Integrand size = 21, antiderivative size = 139 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {2 (a-a \cos (c+d x))^6}{3 a^9 d}-\frac {16 (a-a \cos (c+d x))^7}{7 a^{10} d}+\frac {25 (a-a \cos (c+d x))^8}{8 a^{11} d}-\frac {19 (a-a \cos (c+d x))^9}{9 a^{12} d}+\frac {7 (a-a \cos (c+d x))^{10}}{10 a^{13} d}-\frac {(a-a \cos (c+d x))^{11}}{11 a^{14} d} \]
2/3*(a-a*cos(d*x+c))^6/a^9/d-16/7*(a-a*cos(d*x+c))^7/a^10/d+25/8*(a-a*cos( d*x+c))^8/a^11/d-19/9*(a-a*cos(d*x+c))^9/a^12/d+7/10*(a-a*cos(d*x+c))^10/a ^13/d-1/11*(a-a*cos(d*x+c))^11/a^14/d
Time = 3.90 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.86 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {-1615571+2273040 \cos (c+d x)-1496880 \cos (2 (c+d x))+535920 \cos (3 (c+d x))+110880 \cos (4 (c+d x))-293832 \cos (5 (c+d x))+212520 \cos (6 (c+d x))-67320 \cos (7 (c+d x))-27720 \cos (8 (c+d x))+40040 \cos (9 (c+d x))-16632 \cos (10 (c+d x))+2520 \cos (11 (c+d x))}{28385280 a^3 d} \]
(-1615571 + 2273040*Cos[c + d*x] - 1496880*Cos[2*(c + d*x)] + 535920*Cos[3 *(c + d*x)] + 110880*Cos[4*(c + d*x)] - 293832*Cos[5*(c + d*x)] + 212520*C os[6*(c + d*x)] - 67320*Cos[7*(c + d*x)] - 27720*Cos[8*(c + d*x)] + 40040* Cos[9*(c + d*x)] - 16632*Cos[10*(c + d*x)] + 2520*Cos[11*(c + d*x)])/(2838 5280*a^3*d)
Time = 0.46 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.89, 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, 99, 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.
\(\displaystyle \int \frac {\sin ^{11}(c+d x)}{(a \sec (c+d x)+a)^3} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\cos \left (c+d x-\frac {\pi }{2}\right )^{11}}{\left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3}dx\) |
\(\Big \downarrow \) 4360 |
\(\displaystyle \int -\frac {\sin ^{11}(c+d x) \cos ^3(c+d x)}{(a (-\cos (c+d x))-a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int -\frac {\cos ^3(c+d x) \sin ^{11}(c+d x)}{(\cos (c+d x) a+a)^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \int \frac {\sin ^{11}(c+d x) \cos ^3(c+d x)}{(a \cos (c+d x)+a)^3}dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -\frac {\sin \left (c+d x+\frac {\pi }{2}\right )^3 \cos \left (c+d x+\frac {\pi }{2}\right )^{11}}{\left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^{11} \sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}{\left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}dx\) |
\(\Big \downarrow \) 3315 |
\(\displaystyle -\frac {\int \cos ^3(c+d x) (a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2d(a \cos (c+d x))}{a^{11} d}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {\int a^3 \cos ^3(c+d x) (a-a \cos (c+d x))^5 (\cos (c+d x) a+a)^2d(a \cos (c+d x))}{a^{14} d}\) |
\(\Big \downarrow \) 99 |
\(\displaystyle -\frac {\int \left (-(a-a \cos (c+d x))^{10}+7 a (a-a \cos (c+d x))^9-19 a^2 (a-a \cos (c+d x))^8+25 a^3 (a-a \cos (c+d x))^7-16 a^4 (a-a \cos (c+d x))^6+4 a^5 (a-a \cos (c+d x))^5\right )d(a \cos (c+d x))}{a^{14} d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\frac {2}{3} a^5 (a-a \cos (c+d x))^6+\frac {16}{7} a^4 (a-a \cos (c+d x))^7-\frac {25}{8} a^3 (a-a \cos (c+d x))^8+\frac {19}{9} a^2 (a-a \cos (c+d x))^9+\frac {1}{11} (a-a \cos (c+d x))^{11}-\frac {7}{10} a (a-a \cos (c+d x))^{10}}{a^{14} d}\) |
-(((-2*a^5*(a - a*Cos[c + d*x])^6)/3 + (16*a^4*(a - a*Cos[c + d*x])^7)/7 - (25*a^3*(a - a*Cos[c + d*x])^8)/8 + (19*a^2*(a - a*Cos[c + d*x])^9)/9 - ( 7*a*(a - a*Cos[c + d*x])^10)/10 + (a - a*Cos[c + d*x])^11/11)/(a^14*d))
3.1.91.3.1 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[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && IntegersQ[m, n] && (IntegerQ[p] | | (GtQ[m, 0] && GeQ[n, -1]))
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 = 1.04 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64
method | result | size |
derivativedivides | \(\frac {\frac {\cos \left (d x +c \right )^{11}}{11}-\frac {3 \cos \left (d x +c \right )^{10}}{10}+\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {5 \cos \left (d x +c \right )^{8}}{8}-\frac {5 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{6}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(89\) |
default | \(\frac {\frac {\cos \left (d x +c \right )^{11}}{11}-\frac {3 \cos \left (d x +c \right )^{10}}{10}+\frac {\cos \left (d x +c \right )^{9}}{9}+\frac {5 \cos \left (d x +c \right )^{8}}{8}-\frac {5 \cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{6}}{6}+\frac {3 \cos \left (d x +c \right )^{5}}{5}-\frac {\cos \left (d x +c \right )^{4}}{4}}{d \,a^{3}}\) | \(89\) |
