Integrand size = 27, antiderivative size = 125 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5 a x}{128}-\frac {a \cos ^7(c+d x)}{7 d}+\frac {a \cos ^9(c+d x)}{9 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {a \cos ^7(c+d x) \sin (c+d x)}{8 d} \] Output:
5/128*a*x-1/7*a*cos(d*x+c)^7/d+1/9*a*cos(d*x+c)^9/d+5/128*a*cos(d*x+c)*sin (d*x+c)/d+5/192*a*cos(d*x+c)^3*sin(d*x+c)/d+1/48*a*cos(d*x+c)^5*sin(d*x+c) /d-1/8*a*cos(d*x+c)^7*sin(d*x+c)/d
Time = 0.24 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a (2520 d x-1512 \cos (c+d x)-672 \cos (3 (c+d x))+108 \cos (7 (c+d x))+28 \cos (9 (c+d x))+1008 \sin (2 (c+d x))-504 \sin (4 (c+d x))-336 \sin (6 (c+d x))-63 \sin (8 (c+d x)))}{64512 d} \] Input:
Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
(a*(2520*d*x - 1512*Cos[c + d*x] - 672*Cos[3*(c + d*x)] + 108*Cos[7*(c + d *x)] + 28*Cos[9*(c + d*x)] + 1008*Sin[2*(c + d*x)] - 504*Sin[4*(c + d*x)] - 336*Sin[6*(c + d*x)] - 63*Sin[8*(c + d*x)]))/(64512*d)
Time = 0.66 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.10, number of steps used = 15, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3317, 3042, 3045, 244, 2009, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24}
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.
| \(\displaystyle \int \sin ^2(c+d x) \cos ^6(c+d x) (a \sin (c+d x)+a) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x)^2 \cos (c+d x)^6 (a \sin (c+d x)+a)dx\) |
| \(\Big \downarrow \) 3317 |
| \(\displaystyle a \int \cos ^6(c+d x) \sin ^3(c+d x)dx+a \int \cos ^6(c+d x) \sin ^2(c+d x)dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^2dx+a \int \cos (c+d x)^6 \sin (c+d x)^3dx\) |
| \(\Big \downarrow \) 3045 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {a \int \cos ^6(c+d x) \left (1-\cos ^2(c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 244 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {a \int \left (\cos ^6(c+d x)-\cos ^8(c+d x)\right )d\cos (c+d x)}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle a \int \cos (c+d x)^6 \sin (c+d x)^2dx-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3048 |
| \(\displaystyle a \left (\frac {1}{8} \int \cos ^6(c+d x)dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {1}{8} \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle a \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle a \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle a \left (\frac {1}{8} \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\sin (c+d x) \cos ^7(c+d x)}{8 d}\right )-\frac {a \left (\frac {1}{7} \cos ^7(c+d x)-\frac {1}{9} \cos ^9(c+d x)\right )}{d}\) |
Input:
Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x]),x]
Output:
-((a*(Cos[c + d*x]^7/7 - Cos[c + d*x]^9/9))/d) + a*(-1/8*(Cos[c + d*x]^7*S in[c + d*x])/d + ((Cos[c + d*x]^5*Sin[c + d*x])/(6*d) + (5*((Cos[c + d*x]^ 3*Sin[c + d*x])/(4*d) + (3*(x/2 + (Cos[c + d*x]*Sin[c + d*x])/(2*d)))/4))/ 6)/8)
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_)]*(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_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m _), x_Symbol] :> Simp[(-a)*(b*Cos[e + f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Simp[a^2*((m - 1)/(m + n)) Int[(b*Cos[e + f*x])^n *(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*m, 2*n]
Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d* x]*((b*Sin[c + d*x])^(n - 1)/(d*n)), x] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n _.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[a Int[(g*Co s[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Simp[b/d Int[(g*Cos[e + f*x])^ p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]
Time = 65.90 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.77
| method | result | size |
| parallelrisch | \(-\frac {\left (-5 d x +\sin \left (4 d x +4 c \right )+\frac {2 \sin \left (6 d x +6 c \right )}{3}+\frac {\sin \left (8 d x +8 c \right )}{8}+3 \cos \left (d x +c \right )+\frac {4 \cos \left (3 d x +3 c \right )}{3}-\frac {3 \cos \left (7 d x +7 c \right )}{14}-\frac {\cos \left (9 d x +9 c \right )}{18}-2 \sin \left (2 d x +2 c \right )+\frac {256}{63}\right ) a}{128 d}\) | \(96\) |
| derivativedivides | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+a \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(98\) |
| default | \(\frac {a \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )+a \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(98\) |
