Integrand size = 21, antiderivative size = 109 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\cos ^4(c+d x)}{4 a^3 d}+\frac {3 \cos ^5(c+d x)}{5 a^3 d}-\frac {\cos ^6(c+d x)}{3 a^3 d}-\frac {2 \cos ^7(c+d x)}{7 a^3 d}+\frac {3 \cos ^8(c+d x)}{8 a^3 d}-\frac {\cos ^9(c+d x)}{9 a^3 d} \]
-1/4*cos(d*x+c)^4/a^3/d+3/5*cos(d*x+c)^5/a^3/d-1/3*cos(d*x+c)^6/a^3/d-2/7* cos(d*x+c)^7/a^3/d+3/8*cos(d*x+c)^8/a^3/d-1/9*cos(d*x+c)^9/a^3/d
Time = 1.86 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.92 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {34771-52920 \cos (c+d x)+37800 \cos (2 (c+d x))-18480 \cos (3 (c+d x))+3780 \cos (4 (c+d x))+3024 \cos (5 (c+d x))-4200 \cos (6 (c+d x))+2700 \cos (7 (c+d x))-945 \cos (8 (c+d x))+140 \cos (9 (c+d x))}{322560 a^3 d} \]
-1/322560*(34771 - 52920*Cos[c + d*x] + 37800*Cos[2*(c + d*x)] - 18480*Cos [3*(c + d*x)] + 3780*Cos[4*(c + d*x)] + 3024*Cos[5*(c + d*x)] - 4200*Cos[6 *(c + d*x)] + 2700*Cos[7*(c + d*x)] - 945*Cos[8*(c + d*x)] + 140*Cos[9*(c + d*x)])/(a^3*d)
Time = 0.43 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.91, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.476, Rules used = {3042, 4360, 25, 25, 3042, 25, 3315, 27, 84, 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.
\(\displaystyle \int \frac {\sin ^9(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^9}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sin ^9(c+d x) \cos ^3(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^9(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sin ^9(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^9}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^9 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}{\left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle -\frac {\int \cos ^3(c+d x) (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)d(a \cos (c+d x))}{a^9 d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int a^3 \cos ^3(c+d x) (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)d(a \cos (c+d x))}{a^{12} d}\) |
\(\Big \downarrow \) 84 |
\(\displaystyle -\frac {\int \left (\cos ^8(c+d x) a^8-3 \cos ^7(c+d x) a^8+2 \cos ^6(c+d x) a^8+2 \cos ^5(c+d x) a^8-3 \cos ^4(c+d x) a^8+\cos ^3(c+d x) a^8\right )d(a \cos (c+d x))}{a^{12} d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {1}{9} a^9 \cos ^9(c+d x)-\frac {3}{8} a^9 \cos ^8(c+d x)+\frac {2}{7} a^9 \cos ^7(c+d x)+\frac {1}{3} a^9 \cos ^6(c+d x)-\frac {3}{5} a^9 \cos ^5(c+d x)+\frac {1}{4} a^9 \cos ^4(c+d x)}{a^{12} d}\) |
-(((a^9*Cos[c + d*x]^4)/4 - (3*a^9*Cos[c + d*x]^5)/5 + (a^9*Cos[c + d*x]^6 )/3 + (2*a^9*Cos[c + d*x]^7)/7 - (3*a^9*Cos[c + d*x]^8)/8 + (a^9*Cos[c + d *x]^9)/9)/(a^12*d))
3.1.92.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_] : > Int[ExpandIntegrand[(a + b*x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && EqQ[b*e + a*f, 0] && !(ILtQ[n + p + 2, 0 ] && GtQ[n + 2*p, 0])
Int[cos[(e_.) + (f_.)*(x_)]^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_ .)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Simp[1/(b^p* f) Subst[Int[(a + x)^(m + (p - 1)/2)*(a - x)^((p - 1)/2)*(c + (d/b)*x)^n, x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, c, d, m, n}, x] && Intege rQ[(p - 1)/2] && EqQ[a^2 - b^2, 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 = 0.99 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {3 \cos \left (d x +c \right )^{8}}{8}-\frac {2 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{3}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(69\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {3 \cos \left (d x +c \right )^{8}}{8}-\frac {2 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{3}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(69\) |