parallelrisch | \(\frac {13860 \cos \left (4 d x +4 c \right )+66990 \cos \left (3 d x +3 c \right )+315 \cos \left (11 d x +11 c \right )+284130 \cos \left (d x +c \right )-3465 \cos \left (8 d x +8 c \right )-8415 \cos \left (7 d x +7 c \right )+26565 \cos \left (6 d x +6 c \right )-36729 \cos \left (5 d x +5 c \right )-187110 \cos \left (2 d x +2 c \right )+5005 \cos \left (9 d x +9 c \right )-2079 \cos \left (10 d x +10 c \right )+463525}{3548160 a^{3} d}\) | \(129\) |
risch | \(\frac {41 \cos \left (d x +c \right )}{512 a^{3} d}+\frac {\cos \left (11 d x +11 c \right )}{11264 d \,a^{3}}-\frac {3 \cos \left (10 d x +10 c \right )}{5120 d \,a^{3}}+\frac {13 \cos \left (9 d x +9 c \right )}{9216 d \,a^{3}}-\frac {\cos \left (8 d x +8 c \right )}{1024 d \,a^{3}}-\frac {17 \cos \left (7 d x +7 c \right )}{7168 d \,a^{3}}+\frac {23 \cos \left (6 d x +6 c \right )}{3072 d \,a^{3}}-\frac {53 \cos \left (5 d x +5 c \right )}{5120 d \,a^{3}}+\frac {\cos \left (4 d x +4 c \right )}{256 d \,a^{3}}+\frac {29 \cos \left (3 d x +3 c \right )}{1536 d \,a^{3}}-\frac {27 \cos \left (2 d x +2 c \right )}{512 d \,a^{3}}\) | \(186\) |
1/d/a^3*(1/11*cos(d*x+c)^11-3/10*cos(d*x+c)^10+1/9*cos(d*x+c)^9+5/8*cos(d* x+c)^8-5/7*cos(d*x+c)^7-1/6*cos(d*x+c)^6+3/5*cos(d*x+c)^5-1/4*cos(d*x+c)^4 )
Time = 0.28 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {2520 \, \cos \left (d x + c\right )^{11} - 8316 \, \cos \left (d x + c\right )^{10} + 3080 \, \cos \left (d x + c\right )^{9} + 17325 \, \cos \left (d x + c\right )^{8} - 19800 \, \cos \left (d x + c\right )^{7} - 4620 \, \cos \left (d x + c\right )^{6} + 16632 \, \cos \left (d x + c\right )^{5} - 6930 \, \cos \left (d x + c\right )^{4}}{27720 \, a^{3} d} \]
1/27720*(2520*cos(d*x + c)^11 - 8316*cos(d*x + c)^10 + 3080*cos(d*x + c)^9 + 17325*cos(d*x + c)^8 - 19800*cos(d*x + c)^7 - 4620*cos(d*x + c)^6 + 166 32*cos(d*x + c)^5 - 6930*cos(d*x + c)^4)/(a^3*d)
Timed out. \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\text {Timed out} \]
Time = 0.22 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.64 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {2520 \, \cos \left (d x + c\right )^{11} - 8316 \, \cos \left (d x + c\right )^{10} + 3080 \, \cos \left (d x + c\right )^{9} + 17325 \, \cos \left (d x + c\right )^{8} - 19800 \, \cos \left (d x + c\right )^{7} - 4620 \, \cos \left (d x + c\right )^{6} + 16632 \, \cos \left (d x + c\right )^{5} - 6930 \, \cos \left (d x + c\right )^{4}}{27720 \, a^{3} d} \]
1/27720*(2520*cos(d*x + c)^11 - 8316*cos(d*x + c)^10 + 3080*cos(d*x + c)^9 + 17325*cos(d*x + c)^8 - 19800*cos(d*x + c)^7 - 4620*cos(d*x + c)^6 + 166 32*cos(d*x + c)^5 - 6930*cos(d*x + c)^4)/(a^3*d)
Time = 0.41 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.49 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=\frac {32 \, {\left (\frac {209 \, {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {1045 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3135 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {6270 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {8778 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {13398 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {2310 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {9240 \, {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - 19\right )}}{3465 \, a^{3} d {\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{11}} \]
32/3465*(209*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1045*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 + 3135*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 - 6270*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 + 8778*(cos(d*x + c) - 1 )^5/(cos(d*x + c) + 1)^5 - 13398*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 2310*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 - 9240*(cos(d*x + c) - 1 )^8/(cos(d*x + c) + 1)^8 - 19)/(a^3*d*((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^11)
Time = 13.48 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.79 \[ \int \frac {\sin ^{11}(c+d x)}{(a+a \sec (c+d x))^3} \, dx=-\frac {\frac {{\cos \left (c+d\,x\right )}^4}{4\,a^3}-\frac {3\,{\cos \left (c+d\,x\right )}^5}{5\,a^3}+\frac {{\cos \left (c+d\,x\right )}^6}{6\,a^3}+\frac {5\,{\cos \left (c+d\,x\right )}^7}{7\,a^3}-\frac {5\,{\cos \left (c+d\,x\right )}^8}{8\,a^3}-\frac {{\cos \left (c+d\,x\right )}^9}{9\,a^3}+\frac {3\,{\cos \left (c+d\,x\right )}^{10}}{10\,a^3}-\frac {{\cos \left (c+d\,x\right )}^{11}}{11\,a^3}}{d} \]