Integrand size = 21, antiderivative size = 131 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {8 a^3 \cos (c+d x)}{d}+\frac {3 a^3 \cos ^2(c+d x)}{d}-\frac {2 a^3 \cos ^3(c+d x)}{d}-\frac {2 a^3 \cos ^4(c+d x)}{d}+\frac {a^3 \cos ^6(c+d x)}{2 d}+\frac {a^3 \cos ^7(c+d x)}{7 d}+\frac {3 a^3 \sec (c+d x)}{d}+\frac {a^3 \sec ^2(c+d x)}{2 d} \] Output:
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 = 0.37 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.81 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {a^3 (679+14014 \cos (c+d x)+42 \cos (2 (c+d x))+2548 \cos (3 (c+d x))+196 \cos (4 (c+d x))-188 \cos (5 (c+d x))-56 \cos (6 (c+d x))+9 \cos (7 (c+d x))+7 \cos (8 (c+d x))+\cos (9 (c+d x))) \sec ^2(c+d x)}{1792 d} \] Input:
Integrate[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^7,x]
Output:
(a^3*(679 + 14014*Cos[c + d*x] + 42*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 = 0.44 (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 |
| \(\displaystyle -\frac {-\frac {1}{7} a^7 \cos ^7(c+d x)-\frac {1}{2} a^7 \cos ^6(c+d x)+2 a^7 \cos ^4(c+d x)+2 a^7 \cos ^3(c+d x)-3 a^7 \cos ^2(c+d x)-8 a^7 \cos (c+d x)-\frac {1}{2} a^7 \sec ^2(c+d x)-3 a^7 \sec (c+d x)}{a^4 d}\) |
Input:
Int[(a + a*Sec[c + d*x])^3*Sin[c + d*x]^7,x]
Output:
-((-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))
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 = 1.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.92
| method | result | size |
| parallelrisch | \(-\frac {a^{3} \left (-14014 \cos \left (d x +c \right )-2548 \cos \left (3 d x +3 c \right )+188 \cos \left (5 d x +5 c \right )-\cos \left (9 d x +9 c \right )-9 \cos \left (7 d x +7 c \right )-7800 \cos \left (2 d x +2 c \right )-7 \cos \left (8 d x +8 c \right )+56 \cos \left (6 d x +6 c \right )-196 \cos \left (4 d x +4 c \right )-8437\right )}{896 d \left (1+\cos \left (2 d x +2 c \right )\right )}\) | \(120\) |
| 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 | \(-\frac {29 a^{3} {\mathrm e}^{3 i \left (d x +c \right )}}{128 d}+\frac {47 a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{128 d}+\frac {421 a^{3} {\mathrm e}^{i \left (d x +c \right )}}{128 d}+\frac {421 a^{3} {\mathrm e}^{-i \left (d x +c \right )}}{128 d}+\frac {47 a^{3} {\mathrm e}^{-2 i \left (d x +c \right )}}{128 d}-\frac {29 a^{3} {\mathrm e}^{-3 i \left (d x +c \right )}}{128 d}+\frac {2 a^{3} \left (3 \,{\mathrm e}^{3 i \left (d x +c \right )}+{\mathrm e}^{2 i \left (d x +c \right )}+3 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}+\frac {a^{3} \cos \left (7 d x +7 c \right )}{448 d}+\frac {a^{3} \cos \left (6 d x +6 c \right )}{64 d}+\frac {a^{3} \cos \left (5 d x +5 c \right )}{64 d}-\frac {5 a^{3} \cos \left (4 d x +4 c \right )}{32 d}\) | \(225\) |
Input:
int((a+a*sec(d*x+c))^3*sin(d*x+c)^7,x,method=_RETURNVERBOSE)
Output:
-1/896/d*a^3*(-14014*cos(d*x+c)-2548*cos(3*d*x+3*c)+188*cos(5*d*x+5*c)-cos (9*d*x+9*c)-9*cos(7*d*x+7*c)-7800*cos(2*d*x+2*c)-7*cos(8*d*x+8*c)+56*cos(6 *d*x+6*c)-196*cos(4*d*x+4*c)-8437)/(1+cos(2*d*x+2*c))
Time = 0.09 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.92 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {32 \, a^{3} \cos \left (d x + c\right )^{9} + 112 \, a^{3} \cos \left (d x + c\right )^{8} - 448 \, a^{3} \cos \left (d x + c\right )^{6} - 448 \, a^{3} \cos \left (d x + c\right )^{5} + 672 \, a^{3} \cos \left (d x + c\right )^{4} + 1792 \, a^{3} \cos \left (d x + c\right )^{3} - 203 \, a^{3} \cos \left (d x + c\right )^{2} + 672 \, a^{3} \cos \left (d x + c\right ) + 112 \, a^{3}}{224 \, d \cos \left (d x + c\right )^{2}} \] Input:
integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^7,x, algorithm="fricas")
Output:
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. \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\text {Timed out} \] Input:
integrate((a+a*sec(d*x+c))**3*sin(d*x+c)**7,x)
Output:
Timed out
Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{7} + 7 \, a^{3} \cos \left (d x + c\right )^{6} - 28 \, a^{3} \cos \left (d x + c\right )^{4} - 28 \, a^{3} \cos \left (d x + c\right )^{3} + 42 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3} \cos \left (d x + c\right ) + \frac {7 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{14 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^7,x, algorithm="maxima")
Output:
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 = 0.17 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {2 \, a^{3} \cos \left (d x + c\right )^{7} + 7 \, a^{3} \cos \left (d x + c\right )^{6} - 28 \, a^{3} \cos \left (d x + c\right )^{4} - 28 \, a^{3} \cos \left (d x + c\right )^{3} + 42 \, a^{3} \cos \left (d x + c\right )^{2} + 112 \, a^{3} \cos \left (d x + c\right ) + \frac {7 \, {\left (6 \, a^{3} \cos \left (d x + c\right ) + a^{3}\right )}}{\cos \left (d x + c\right )^{2}}}{14 \, d} \] Input:
integrate((a+a*sec(d*x+c))^3*sin(d*x+c)^7,x, algorithm="giac")
Output:
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 = 10.46 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.82 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {\frac {3\,a^3\,\cos \left (c+d\,x\right )+\frac {a^3}{2}}{{\cos \left (c+d\,x\right )}^2}+8\,a^3\,\cos \left (c+d\,x\right )+3\,a^3\,{\cos \left (c+d\,x\right )}^2-2\,a^3\,{\cos \left (c+d\,x\right )}^3-2\,a^3\,{\cos \left (c+d\,x\right )}^4+\frac {a^3\,{\cos \left (c+d\,x\right )}^6}{2}+\frac {a^3\,{\cos \left (c+d\,x\right )}^7}{7}}{d} \] Input:
int(sin(c + d*x)^7*(a + a/cos(c + d*x))^3,x)
Output:
((3*a^3*cos(c + d*x) + a^3/2)/cos(c + d*x)^2 + 8*a^3*cos(c + d*x) + 3*a^3* cos(c + d*x)^2 - 2*a^3*cos(c + d*x)^3 - 2*a^3*cos(c + d*x)^4 + (a^3*cos(c + d*x)^6)/2 + (a^3*cos(c + d*x)^7)/7)/d
Time = 0.19 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.47 \[ \int (a+a \sec (c+d x))^3 \sin ^7(c+d x) \, dx=\frac {a^{3} \left (-10 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{8}-2 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{6}-4 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{4}-16 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{2}+32 \cos \left (d x +c \right )^{2}-35 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}-640 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}+640 \cos \left (d x +c \right )-42 \sin \left (d x +c \right )^{8}-42 \sin \left (d x +c \right )^{6}-252 \sin \left (d x +c \right )^{4}+1008 \sin \left (d x +c \right )^{2}-672\right )}{70 \cos \left (d x +c \right ) d \left (\sin \left (d x +c \right )^{2}-1\right )} \] Input:
int((a+a*sec(d*x+c))^3*sin(d*x+c)^7,x)
Output:
(a**3*( - 10*cos(c + d*x)**2*sin(c + d*x)**8 - 2*cos(c + d*x)**2*sin(c + d *x)**6 - 4*cos(c + d*x)**2*sin(c + d*x)**4 - 16*cos(c + d*x)**2*sin(c + d* x)**2 + 32*cos(c + d*x)**2 - 35*cos(c + d*x)*sin(c + d*x)**8 - 640*cos(c + d*x)*sin(c + d*x)**2 + 640*cos(c + d*x) - 42*sin(c + d*x)**8 - 42*sin(c + d*x)**6 - 252*sin(c + d*x)**4 + 1008*sin(c + d*x)**2 - 672))/(70*cos(c + d*x)*d*(sin(c + d*x)**2 - 1))