Integrand size = 27, antiderivative size = 152 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5 a b x}{64}-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{252 d}+\frac {5 a b \cos (c+d x) \sin (c+d x)}{64 d}+\frac {5 a b \cos ^3(c+d x) \sin (c+d x)}{96 d}+\frac {a b \cos ^5(c+d x) \sin (c+d x)}{24 d}-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{36 d}-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d} \] Output:
5/64*a*b*x-1/252*(a^2+8*b^2)*cos(d*x+c)^7/d+5/64*a*b*cos(d*x+c)*sin(d*x+c) /d+5/96*a*b*cos(d*x+c)^3*sin(d*x+c)/d+1/24*a*b*cos(d*x+c)^5*sin(d*x+c)/d-1 /36*a*cos(d*x+c)^7*(a+b*sin(d*x+c))/d-1/9*cos(d*x+c)^7*(a+b*sin(d*x+c))^2/ d
Time = 0.94 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.06 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {-1260 a b c-1260 a b d x+126 \left (10 a^2+3 b^2\right ) \cos (c+d x)+84 \left (9 a^2+2 b^2\right ) \cos (3 (c+d x))+252 a^2 \cos (5 (c+d x))+36 a^2 \cos (7 (c+d x))-27 b^2 \cos (7 (c+d x))-7 b^2 \cos (9 (c+d x))-504 a b \sin (2 (c+d x))+252 a b \sin (4 (c+d x))+168 a b \sin (6 (c+d x))+\frac {63}{2} a b \sin (8 (c+d x))}{16128 d} \] Input:
Integrate[Cos[c + d*x]^6*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]
Output:
-1/16128*(-1260*a*b*c - 1260*a*b*d*x + 126*(10*a^2 + 3*b^2)*Cos[c + d*x] + 84*(9*a^2 + 2*b^2)*Cos[3*(c + d*x)] + 252*a^2*Cos[5*(c + d*x)] + 36*a^2*C os[7*(c + d*x)] - 27*b^2*Cos[7*(c + d*x)] - 7*b^2*Cos[9*(c + d*x)] - 504*a *b*Sin[2*(c + d*x)] + 252*a*b*Sin[4*(c + d*x)] + 168*a*b*Sin[6*(c + d*x)] + (63*a*b*Sin[8*(c + d*x)])/2)/d
Time = 0.81 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.11, number of steps used = 14, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {3042, 3341, 27, 3042, 3341, 3042, 3148, 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.
| \(\displaystyle \int \sin (c+d x) \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sin (c+d x) \cos (c+d x)^6 (a+b \sin (c+d x))^2dx\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{9} \int 2 \cos ^6(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))dx-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{9} \int \cos ^6(c+d x) (b+a \sin (c+d x)) (a+b \sin (c+d x))dx-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \int \cos (c+d x)^6 (b+a \sin (c+d x)) (a+b \sin (c+d x))dx-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \int \cos ^6(c+d x) \left (9 a b+\left (a^2+8 b^2\right ) \sin (c+d x)\right )dx-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \int \cos (c+d x)^6 \left (9 a b+\left (a^2+8 b^2\right ) \sin (c+d x)\right )dx-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \int \cos ^6(c+d x)dx-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {2}{9} \left (\frac {1}{8} \left (9 a b \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {\left (a^2+8 b^2\right ) \cos ^7(c+d x)}{7 d}\right )-\frac {a \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\right )-\frac {\cos ^7(c+d x) (a+b \sin (c+d x))^2}{9 d}\) |
Input:
Int[Cos[c + d*x]^6*Sin[c + d*x]*(a + b*Sin[c + d*x])^2,x]
Output:
-1/9*(Cos[c + d*x]^7*(a + b*Sin[c + d*x])^2)/d + (2*(-1/8*(a*Cos[c + d*x]^ 7*(a + b*Sin[c + d*x]))/d + (-1/7*((a^2 + 8*b^2)*Cos[c + d*x]^7)/d + 9*a*b *((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))/9
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)]), x_Symbol] :> Simp[(-b)*((g*Cos[e + f*x])^(p + 1)/(f*g*(p + 1))), x] + Simp[a Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x] && (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d)* (g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^m/(f*g*(m + p + 1))), x] + S imp[1/(m + p + 1) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1)*Sim p[a*c*(m + p + 1) + b*d*m + (a*d*m + b*c*(m + p + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && !LtQ[p, -1] && IntegerQ[2*m] && !(EqQ[m, 1] && NeQ[c^2 - d^2, 0] && S implerQ[c + d*x, a + b*x])
Time = 0.37 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.76
