Integrand size = 21, antiderivative size = 131 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {8 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{d}-\frac {2 a^3 \cos ^4(c+d x)}{d}+\frac {a^3 \cos ^6(c+d x)}{2 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \]
8*a^3*cos(d*x+c)/d+3*a^3*cos(d*x+c)^2/d-2*a^3*cos(d*x+c)^3/d-2*a^3*cos(d*x +c)^4/d+1/2*a^3*cos(d*x+c)^6/d+1/7*a^3*cos(d*x+c)^7/d+3*a^3*sec(d*x+c)/d+1 /2*a^3*sec(d*x+c)^2/d
Time = 0.59 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {a^3 (427+14014 \cos (c+d x)-210 \cos (2 (c+d x))+2548 \cos (3 (c+d x))+196 \cos (4 (c+d x))-188 \cos (5 (c+d x))-56 \cos (6 (c+d x))+9 \cos (7 (c+d x))+7 \cos (8 (c+d x))+\cos (9 (c+d x))) \sec ^2(c+d x)}{1792 d} \]
(a^3*(427 + 14014*Cos[c + d*x] - 210*Cos[2*(c + d*x)] + 2548*Cos[3*(c + d* x)] + 196*Cos[4*(c + d*x)] - 188*Cos[5*(c + d*x)] - 56*Cos[6*(c + d*x)] + 9*Cos[7*(c + d*x)] + 7*Cos[8*(c + d*x)] + Cos[9*(c + d*x)])*Sec[c + d*x]^2 )/(1792*d)
Time = 0.43 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.88, 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.
| \(\displaystyle \int \sin ^7(c+d x) (a \sec (c+d x)+a)^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos \left (c+d x-\frac {\pi }{2}\right )^7 \left (a-a \csc \left (c+d x-\frac {\pi }{2}\right )\right )^3dx\) |
| \(\Big \downarrow \) 4360 |
| \(\displaystyle \int \sin ^4(c+d x) \tan ^3(c+d x) \left (-(a (-\cos (c+d x))-a)^3\right )dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int -(\cos (c+d x) a+a)^3 \sin ^4(c+d x) \tan ^3(c+d x)dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \sin ^4(c+d x) \tan ^3(c+d x) (a \cos (c+d x)+a)^3dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\cos \left (c+d x+\frac {\pi }{2}\right )^7 \left (a \sin \left (c+d x+\frac {\pi }{2}\right )+a\right )^3}{\sin \left (c+d x+\frac {\pi }{2}\right )^3}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\cos \left (\frac {1}{2} (2 c+\pi )+d x\right )^7 \left (\sin \left (\frac {1}{2} (2 c+\pi )+d x\right ) a+a\right )^3}{\sin \left (\frac {1}{2} (2 c+\pi )+d x\right )^3}dx\) |
| \(\Big \downarrow \) 3315 |
| \(\displaystyle -\frac {\int (a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^6 \sec ^3(c+d x)d(a \cos (c+d x))}{a^7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {(a-a \cos (c+d x))^3 (\cos (c+d x) a+a)^6 \sec ^3(c+d x)}{a^3}d(a \cos (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 99 |
| \(\displaystyle -\frac {\int \left (-\cos ^6(c+d x) a^6-3 \cos ^5(c+d x) a^6+8 \cos ^3(c+d x) a^6+\sec ^3(c+d x) a^6+6 \cos ^2(c+d x) a^6+3 \sec ^2(c+d x) a^6-6 \cos (c+d x) a^6-8 a^6\right )d(a \cos (c+d x))}{a^4 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)-\frac {1}{2} a^7 \cos ^6(c+d x)+2 a^7 \cos ^4(c+d x)+2 a^7 \cos ^3(c+d x)-3 a^7 \cos ^2(c+d x)-8 a^7 \cos (c+d x)-\frac {1}{2} a^7 \sec ^2(c+d x)-3 a^7 \sec (c+d x)}{a^4 d}\) |
-((-8*a^7*Cos[c + d*x] - 3*a^7*Cos[c + d*x]^2 + 2*a^7*Cos[c + d*x]^3 + 2*a ^7*Cos[c + d*x]^4 - (a^7*Cos[c + d*x]^6)/2 - (a^7*Cos[c + d*x]^7)/7 - 3*a^ 7*Sec[c + d*x] - (a^7*Sec[c + d*x]^2)/2)/(a^4*d))
3.1.39.3.1 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 = 2.36 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.90
| method | result | size |
| parallelrisch | \(\frac {a^{3} \left (196 \cos \left (4 d x +4 c \right )+2548 \cos \left (3 d x +3 c \right )+14014 \cos \left (d x +c \right )+7 \cos \left (8 d x +8 c \right )+9 \cos \left (7 d x +7 c \right )-56 \cos \left (6 d x +6 c \right )-188 \cos \left (5 d x +5 c \right )+7800 \cos \left (2 d x +2 c \right )+\cos \left (9 d x +9 c \right )+8437\right )}{896 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(118\) |
| derivativedivides | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(214\) |
| default | \(\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )+3 a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )+3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )-\frac {a^{3} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7}}{d}\) | \(214\) |