parallelrisch | \(\frac {-3780 \cos \left (4 d x +4 c \right )+18480 \cos \left (3 d x +3 c \right )+52920 \cos \left (d x +c \right )+945 \cos \left (8 d x +8 c \right )-2700 \cos \left (7 d x +7 c \right )+4200 \cos \left (6 d x +6 c \right )-3024 \cos \left (5 d x +5 c \right )+101971-37800 \cos \left (2 d x +2 c \right )-140 \cos \left (9 d x +9 c \right )}{322560 a^{3} d}\) | \(107\) |
risch | \(\frac {21 \cos \left (d x +c \right )}{128 a^{3} d}-\frac {\cos \left (9 d x +9 c \right )}{2304 d \,a^{3}}+\frac {3 \cos \left (8 d x +8 c \right )}{1024 d \,a^{3}}-\frac {15 \cos \left (7 d x +7 c \right )}{1792 d \,a^{3}}+\frac {5 \cos \left (6 d x +6 c \right )}{384 d \,a^{3}}-\frac {3 \cos \left (5 d x +5 c \right )}{320 d \,a^{3}}-\frac {3 \cos \left (4 d x +4 c \right )}{256 d \,a^{3}}+\frac {11 \cos \left (3 d x +3 c \right )}{192 d \,a^{3}}-\frac {15 \cos \left (2 d x +2 c \right )}{128 d \,a^{3}}\) | \(152\) |
norman | \(\frac {\frac {128}{315 a d}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{a d}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 a d}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}+\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d a}+\frac {512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d a}+\frac {512 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d a}+\frac {256 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9} a^{2}}\) | \(162\) |
1/d/a^3*(-1/9*cos(d*x+c)^9+3/8*cos(d*x+c)^8-2/7*cos(d*x+c)^7-1/3*cos(d*x+c )^6+3/5*cos(d*x+c)^5-1/4*cos(d*x+c)^4)
Time = 0.26 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 945 \, \cos \left (d x + c\right )^{8} + 720 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 1512 \, \cos \left (d x + c\right )^{5} + 630 \, \cos \left (d x + c\right )^{4}}{2520 \, a^{3} d} \]
-1/2520*(280*cos(d*x + c)^9 - 945*cos(d*x + c)^8 + 720*cos(d*x + c)^7 + 84 0*cos(d*x + c)^6 - 1512*cos(d*x + c)^5 + 630*cos(d*x + c)^4)/(a^3*d)
Timed out. \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.63 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 945 \, \cos \left (d x + c\right )^{8} + 720 \, \cos \left (d x + c\right )^{7} + 840 \, \cos \left (d x + c\right )^{6} - 1512 \, \cos \left (d x + c\right )^{5} + 630 \, \cos \left (d x + c\right )^{4}}{2520 \, a^{3} d} \]
-1/2520*(280*cos(d*x + c)^9 - 945*cos(d*x + c)^8 + 720*cos(d*x + c)^7 + 84 0*cos(d*x + c)^6 - 1512*cos(d*x + c)^5 + 630*cos(d*x + c)^4)/(a^3*d)
Time = 0.42 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.70 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {32 \, {\left (\frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {144 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {336 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {504 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {630 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {105 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {315 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - 4\right )}}{315 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \]
32/315*(36*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 144*(cos(d*x + c) - 1)^ 2/(cos(d*x + c) + 1)^2 + 336*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 5 04*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 630*(cos(d*x + c) - 1)^5/(c os(d*x + c) + 1)^5 - 105*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 + 315*( cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 - 4)/(a^3*d*((cos(d*x + c) - 1)/( cos(d*x + c) + 1) - 1)^9)
Time = 13.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.77 \[ \int \frac {\sin ^9(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{3\,a^3}+\frac {2\,{\cos \left (c+d\,x\right )}^7}{7\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^8}{8\,a^3}+\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^3}}{d} \]