\[\frac {-\frac {a^{2} \cos \left (d x +c \right )^{7}}{7}+2 a b \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+b^{2} \left (-\frac {\cos \left (d x +c \right )^{7} \sin \left (d x +c \right )^{2}}{9}-\frac {2 \cos \left (d x +c \right )^{7}}{63}\right )}{d}\]
Input:
int(cos(d*x+c)^6*sin(d*x+c)*(a+b*sin(d*x+c))^2,x)
Output:
1/d*(-1/7*a^2*cos(d*x+c)^7+2*a*b*(-1/8*cos(d*x+c)^7*sin(d*x+c)+1/48*(cos(d *x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d*x+5/128*c)+b^ 2*(-1/9*cos(d*x+c)^7*sin(d*x+c)^2-2/63*cos(d*x+c)^7))
Time = 0.09 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.64 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {448 \, b^{2} \cos \left (d x + c\right )^{9} - 576 \, {\left (a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{7} + 315 \, a b d x - 21 \, {\left (48 \, a b \cos \left (d x + c\right )^{7} - 8 \, a b \cos \left (d x + c\right )^{5} - 10 \, a b \cos \left (d x + c\right )^{3} - 15 \, a b \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4032 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="fricas" )
Output:
1/4032*(448*b^2*cos(d*x + c)^9 - 576*(a^2 + b^2)*cos(d*x + c)^7 + 315*a*b* d*x - 21*(48*a*b*cos(d*x + c)^7 - 8*a*b*cos(d*x + c)^5 - 10*a*b*cos(d*x + c)^3 - 15*a*b*cos(d*x + c))*sin(d*x + c))/d
Time = 0.94 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.86 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} - \frac {a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 a b x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {5 a b x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a b x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {5 a b x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a b x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {5 a b \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {55 a b \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac {73 a b \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac {5 a b \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {b^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {2 b^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \sin {\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*sin(d*x+c)*(a+b*sin(d*x+c))**2,x)
Output:
Piecewise((-a**2*cos(c + d*x)**7/(7*d) + 5*a*b*x*sin(c + d*x)**8/64 + 5*a* b*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 15*a*b*x*sin(c + d*x)**4*cos(c + d*x)**4/32 + 5*a*b*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 5*a*b*x*cos(c + d*x)**8/64 + 5*a*b*sin(c + d*x)**7*cos(c + d*x)/(64*d) + 55*a*b*sin(c + d* x)**5*cos(c + d*x)**3/(192*d) + 73*a*b*sin(c + d*x)**3*cos(c + d*x)**5/(19 2*d) - 5*a*b*sin(c + d*x)*cos(c + d*x)**7/(64*d) - b**2*sin(c + d*x)**2*co s(c + d*x)**7/(7*d) - 2*b**2*cos(c + d*x)**9/(63*d), Ne(d, 0)), (x*(a + b* sin(c))**2*sin(c)*cos(c)**6, True))
Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {4608 \, a^{2} \cos \left (d x + c\right )^{7} - 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b - 512 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} b^{2}}{32256 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="maxima" )
Output:
-1/32256*(4608*a^2*cos(d*x + c)^7 - 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*b - 512*(7*cos(d*x + c )^9 - 9*cos(d*x + c)^7)*b^2)/d
Time = 0.27 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.16 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5}{64} \, a b x + \frac {b^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {a b \sin \left (8 \, d x + 8 \, c\right )}{512 \, d} - \frac {a b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} - \frac {a b \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {a b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} - \frac {{\left (4 \, a^{2} - 3 \, b^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (10 \, a^{2} + 3 \, b^{2}\right )} \cos \left (d x + c\right )}{128 \, d} \] Input:
integrate(cos(d*x+c)^6*sin(d*x+c)*(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
5/64*a*b*x + 1/2304*b^2*cos(9*d*x + 9*c)/d - 1/64*a^2*cos(5*d*x + 5*c)/d - 1/512*a*b*sin(8*d*x + 8*c)/d - 1/96*a*b*sin(6*d*x + 6*c)/d - 1/64*a*b*sin (4*d*x + 4*c)/d + 1/32*a*b*sin(2*d*x + 2*c)/d - 1/1792*(4*a^2 - 3*b^2)*cos (7*d*x + 7*c)/d - 1/192*(9*a^2 + 2*b^2)*cos(3*d*x + 3*c)/d - 1/128*(10*a^2 + 3*b^2)*cos(d*x + c)/d
Time = 23.29 (sec) , antiderivative size = 332, normalized size of antiderivative = 2.18 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5\,a\,b\,x}{64}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (4\,a^2+4\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {4\,a^2}{7}+\frac {4\,b^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (12\,a^2+12\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (16\,a^2-12\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (12\,a^2-\frac {20\,b^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (20\,a^2+20\,b^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {44\,a^2}{7}-\frac {12\,b^2}{7}\right )+2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {2\,a^2}{7}+\frac {4\,b^2}{63}-\frac {191\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{48}+\frac {83\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{16}-\frac {145\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{16}+\frac {145\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{16}-\frac {83\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{16}+\frac {191\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{48}-\frac {5\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{32}+\frac {5\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{32}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^9} \] Input:
int(cos(c + d*x)^6*sin(c + d*x)*(a + b*sin(c + d*x))^2,x)
Output:
(5*a*b*x)/64 - (tan(c/2 + (d*x)/2)^14*(4*a^2 + 4*b^2) + tan(c/2 + (d*x)/2) ^2*((4*a^2)/7 + (4*b^2)/7) + tan(c/2 + (d*x)/2)^6*(12*a^2 + 12*b^2) + tan( c/2 + (d*x)/2)^8*(16*a^2 - 12*b^2) + tan(c/2 + (d*x)/2)^12*(12*a^2 - (20*b ^2)/3) + tan(c/2 + (d*x)/2)^10*(20*a^2 + 20*b^2) + tan(c/2 + (d*x)/2)^4*(( 44*a^2)/7 - (12*b^2)/7) + 2*a^2*tan(c/2 + (d*x)/2)^16 + (2*a^2)/7 + (4*b^2 )/63 - (191*a*b*tan(c/2 + (d*x)/2)^3)/48 + (83*a*b*tan(c/2 + (d*x)/2)^5)/1 6 - (145*a*b*tan(c/2 + (d*x)/2)^7)/16 + (145*a*b*tan(c/2 + (d*x)/2)^11)/16 - (83*a*b*tan(c/2 + (d*x)/2)^13)/16 + (191*a*b*tan(c/2 + (d*x)/2)^15)/48 - (5*a*b*tan(c/2 + (d*x)/2)^17)/32 + (5*a*b*tan(c/2 + (d*x)/2))/32)/(d*(ta n(c/2 + (d*x)/2)^2 + 1)^9)
Time = 0.18 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.62 \[ \int \cos ^6(c+d x) \sin (c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {448 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{8} b^{2}+1008 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} a b +576 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{2}-1216 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} b^{2}-2856 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a b -1728 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{2}+960 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{2}+2478 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a b +1728 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{2}-64 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} b^{2}-315 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a b -576 \cos \left (d x +c \right ) a^{2}-128 \cos \left (d x +c \right ) b^{2}+576 a^{2}+315 a b d x +128 b^{2}}{4032 d} \] Input:
int(cos(d*x+c)^6*sin(d*x+c)*(a+b*sin(d*x+c))^2,x)
Output:
(448*cos(c + d*x)*sin(c + d*x)**8*b**2 + 1008*cos(c + d*x)*sin(c + d*x)**7 *a*b + 576*cos(c + d*x)*sin(c + d*x)**6*a**2 - 1216*cos(c + d*x)*sin(c + d *x)**6*b**2 - 2856*cos(c + d*x)*sin(c + d*x)**5*a*b - 1728*cos(c + d*x)*si n(c + d*x)**4*a**2 + 960*cos(c + d*x)*sin(c + d*x)**4*b**2 + 2478*cos(c + d*x)*sin(c + d*x)**3*a*b + 1728*cos(c + d*x)*sin(c + d*x)**2*a**2 - 64*cos (c + d*x)*sin(c + d*x)**2*b**2 - 315*cos(c + d*x)*sin(c + d*x)*a*b - 576*c os(c + d*x)*a**2 - 128*cos(c + d*x)*b**2 + 576*a**2 + 315*a*b*d*x + 128*b* *2)/(4032*d)