*I*e)*weierstrassPInverse(4, 0 , cos(d*x + c) + I*sin(d*x + c)) + 65*I*sqrt(-2*I*e)*weierstrassPInverse(4 , 0, cos(d*x + c) - I*sin(d*x + c)))/(a^2*d*e^4)
Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
\[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int { \frac {1}{\left (e \csc \left (d x + c\right )\right )^{\frac {7}{2}} {\left (a \sec \left (d x + c\right ) + a\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {1}{(e \csc (c+d x))^{7/2} (a+a \sec (c+d x))^2} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{a^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{7/2}\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]