| parts | \(-\frac {a^{3} \left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )}{7 d}+\frac {a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{2 \cos \left (d x +c \right )^{2}}+\frac {\sin \left (d x +c \right )^{6}}{2}+\frac {3 \sin \left (d x +c \right )^{4}}{4}+\frac {3 \sin \left (d x +c \right )^{2}}{2}+3 \ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (-\frac {\sin \left (d x +c \right )^{6}}{6}-\frac {\sin \left (d x +c \right )^{4}}{4}-\frac {\sin \left (d x +c \right )^{2}}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}+\frac {3 a^{3} \left (\frac {\sin \left (d x +c \right )^{8}}{\cos \left (d x +c \right )}+\left (\frac {16}{5}+\sin \left (d x +c \right )^{6}+\frac {6 \sin \left (d x +c \right )^{4}}{5}+\frac {8 \sin \left (d x +c \right )^{2}}{5}\right ) \cos \left (d x +c \right )\right )}{d}\) | \(222\) |
| risch | \(-\frac {29 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{128 d}+\frac {47 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {421 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {421 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {47 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {29 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{128 d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \cos \left (6 d x +6 c \right )}{64 d}+\frac {a^{3} \cos \left (5 d x +5 c \right )}{64 d}-\frac {5 a^{3} \cos \left (4 d x +4 c \right )}{32 d}\) | \(225\) |
1/896/d*a^3*(196*cos(4*d*x+4*c)+2548*cos(3*d*x+3*c)+14014*cos(d*x+c)+7*cos (8*d*x+8*c)+9*cos(7*d*x+7*c)-56*cos(6*d*x+6*c)-188*cos(5*d*x+5*c)+7800*cos (2*d*x+2*c)+cos(9*d*x+9*c)+8437)/(1+cos(2*d*x+2*c))
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {32 \, a^{3} \cos \left (d x + c\right )^{9} + 112 \, a^{3} \cos \left (d x + c\right )^{8} - 448 \, a^{3} \cos \left (d x + c\right )^{6} - 448 \, a^{3} \cos \left (d x + c\right )^{5} + 672 \, a^{3} \cos \left (d x + c\right )^{4} + 1792 \, a^{3} \cos \left (d x + c\right )^{3} - 203 \, a^{3} \cos \left (d x + c\right )^{2} + 672 \, a^{3} \cos \left (d x + c\right ) + 112 \, a^{3}}{224 \, d \cos \left (d x + c\right )^{2}} \]
1/224*(32*a^3*cos(d*x + c)^9 + 112*a^3*cos(d*x + c)^8 - 448*a^3*cos(d*x + c)^6 - 448*a^3*cos(d*x + c)^5 + 672*a^3*cos(d*x + c)^4 + 1792*a^3*cos(d*x + c)^3 - 203*a^3*cos(d*x + c)^2 + 672*a^3*cos(d*x + c) + 112*a^3)/(d*cos(d *x + c)^2)
Timed out. \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\text {Timed out} \]
Time = 0.20 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{7} + 7 \, a^{3} \cos \left (d x + c\right )^{6} - 28 \, a^{3} \cos \left (d x + c\right )^{4} - 28 \, a^{3} \cos \left (d x + c\right )^{3} + 42 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3} \cos \left (d x + c\right ) + \frac {7 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{14 \, d} \]
1/14*(2*a^3*cos(d*x + c)^7 + 7*a^3*cos(d*x + c)^6 - 28*a^3*cos(d*x + c)^4 - 28*a^3*cos(d*x + c)^3 + 42*a^3*cos(d*x + c)^2 + 112*a^3*cos(d*x + c) + 7 *(6*a^3*cos(d*x + c) + a^3)/cos(d*x + c)^2)/d
Time = 0.46 (sec) , antiderivative size = 239, normalized size of antiderivative = 1.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {2 \, {\left (\frac {7 \, {\left (3 \, a^{3} + \frac {2 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{2}} - \frac {43 \, a^{3} - \frac {273 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {672 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {630 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {343 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {105 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {14 \, a^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1\right )}^{7}}\right )}}{7 \, d} \]
2/7*(7*(3*a^3 + 2*a^3*(cos(d*x + c) - 1)/(cos(d*x + c) + 1))/((cos(d*x + c ) - 1)/(cos(d*x + c) + 1) + 1)^2 - (43*a^3 - 273*a^3*(cos(d*x + c) - 1)/(c os(d*x + c) + 1) + 672*a^3*(cos(d*x + c) - 1)^2/(cos(d*x + c) + 1)^2 - 630 *a^3*(cos(d*x + c) - 1)^3/(cos(d*x + c) + 1)^3 + 343*a^3*(cos(d*x + c) - 1 )^4/(cos(d*x + c) + 1)^4 - 105*a^3*(cos(d*x + c) - 1)^5/(cos(d*x + c) + 1) ^5 + 14*a^3*(cos(d*x + c) - 1)^6/(cos(d*x + c) + 1)^6)/((cos(d*x + c) - 1) /(cos(d*x + c) + 1) - 1)^7)/d
Time = 13.50 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+8\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-2\,a^3\,{\cos \left (c+d\,x\right )}^3-2\,a^3\,{\cos \left (c+d\,x\right )}^4+\frac {a^3\,{\cos \left (c+d\,x\right )}^6}{2}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}}{d} \]