| risch | \(\frac {5 a x}{128}-\frac {3 a \cos \left (d x +c \right )}{128 d}+\frac {a \cos \left (9 d x +9 c \right )}{2304 d}-\frac {a \sin \left (8 d x +8 c \right )}{1024 d}+\frac {3 a \cos \left (7 d x +7 c \right )}{1792 d}-\frac {a \sin \left (6 d x +6 c \right )}{192 d}-\frac {a \sin \left (4 d x +4 c \right )}{128 d}-\frac {a \cos \left (3 d x +3 c \right )}{96 d}+\frac {a \sin \left (2 d x +2 c \right )}{64 d}\) | \(123\) |
| norman | \(\frac {-\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{7 d}-\frac {4 a}{63 d}+\frac {12 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{7 d}+\frac {5 a x}{128}+\frac {12 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {145 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{32 d}+\frac {20 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{3 d}-\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}+\frac {315 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{64}+\frac {315 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{64}+\frac {105 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{32}+\frac {45 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{128}+\frac {45 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{32}+\frac {105 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{32}+\frac {45 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}}{128}+\frac {5 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{18}}{128}-\frac {12 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {191 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{96 d}+\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{17}}{64 d}-\frac {4 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {145 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{32 d}+\frac {83 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{32 d}-\frac {83 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{32 d}+\frac {45 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{32}-\frac {20 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}+\frac {191 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{96 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{9}}\) | \(416\) |
| orering | \(\text {Expression too large to display}\) | \(3385\) |
Input:
int(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
-1/128*(-5*d*x+sin(4*d*x+4*c)+2/3*sin(6*d*x+6*c)+1/8*sin(8*d*x+8*c)+3*cos( d*x+c)+4/3*cos(3*d*x+3*c)-3/14*cos(7*d*x+7*c)-1/18*cos(9*d*x+9*c)-2*sin(2* d*x+2*c)+256/63)*a/d
Time = 0.10 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {896 \, a \cos \left (d x + c\right )^{9} - 1152 \, a \cos \left (d x + c\right )^{7} + 315 \, a d x - 21 \, {\left (48 \, a \cos \left (d x + c\right )^{7} - 8 \, a \cos \left (d x + c\right )^{5} - 10 \, a \cos \left (d x + c\right )^{3} - 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="fricas" )
Output:
1/8064*(896*a*cos(d*x + c)^9 - 1152*a*cos(d*x + c)^7 + 315*a*d*x - 21*(48* a*cos(d*x + c)^7 - 8*a*cos(d*x + c)^5 - 10*a*cos(d*x + c)^3 - 15*a*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 248 vs. \(2 (116) = 232\).
Time = 0.96 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.98 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\begin {cases} \frac {5 a x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 a x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 a x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 a x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 a \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 a \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 a \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {a \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 a \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {2 a \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right ) \sin ^{2}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)**2*(a+a*sin(d*x+c)),x)
Output:
Piecewise((5*a*x*sin(c + d*x)**8/128 + 5*a*x*sin(c + d*x)**6*cos(c + d*x)* *2/32 + 15*a*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 5*a*x*sin(c + d*x)**2* cos(c + d*x)**6/32 + 5*a*x*cos(c + d*x)**8/128 + 5*a*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*a*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 73*a*sin( c + d*x)**3*cos(c + d*x)**5/(384*d) - a*sin(c + d*x)**2*cos(c + d*x)**7/(7 *d) - 5*a*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 2*a*cos(c + d*x)**9/(63*d ), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**2*cos(c)**6, True))
Time = 0.04 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {1024 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a + 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{64512 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="maxima" )
Output:
