Integrand size = 19, antiderivative size = 152 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a \cos (c+d x)}{d}+\frac {2 a \cos ^2(c+d x)}{d}+\frac {4 a \cos ^3(c+d x)}{3 d}-\frac {3 a \cos ^4(c+d x)}{2 d}-\frac {6 a \cos ^5(c+d x)}{5 d}+\frac {2 a \cos ^6(c+d x)}{3 d}+\frac {4 a \cos ^7(c+d x)}{7 d}-\frac {a \cos ^8(c+d x)}{8 d}-\frac {a \cos ^9(c+d x)}{9 d}-\frac {a \log (\cos (c+d x))}{d} \]
-a*cos(d*x+c)/d+2*a*cos(d*x+c)^2/d+4/3*a*cos(d*x+c)^3/d-3/2*a*cos(d*x+c)^4 /d-6/5*a*cos(d*x+c)^5/d+2/3*a*cos(d*x+c)^6/d+4/7*a*cos(d*x+c)^7/d-1/8*a*co s(d*x+c)^8/d-1/9*a*cos(d*x+c)^9/d-a*ln(cos(d*x+c))/d
Time = 0.48 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.70 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a \left (39690 \cos (c+d x)-161280 \cos ^2(c+d x)+120960 \cos ^4(c+d x)-53760 \cos ^6(c+d x)+10080 \cos ^8(c+d x)-8820 \cos (3 (c+d x))+2268 \cos (5 (c+d x))-405 \cos (7 (c+d x))+35 \cos (9 (c+d x))+80640 \log (\cos (c+d x))\right )}{80640 d} \]
-1/80640*(a*(39690*Cos[c + d*x] - 161280*Cos[c + d*x]^2 + 120960*Cos[c + d *x]^4 - 53760*Cos[c + d*x]^6 + 10080*Cos[c + d*x]^8 - 8820*Cos[3*(c + d*x) ] + 2268*Cos[5*(c + d*x)] - 405*Cos[7*(c + d*x)] + 35*Cos[9*(c + d*x)] + 8 0640*Log[Cos[c + d*x]]))/d
Time = 0.38 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.99, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.526, 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.
| \(\displaystyle \int \sin ^9(c+d x) (a \sec (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^9 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int -\left (\sin ^8(c+d x) \tan (c+d x) (a (-\cos (c+d x))-a)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -\left ((\cos (c+d x) a+a) \sin ^8(c+d x) \tan (c+d x)\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^8(c+d x) \tan (c+d x) (a \cos (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^9 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )}{\sin \left (c+d x+\frac {\pi }{2}\right )}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^9 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle -\frac {\int (a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^5 \sec (c+d x)d(a \cos (c+d x))}{a^9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^4 (\cos (c+d x) a+a)^5 \sec (c+d x)}{a}d(a \cos (c+d x))}{a^8 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (\cos ^8(c+d x) a^8+\cos ^7(c+d x) a^8-4 \cos ^6(c+d x) a^8-4 \cos ^5(c+d x) a^8+6 \cos ^4(c+d x) a^8+6 \cos ^3(c+d x) a^8-4 \cos ^2(c+d x) a^8-4 \cos (c+d x) a^8+\sec (c+d x) a^8+a^8\right )d(a \cos (c+d x))}{a^8 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\frac {1}{9} a^9 \cos ^9(c+d x)+\frac {1}{8} a^9 \cos ^8(c+d x)-\frac {4}{7} a^9 \cos ^7(c+d x)-\frac {2}{3} a^9 \cos ^6(c+d x)+\frac {6}{5} a^9 \cos ^5(c+d x)+\frac {3}{2} a^9 \cos ^4(c+d x)-\frac {4}{3} a^9 \cos ^3(c+d x)-2 a^9 \cos ^2(c+d x)+a^9 \cos (c+d x)+a^9 \log (a \cos (c+d x))}{a^8 d}\) |
-((a^9*Cos[c + d*x] - 2*a^9*Cos[c + d*x]^2 - (4*a^9*Cos[c + d*x]^3)/3 + (3 *a^9*Cos[c + d*x]^4)/2 + (6*a^9*Cos[c + d*x]^5)/5 - (2*a^9*Cos[c + d*x]^6) /3 - (4*a^9*Cos[c + d*x]^7)/7 + (a^9*Cos[c + d*x]^8)/8 + (a^9*Cos[c + d*x] ^9)/9 + a^9*Log[a*Cos[c + d*x]])/(a^8*d))
3.1.1.3.1 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 = 1.58 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(107\) |
| default | \(\frac {a \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9}}{d}\) | \(107\) |
| parts | \(-\frac {a \left (\frac {128}{35}+\sin \left (d x +c \right )^{8}+\frac {8 \sin \left (d x +c \right )^{6}}{7}+\frac {48 \sin \left (d x +c \right )^{4}}{35}+\frac {64 \sin \left (d x +c \right )^{2}}{35}\right ) \cos \left (d x +c \right )}{9 d}+\frac {a \left (-\frac {\sin \left (d x +c \right )^{8}}{8}-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(109\) |
| parallelrisch | \(-\frac {a \left (-322560 \ln \left (\sec \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )+322560 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )+322560 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )+269777+28980 \cos \left (4 d x +4 c \right )-35280 \cos \left (3 d x +3 c \right )+158760 \cos \left (d x +c \right )+315 \cos \left (8 d x +8 c \right )-1620 \cos \left (7 d x +7 c \right )-4200 \cos \left (6 d x +6 c \right )+9072 \cos \left (5 d x +5 c \right )-163800 \cos \left (2 d x +2 c \right )+140 \cos \left (9 d x +9 c \right )\right )}{322560 d}\) | \(147\) |
