Integrand size = 21, antiderivative size = 91 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {\cos ^3(c+d x)}{3 a d}-\frac {3 \cos ^5(c+d x)}{5 a d}+\frac {3 \cos ^7(c+d x)}{7 a d}-\frac {\cos ^9(c+d x)}{9 a d}+\frac {\sin ^8(c+d x)}{8 a d} \]
1/3*cos(d*x+c)^3/a/d-3/5*cos(d*x+c)^5/a/d+3/7*cos(d*x+c)^7/a/d-1/9*cos(d*x +c)^9/a/d+1/8*sin(d*x+c)^8/a/d
Time = 3.22 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.68 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {(4258+6995 \cos (c+d x)+3650 \cos (2 (c+d x))+1085 \cos (3 (c+d x))+140 \cos (4 (c+d x))) \sin ^{10}\left (\frac {1}{2} (c+d x)\right )}{315 a d} \]
((4258 + 6995*Cos[c + d*x] + 3650*Cos[2*(c + d*x)] + 1085*Cos[3*(c + d*x)] + 140*Cos[4*(c + d*x)])*Sin[(c + d*x)/2]^10)/(315*a*d)
Time = 0.52 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.82, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {3042, 4360, 25, 25, 3042, 25, 3314, 25, 3042, 3044, 15, 3045, 244, 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.
\(\displaystyle \int \frac {\sin ^9(c+d x)}{a \sec (c+d x)+a} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^9}{a-a \csc \left (c+d x-\frac {\pi }{2}\right )}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sin ^9(c+d x) \cos (c+d x)}{a (-\cos (c+d x))-a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cos (c+d x) \sin ^9(c+d x)}{\cos (c+d x) a+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sin ^9(c+d x) \cos (c+d x)}{a \cos (c+d x)+a}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right ) \cos \left (c+d x+\frac {\pi }{2}\right )^9}{a \sin \left (c+d x+\frac {\pi }{2}\right )+a}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^9 \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a}dx\) |
\(\Big \downarrow \) 3314 |
\(\displaystyle \frac {\int -\cos ^2(c+d x) \sin ^7(c+d x)dx}{a}-\frac {\int -\cos (c+d x) \sin ^7(c+d x)dx}{a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \cos (c+d x) \sin ^7(c+d x)dx}{a}-\frac {\int \cos ^2(c+d x) \sin ^7(c+d x)dx}{a}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int \cos (c+d x) \sin (c+d x)^7dx}{a}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^7dx}{a}\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {\int \sin ^7(c+d x)d\sin (c+d x)}{a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^7dx}{a}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {\sin ^8(c+d x)}{8 a d}-\frac {\int \cos (c+d x)^2 \sin (c+d x)^7dx}{a}\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle \frac {\int \cos ^2(c+d x) \left (1-\cos ^2(c+d x)\right )^3d\cos (c+d x)}{a d}+\frac {\sin ^8(c+d x)}{8 a d}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {\int \left (-\cos ^8(c+d x)+3 \cos ^6(c+d x)-3 \cos ^4(c+d x)+\cos ^2(c+d x)\right )d\cos (c+d x)}{a d}+\frac {\sin ^8(c+d x)}{8 a d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sin ^8(c+d x)}{8 a d}+\frac {-\frac {1}{9} \cos ^9(c+d x)+\frac {3}{7} \cos ^7(c+d x)-\frac {3}{5} \cos ^5(c+d x)+\frac {1}{3} \cos ^3(c+d x)}{a d}\) |
(Cos[c + d*x]^3/3 - (3*Cos[c + d*x]^5)/5 + (3*Cos[c + d*x]^7)/7 - Cos[c + d*x]^9/9)/(a*d) + Sin[c + d*x]^8/(8*a*d)
3.1.57.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, n])
Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(a*f)^(-1) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Cos[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(IntegerQ[(m - 1)/2] && GtQ[m, 0] && LeQ[m, n])
Int[(cos[(e_.) + (f_.)*(x_)]^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.))/(( a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[1/a Int[Cos[e + f *x]^(p - 2)*(d*Sin[e + f*x])^n, x], x] - Simp[1/(b*d) Int[Cos[e + f*x]^(p - 2)*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, n, p}, x] & & IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0] && IntegerQ[n] && (LtQ[0, n, (p + 1)/2] || (LeQ[p, -n] && LtQ[-n, 2*p - 3]) || (GtQ[n, 0] && LeQ[n, -p]))
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 = 0.83 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98
method | result | size |
derivativedivides | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {\cos \left (d x +c \right )^{8}}{8}+\frac {3 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}-\frac {3 \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(89\) |
default | \(\frac {-\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {\cos \left (d x +c \right )^{8}}{8}+\frac {3 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{2}-\frac {3 \cos \left (d x +c \right )^{5}}{5}+\frac {3 \cos \left (d x +c \right )^{4}}{4}+\frac {\cos \left (d x +c \right )^{3}}{3}-\frac {\cos \left (d x +c \right )^{2}}{2}}{d a}\) | \(89\) |
parallelrisch | \(\frac {8820 \cos \left (4 d x +4 c \right )+17640 \cos \left (d x +c \right )+27409+315 \cos \left (8 d x +8 c \right )+900 \cos \left (7 d x +7 c \right )-2520 \cos \left (6 d x +6 c \right )-2016 \cos \left (5 d x +5 c \right )-17640 \cos \left (2 d x +2 c \right )-140 \cos \left (9 d x +9 c \right )}{322560 d a}\) | \(96\) |
norman | \(\frac {\frac {32}{315 a d}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d a}+\frac {32 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{35 d a}+\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{35 d a}+\frac {128 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{15 d a}+\frac {64 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{5 d a}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(121\) |
risch | \(\frac {7 \cos \left (d x +c \right )}{128 a d}-\frac {\cos \left (9 d x +9 c \right )}{2304 a d}+\frac {\cos \left (8 d x +8 c \right )}{1024 a d}+\frac {5 \cos \left (7 d x +7 c \right )}{1792 a d}-\frac {\cos \left (6 d x +6 c \right )}{128 a d}-\frac {\cos \left (5 d x +5 c \right )}{160 a d}+\frac {7 \cos \left (4 d x +4 c \right )}{256 a d}-\frac {7 \cos \left (2 d x +2 c \right )}{128 a d}\) | \(135\) |
1/d/a*(-1/9*cos(d*x+c)^9+1/8*cos(d*x+c)^8+3/7*cos(d*x+c)^7-1/2*cos(d*x+c)^ 6-3/5*cos(d*x+c)^5+3/4*cos(d*x+c)^4+1/3*cos(d*x+c)^3-1/2*cos(d*x+c)^2)
Time = 0.27 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 1080 \, \cos \left (d x + c\right )^{7} + 1260 \, \cos \left (d x + c\right )^{6} + 1512 \, \cos \left (d x + c\right )^{5} - 1890 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 1260 \, \cos \left (d x + c\right )^{2}}{2520 \, a d} \]
-1/2520*(280*cos(d*x + c)^9 - 315*cos(d*x + c)^8 - 1080*cos(d*x + c)^7 + 1 260*cos(d*x + c)^6 + 1512*cos(d*x + c)^5 - 1890*cos(d*x + c)^4 - 840*cos(d *x + c)^3 + 1260*cos(d*x + c)^2)/(a*d)
Timed out. \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.98 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {280 \, \cos \left (d x + c\right )^{9} - 315 \, \cos \left (d x + c\right )^{8} - 1080 \, \cos \left (d x + c\right )^{7} + 1260 \, \cos \left (d x + c\right )^{6} + 1512 \, \cos \left (d x + c\right )^{5} - 1890 \, \cos \left (d x + c\right )^{4} - 840 \, \cos \left (d x + c\right )^{3} + 1260 \, \cos \left (d x + c\right )^{2}}{2520 \, a d} \]
-1/2520*(280*cos(d*x + c)^9 - 315*cos(d*x + c)^8 - 1080*cos(d*x + c)^7 + 1 260*cos(d*x + c)^6 + 1512*cos(d*x + c)^5 - 1890*cos(d*x + c)^4 - 840*cos(d *x + c)^3 + 1260*cos(d*x + c)^2)/(a*d)
Time = 0.31 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.55 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=\frac {32 \, {\left (\frac {9 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {36 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {84 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {126 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {630 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - 1\right )}}{315 \, a d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}} \]
32/315*(9*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 36*(cos(d*x + c) - 1)^2/ (cos(d*x + c) + 1)^2 + 84*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 126* (cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 630*(cos(d*x + c) - 1)^5/(cos( d*x + c) + 1)^5 - 1)/(a*d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)
Time = 0.09 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.21 \[ \int \frac {\sin ^9(c+d x)}{a+a \sec (c+d x)} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^2}{2\,a}-\frac {{\cos \left (c+d\,x\right )}^3}{3\,a}-\frac {3\,{\cos \left (c+d\,x\right )}^4}{4\,a}+\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a}+\frac {{\cos \left (c+d\,x\right )}^6}{2\,a}-\frac {3\,{\cos \left (c+d\,x\right )}^7}{7\,a}-\frac {{\cos \left (c+d\,x\right )}^8}{8\,a}+\frac {{\cos \left (c+d\,x\right )}^9}{9\,a}}{d} \]