Integrand size = 21, antiderivative size = 146 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5}{128} \left (8 a^2+b^2\right ) x-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \] Output:
5/128*(8*a^2+b^2)*x-9/56*a*b*cos(d*x+c)^7/d+5/128*(8*a^2+b^2)*cos(d*x+c)*s in(d*x+c)/d+5/192*(8*a^2+b^2)*cos(d*x+c)^3*sin(d*x+c)/d+1/48*(8*a^2+b^2)*c os(d*x+c)^5*sin(d*x+c)/d-1/8*b*cos(d*x+c)^7*(a+b*sin(d*x+c))/d
Time = 0.67 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {840 \left (8 a^2+b^2\right ) (c+d x)-3360 a b \cos (c+d x)-2016 a b \cos (3 (c+d x))-672 a b \cos (5 (c+d x))-96 a b \cos (7 (c+d x))+336 \left (15 a^2+b^2\right ) \sin (2 (c+d x))+168 \left (6 a^2-b^2\right ) \sin (4 (c+d x))+112 (a-b) (a+b) \sin (6 (c+d x))-21 b^2 \sin (8 (c+d x))}{21504 d} \] Input:
Integrate[Cos[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]
Output:
(840*(8*a^2 + b^2)*(c + d*x) - 3360*a*b*Cos[c + d*x] - 2016*a*b*Cos[3*(c + d*x)] - 672*a*b*Cos[5*(c + d*x)] - 96*a*b*Cos[7*(c + d*x)] + 336*(15*a^2 + b^2)*Sin[2*(c + d*x)] + 168*(6*a^2 - b^2)*Sin[4*(c + d*x)] + 112*(a - b) *(a + b)*Sin[6*(c + d*x)] - 21*b^2*Sin[8*(c + d*x)])/(21504*d)
Time = 0.62 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.93, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.524, Rules used = {3042, 3171, 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.
| \(\displaystyle \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^6 (a+b \sin (c+d x))^2dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {1}{8} \int \cos ^6(c+d x) \left (8 a^2+9 b \sin (c+d x) a+b^2\right )dx-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \int \cos (c+d x)^6 \left (8 a^2+9 b \sin (c+d x) a+b^2\right )dx-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \int \cos ^6(c+d x)dx-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^6dx-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \left (\frac {5}{6} \int \cos ^4(c+d x)dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \left (\frac {5}{6} \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )+\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}\right )-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{8} \left (\left (8 a^2+b^2\right ) \left (\frac {\sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5}{6} \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )\right )-\frac {9 a b \cos ^7(c+d x)}{7 d}\right )-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\) |
Input:
Int[Cos[c + d*x]^6*(a + b*Sin[c + d*x])^2,x]
Output:
-1/8*(b*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/d + ((-9*a*b*Cos[c + d*x]^7)/ (7*d) + (8*a^2 + b^2)*((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[((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_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m - 1)/(f*g*(m + p))), x] + Simp[1/(m + p) Int[(g*Cos[e + f*x])^p* (a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1) *Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m])
Time = 59.51 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88
| method | result | size |
| derivativedivides | \(\frac {a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {2 a b \cos \left (d x +c \right )^{7}}{7}+b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(128\) |
| default | \(\frac {a^{2} \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {2 a b \cos \left (d x +c \right )^{7}}{7}+b^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{7}}{8}+\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\) | \(128\) |
| parallelrisch | \(\frac {336 \left (15 a^{2}+b^{2}\right ) \sin \left (2 d x +2 c \right )+168 \left (6 a^{2}-b^{2}\right ) \sin \left (4 d x +4 c \right )+112 \left (a^{2}-b^{2}\right ) \sin \left (6 d x +6 c \right )+6720 a^{2} d x +840 b^{2} d x -3360 \cos \left (d x +c \right ) a b -2016 a b \cos \left (3 d x +3 c \right )-672 a b \cos \left (5 d x +5 c \right )-96 a b \cos \left (7 d x +7 c \right )-21 b^{2} \sin \left (8 d x +8 c \right )-6144 a b}{21504 d}\) | \(150\) |
| risch | \(\frac {5 a^{2} x}{16}+\frac {5 b^{2} x}{128}-\frac {5 a b \cos \left (d x +c \right )}{32 d}-\frac {b^{2} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {a b \cos \left (7 d x +7 c \right )}{224 d}+\frac {\sin \left (6 d x +6 c \right ) a^{2}}{192 d}-\frac {\sin \left (6 d x +6 c \right ) b^{2}}{192 d}-\frac {a b \cos \left (5 d x +5 c \right )}{32 d}+\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{64 d}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{128 d}-\frac {3 a b \cos \left (3 d x +3 c \right )}{32 d}+\frac {15 a^{2} \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) b^{2}}{64 d}\) | \(194\) |
| norman | \(\frac {-\frac {\left (88 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{15}}{64 d}+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}+\left (\frac {35 a^{2}}{4}+\frac {35 b^{2}}{32}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (\frac {35 a^{2}}{2}+\frac {35 b^{2}}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}+\left (\frac {175 a^{2}}{8}+\frac {175 b^{2}}{64}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (\frac {35 a^{2}}{2}+\frac {35 b^{2}}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\left (\frac {35 a^{2}}{4}+\frac {35 b^{2}}{32}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{16}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}+\left (\frac {5 a^{2}}{16}+\frac {5 b^{2}}{128}\right ) x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{16}-\frac {5 \left (136 a^{2}+353 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{192 d}-\frac {\left (904 a^{2}-895 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{192 d}-\frac {\left (488 a^{2}+397 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{192 d}-\frac {12 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {20 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}-\frac {12 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{d}-\frac {20 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{7 d}+\frac {\left (488 a^{2}+397 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{192 d}+\frac {\left (88 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {4 a b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{14}}{d}+\frac {\left (904 a^{2}-895 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{192 d}+\frac {5 \left (136 a^{2}+353 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{192 d}-\frac {4 a b}{7 d}+\left (\frac {5 a^{2}}{16}+\frac {5 b^{2}}{128}\right ) x}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{8}}\) | \(570\) |