1/64512*(1024*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a + 21*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 24*sin(4*d*x + 4*c))*a)/d
Time = 0.17 (sec) , antiderivative size = 122, normalized size of antiderivative = 0.98 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5}{128} \, a x + \frac {a \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} + \frac {3 \, a \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {a \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a \cos \left (d x + c\right )}{128 \, d} - \frac {a \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x, algorithm="giac")
Output:
5/128*a*x + 1/2304*a*cos(9*d*x + 9*c)/d + 3/1792*a*cos(7*d*x + 7*c)/d - 1/ 96*a*cos(3*d*x + 3*c)/d - 3/128*a*cos(d*x + c)/d - 1/1024*a*sin(8*d*x + 8* c)/d - 1/192*a*sin(6*d*x + 6*c)/d - 1/128*a*sin(4*d*x + 4*c)/d + 1/64*a*si n(2*d*x + 2*c)/d
Time = 37.39 (sec) , antiderivative size = 386, normalized size of antiderivative = 3.09 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {5\,a\,x}{128}+\frac {\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{64}-\frac {191\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{96}+\left (\frac {a\,\left (11340\,c+11340\,d\,x-32256\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {83\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\left (\frac {a\,\left (26460\,c+26460\,d\,x+53760\right )}{8064}-\frac {105\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-\frac {145\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{32}+\left (\frac {a\,\left (39690\,c+39690\,d\,x-161280\right )}{8064}-\frac {315\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\left (\frac {a\,\left (39690\,c+39690\,d\,x+96768\right )}{8064}-\frac {315\,a\,\left (c+d\,x\right )}{64}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {145\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\left (\frac {a\,\left (26460\,c+26460\,d\,x-96768\right )}{8064}-\frac {105\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-\frac {83\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{32}+\left (\frac {a\,\left (11340\,c+11340\,d\,x+13824\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{32}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {191\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{96}+\left (\frac {a\,\left (2835\,c+2835\,d\,x-4608\right )}{8064}-\frac {45\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {5\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{64}+\frac {a\,\left (315\,c+315\,d\,x-512\right )}{8064}-\frac {5\,a\,\left (c+d\,x\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \] Input:
int(cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x)),x)
Output:
(5*a*x)/128 + ((a*(315*c + 315*d*x - 512))/8064 - (5*a*tan(c/2 + (d*x)/2)) /64 - (5*a*(c + d*x))/128 + tan(c/2 + (d*x)/2)^2*((a*(2835*c + 2835*d*x - 4608))/8064 - (45*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^4*((a*(11340*c + 11340*d*x + 13824))/8064 - (45*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^14*(( a*(11340*c + 11340*d*x - 32256))/8064 - (45*a*(c + d*x))/32) + tan(c/2 + ( d*x)/2)^12*((a*(26460*c + 26460*d*x + 53760))/8064 - (105*a*(c + d*x))/32) + tan(c/2 + (d*x)/2)^6*((a*(26460*c + 26460*d*x - 96768))/8064 - (105*a*( c + d*x))/32) + tan(c/2 + (d*x)/2)^8*((a*(39690*c + 39690*d*x + 96768))/80 64 - (315*a*(c + d*x))/64) + tan(c/2 + (d*x)/2)^10*((a*(39690*c + 39690*d* x - 161280))/8064 - (315*a*(c + d*x))/64) + (191*a*tan(c/2 + (d*x)/2)^3)/9 6 - (83*a*tan(c/2 + (d*x)/2)^5)/32 + (145*a*tan(c/2 + (d*x)/2)^7)/32 - (14 5*a*tan(c/2 + (d*x)/2)^11)/32 + (83*a*tan(c/2 + (d*x)/2)^13)/32 - (191*a*t an(c/2 + (d*x)/2)^15)/96 + (5*a*tan(c/2 + (d*x)/2)^17)/64)/(d*(tan(c/2 + ( d*x)/2)^2 + 1)^9)
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.17 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x)) \, dx=\frac {a \left (896 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7}-2432 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6}-2856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5}+1920 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4}+2478 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3}-128 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right )-256 \cos \left (d x +c \right )+315 d x +256\right )}{8064 d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c)),x)
Output:
(a*(896*cos(c + d*x)*sin(c + d*x)**8 + 1008*cos(c + d*x)*sin(c + d*x)**7 - 2432*cos(c + d*x)*sin(c + d*x)**6 - 2856*cos(c + d*x)*sin(c + d*x)**5 + 1 920*cos(c + d*x)*sin(c + d*x)**4 + 2478*cos(c + d*x)*sin(c + d*x)**3 - 128 *cos(c + d*x)*sin(c + d*x)**2 - 315*cos(c + d*x)*sin(c + d*x) - 256*cos(c + d*x) + 315*d*x + 256))/(8064*d)