| risch | \(i a x +\frac {2 i a c}{d}+\frac {65 a \,{\mathrm e}^{2 i \left (d x +c \right )}}{256 d}+\frac {65 a \,{\mathrm e}^{-2 i \left (d x +c \right )}}{256 d}-\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}-\frac {63 a \cos \left (d x +c \right )}{128 d}-\frac {a \cos \left (9 d x +9 c \right )}{2304 d}-\frac {a \cos \left (8 d x +8 c \right )}{1024 d}+\frac {9 a \cos \left (7 d x +7 c \right )}{1792 d}+\frac {5 a \cos \left (6 d x +6 c \right )}{384 d}-\frac {9 a \cos \left (5 d x +5 c \right )}{320 d}-\frac {23 a \cos \left (4 d x +4 c \right )}{256 d}+\frac {7 a \cos \left (3 d x +3 c \right )}{64 d}\) | \(180\) |
1/d*(a*(-1/8*sin(d*x+c)^8-1/6*sin(d*x+c)^6-1/4*sin(d*x+c)^4-1/2*sin(d*x+c) ^2-ln(cos(d*x+c)))-1/9*a*(128/35+sin(d*x+c)^8+8/7*sin(d*x+c)^6+48/35*sin(d *x+c)^4+64/35*sin(d*x+c)^2)*cos(d*x+c))
Time = 0.31 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.76 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (-\cos \left (d x + c\right )\right )}{2520 \, d} \]
-1/2520*(280*a*cos(d*x + c)^9 + 315*a*cos(d*x + c)^8 - 1440*a*cos(d*x + c) ^7 - 1680*a*cos(d*x + c)^6 + 3024*a*cos(d*x + c)^5 + 3780*a*cos(d*x + c)^4 - 3360*a*cos(d*x + c)^3 - 5040*a*cos(d*x + c)^2 + 2520*a*cos(d*x + c) + 2 520*a*log(-cos(d*x + c)))/d
Timed out. \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=\text {Timed out} \]
Time = 0.19 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.74 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {280 \, a \cos \left (d x + c\right )^{9} + 315 \, a \cos \left (d x + c\right )^{8} - 1440 \, a \cos \left (d x + c\right )^{7} - 1680 \, a \cos \left (d x + c\right )^{6} + 3024 \, a \cos \left (d x + c\right )^{5} + 3780 \, a \cos \left (d x + c\right )^{4} - 3360 \, a \cos \left (d x + c\right )^{3} - 5040 \, a \cos \left (d x + c\right )^{2} + 2520 \, a \cos \left (d x + c\right ) + 2520 \, a \log \left (\cos \left (d x + c\right )\right )}{2520 \, d} \]
-1/2520*(280*a*cos(d*x + c)^9 + 315*a*cos(d*x + c)^8 - 1440*a*cos(d*x + c) ^7 - 1680*a*cos(d*x + c)^6 + 3024*a*cos(d*x + c)^5 + 3780*a*cos(d*x + c)^4 - 3360*a*cos(d*x + c)^3 - 5040*a*cos(d*x + c)^2 + 2520*a*cos(d*x + c) + 2 520*a*log(cos(d*x + c)))/d
Leaf count of result is larger than twice the leaf count of optimal. 293 vs. \(2 (138) = 276\).
Time = 0.33 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.93 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=\frac {2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {9177 \, a - \frac {87633 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {375732 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {953988 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {1594782 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {1336734 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {781956 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {302004 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {69201 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {7129 \, a {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{9}}}{2520 \, d} \]
1/2520*(2520*a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 1)) - 2520 *a*log(abs(-(cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)) + (9177*a - 87633* a*(cos(d*x + c) - 1)/(cos(d*x + c) + 1) + 375732*a*(cos(d*x + c) - 1)^2/(c os(d*x + c) + 1)^2 - 953988*a*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 1594782*a*(cos(d*x + c) - 1)^4/(cos(d*x + c) + 1)^4 - 1336734*a*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1)^5 + 781956*a*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6 - 302004*a*(cos(d*x + c) - 1)^7/(cos(d*x + c) + 1)^7 + 69201*a* (cos(d*x + c) - 1)^8/(cos(d*x + c) + 1)^8 - 7129*a*(cos(d*x + c) - 1)^9/(c os(d*x + c) + 1)^9)/((cos(d*x + c) - 1)/(cos(d*x + c) + 1) - 1)^9)/d
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.73 \[ \int (a+a \sec (c+d x)) \sin ^9(c+d x) \, dx=-\frac {a\,\cos \left (c+d\,x\right )-2\,a\,{\cos \left (c+d\,x\right )}^2-\frac {4\,a\,{\cos \left (c+d\,x\right )}^3}{3}+\frac {3\,a\,{\cos \left (c+d\,x\right )}^4}{2}+\frac {6\,a\,{\cos \left (c+d\,x\right )}^5}{5}-\frac {2\,a\,{\cos \left (c+d\,x\right )}^6}{3}-\frac {4\,a\,{\cos \left (c+d\,x\right )}^7}{7}+\frac {a\,{\cos \left (c+d\,x\right )}^8}{8}+\frac {a\,{\cos \left (c+d\,x\right )}^9}{9}+a\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \]