Integrand size = 21, antiderivative size = 89 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {3 \cot ^5(c+d x)}{5 a^3 d}+\frac {4 \cot ^7(c+d x)}{7 a^3 d}-\frac {\csc ^3(c+d x)}{a^3 d}+\frac {7 \csc ^5(c+d x)}{5 a^3 d}-\frac {4 \csc ^7(c+d x)}{7 a^3 d} \]
3/5*cot(d*x+c)^5/a^3/d+4/7*cot(d*x+c)^7/a^3/d-csc(d*x+c)^3/a^3/d+7/5*csc(d *x+c)^5/a^3/d-4/7*csc(d*x+c)^7/a^3/d
Time = 1.00 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.54 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\csc (c) \csc (c+d x) \sec ^3(c+d x) (-840 \sin (c)+448 \sin (d x)+602 \sin (c+d x)+602 \sin (2 (c+d x))+258 \sin (3 (c+d x))+43 \sin (4 (c+d x))-560 \sin (2 c+d x)+168 \sin (c+2 d x)-280 \sin (3 c+2 d x)-48 \sin (2 c+3 d x)-8 \sin (3 c+4 d x))}{2240 a^3 d (1+\sec (c+d x))^3} \]
(Csc[c]*Csc[c + d*x]*Sec[c + d*x]^3*(-840*Sin[c] + 448*Sin[d*x] + 602*Sin[ c + d*x] + 602*Sin[2*(c + d*x)] + 258*Sin[3*(c + d*x)] + 43*Sin[4*(c + d*x )] - 560*Sin[2*c + d*x] + 168*Sin[c + 2*d*x] - 280*Sin[3*c + 2*d*x] - 48*S in[2*c + 3*d*x] - 8*Sin[3*c + 4*d*x]))/(2240*a^3*d*(1 + Sec[c + d*x])^3)
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.
| \(\displaystyle \int \frac {\csc ^2(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos \left (c+d x-\frac {\pi }{2}\right )^2 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\frac {\cos (c+d x) \cot ^2(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\frac {\cos (c+d x) \cot ^2(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\cos (c+d x) \cot ^2(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^2 \left (a-a \sin \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^2 \left (a-a \sin \left (\frac {1}{2} (2 c-\pi )+d x\right )\right )^3}dx\) |
| \(\Big \downarrow \) 3354 |
| \(\displaystyle -\frac {\int -(a-a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^5(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int (a-a \cos (c+d x))^3 \cot ^3(c+d x) \csc ^5(c+d x)dx}{a^6}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int -\frac {\sin \left (c+d x-\frac {\pi }{2}\right )^3 \left (\sin \left (c+d x-\frac {\pi }{2}\right ) a+a\right )^3}{\cos \left (c+d x-\frac {\pi }{2}\right )^8}dx}{a^6}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\int \frac {\sin \left (\frac {1}{2} (2 c-\pi )+d x\right )^3 \left (\sin \left (\frac {1}{2} (2 c-\pi )+d x\right ) a+a\right )^3}{\cos \left (\frac {1}{2} (2 c-\pi )+d x\right )^8}dx}{a^6}\) |
| \(\Big \downarrow \) 3352 |
| \(\displaystyle -\frac {\int \left (a^3 \cos ^6(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )-3 a^3 \cos ^5(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )+3 a^3 \cos ^4(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )-a^3 \cos ^3(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )\right )dx}{a^6}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^3 \int \cos ^6(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )dx-3 a^3 \int \cos ^5(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )dx+3 a^3 \int \cos ^4(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )dx+a^3 \left (-\int \cos ^3(c+d x) \sec ^8\left (\frac {1}{2} (2 c-\pi )+d x\right )dx\right )}{a^6}\) |
3.2.4.3.1 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]
Time = 0.58 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.65
| method | result | size |
| parallelrisch | \(\frac {-5 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}+14 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}-70 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-35 \cot \left (\frac {d x}{2}+\frac {c}{2}\right )}{560 a^{3} d}\) | \(58\) |
| derivativedivides | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{16 d \,a^{3}}\) | \(60\) |
| default | \(\frac {-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{7}+\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5}-2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-\frac {1}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{16 d \,a^{3}}\) | \(60\) |
| norman | \(\frac {-\frac {1}{16 a d}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8 d a}+\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{40 d a}-\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{112 d a}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{2}}\) | \(82\) |
| risch | \(-\frac {2 i \left (35 \,{\mathrm e}^{6 i \left (d x +c \right )}+70 \,{\mathrm e}^{5 i \left (d x +c \right )}+105 \,{\mathrm e}^{4 i \left (d x +c \right )}+56 \,{\mathrm e}^{3 i \left (d x +c \right )}+21 \,{\mathrm e}^{2 i \left (d x +c \right )}-6 \,{\mathrm e}^{i \left (d x +c \right )}-1\right )}{35 a^{3} d \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{7} \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}\) | \(104\) |
1/560*(-5*tan(1/2*d*x+1/2*c)^7+14*tan(1/2*d*x+1/2*c)^5-70*tan(1/2*d*x+1/2* c)-35*cot(1/2*d*x+1/2*c))/a^3/d
Time = 0.27 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.07 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{3} - 15 \, \cos \left (d x + c\right )^{2} - 18 \, \cos \left (d x + c\right ) - 6}{35 \, {\left (a^{3} d \cos \left (d x + c\right )^{3} + 3 \, a^{3} d \cos \left (d x + c\right )^{2} + 3 \, a^{3} d \cos \left (d x + c\right ) + a^{3} d\right )} \sin \left (d x + c\right )} \]
1/35*(cos(d*x + c)^4 + 3*cos(d*x + c)^3 - 15*cos(d*x + c)^2 - 18*cos(d*x + c) - 6)/((a^3*d*cos(d*x + c)^3 + 3*a^3*d*cos(d*x + c)^2 + 3*a^3*d*cos(d*x + c) + a^3*d)*sin(d*x + c))
\[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {\int \frac {\csc ^{2}{\left (c + d x \right )}}{\sec ^{3}{\left (c + d x \right )} + 3 \sec ^{2}{\left (c + d x \right )} + 3 \sec {\left (c + d x \right )} + 1}\, dx}{a^{3}} \]
Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.01 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {\frac {70 \, \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {14 \, \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {5 \, \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}}}{a^{3}} + \frac {35 \, {\left (\cos \left (d x + c\right ) + 1\right )}}{a^{3} \sin \left (d x + c\right )}}{560 \, d} \]
-1/560*((70*sin(d*x + c)/(cos(d*x + c) + 1) - 14*sin(d*x + c)^5/(cos(d*x + c) + 1)^5 + 5*sin(d*x + c)^7/(cos(d*x + c) + 1)^7)/a^3 + 35*(cos(d*x + c) + 1)/(a^3*sin(d*x + c)))/d
Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.82 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {35}{a^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )} + \frac {5 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 14 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} + 70 \, a^{18} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{a^{21}}}{560 \, d} \]
-1/560*(35/(a^3*tan(1/2*d*x + 1/2*c)) + (5*a^18*tan(1/2*d*x + 1/2*c)^7 - 1 4*a^18*tan(1/2*d*x + 1/2*c)^5 + 70*a^18*tan(1/2*d*x + 1/2*c))/a^21)/d
Time = 13.48 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.94 \[ \int \frac {\csc ^2(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {-16\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+8\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+72\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-34\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+5}{560\,a^3\,d\,{\cos \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )} \]