| orering | \(\text {Expression too large to display}\) | \(4298\) |
Input:
int(cos(d*x+c)^6*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
1/d*(a^2*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5 /16*d*x+5/16*c)-2/7*a*b*cos(d*x+c)^7+b^2*(-1/8*sin(d*x+c)*cos(d*x+c)^7+1/4 8*(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+5/1 28*c))
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {768 \, a b \cos \left (d x + c\right )^{7} - 105 \, {\left (8 \, a^{2} + b^{2}\right )} d x + 7 \, {\left (48 \, b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm="fricas")
Output:
-1/2688*(768*a*b*cos(d*x + c)^7 - 105*(8*a^2 + b^2)*d*x + 7*(48*b^2*cos(d* x + c)^7 - 8*(8*a^2 + b^2)*cos(d*x + c)^5 - 10*(8*a^2 + b^2)*cos(d*x + c)^ 3 - 15*(8*a^2 + b^2)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (141) = 282\).
Time = 0.84 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.73 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**6*(a+b*sin(d*x+c))**2,x)
Output:
Piecewise((5*a**2*x*sin(c + d*x)**6/16 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*a**2*x*cos( c + d*x)**6/16 + 5*a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a**2*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*a**2*sin(c + d*x)*cos(c + d*x)**5/(1 6*d) - 2*a*b*cos(c + d*x)**7/(7*d) + 5*b**2*x*sin(c + d*x)**8/128 + 5*b**2 *x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*b**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 5*b**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 5*b**2*x*cos(c + d*x)**8/128 + 5*b**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*b**2*sin( c + d*x)**5*cos(c + d*x)**3/(384*d) + 73*b**2*sin(c + d*x)**3*cos(c + d*x) **5/(384*d) - 5*b**2*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (x*( a + b*sin(c))**2*cos(c)**6, True))
Time = 0.03 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {6144 \, a b \cos \left (d x + c\right )^{7} + 112 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 7 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{21504 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm="maxima")
Output:
-1/21504*(6144*a*b*cos(d*x + c)^7 + 112*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 6 0*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^2 - 7*(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))*b^2)/d
Time = 0.20 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.11 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5}{128} \, {\left (8 \, a^{2} + b^{2}\right )} x - \frac {a b \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {3 \, a b \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {5 \, a b \cos \left (d x + c\right )}{32 \, d} - \frac {b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{2} - b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, a^{2} - b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, a^{2} + b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^6*(a+b*sin(d*x+c))^2,x, algorithm="giac")
Output:
5/128*(8*a^2 + b^2)*x - 1/224*a*b*cos(7*d*x + 7*c)/d - 1/32*a*b*cos(5*d*x + 5*c)/d - 3/32*a*b*cos(3*d*x + 3*c)/d - 5/32*a*b*cos(d*x + c)/d - 1/1024* b^2*sin(8*d*x + 8*c)/d + 1/192*(a^2 - b^2)*sin(6*d*x + 6*c)/d + 1/128*(6*a ^2 - b^2)*sin(4*d*x + 4*c)/d + 1/64*(15*a^2 + b^2)*sin(2*d*x + 2*c)/d
Time = 15.72 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.22 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5\,a^2\,x}{16}+\frac {5\,b^2\,x}{128}+\frac {5\,a^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{24\,d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{6\,d}+\frac {5\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{192\,d}+\frac {b^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{48\,d}-\frac {b^2\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}-\frac {2\,a\,b\,{\cos \left (c+d\,x\right )}^7}{7\,d}+\frac {5\,a^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{16\,d}+\frac {5\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d} \] Input:
int(cos(c + d*x)^6*(a + b*sin(c + d*x))^2,x)
Output:
(5*a^2*x)/16 + (5*b^2*x)/128 + (5*a^2*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/(6*d) + (5*b^2*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (b^2*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (b^2*cos(c + d* x)^7*sin(c + d*x))/(8*d) - (2*a*b*cos(c + d*x)^7)/(7*d) + (5*a^2*cos(c + d *x)*sin(c + d*x))/(16*d) + (5*b^2*cos(c + d*x)*sin(c + d*x))/(128*d)
Time = 0.18 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.49 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {336 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{7} b^{2}+768 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a b +448 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a^{2}-952 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} b^{2}-2304 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a b -1456 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a^{2}+826 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} b^{2}+2304 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a b +1848 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a^{2}-105 \cos \left (d x +c \right ) \sin \left (d x +c \right ) b^{2}-768 \cos \left (d x +c \right ) a b +840 a^{2} d x +768 a b +105 b^{2} d x}{2688 d} \] Input:
int(cos(d*x+c)^6*(a+b*sin(d*x+c))^2,x)
Output:
(336*cos(c + d*x)*sin(c + d*x)**7*b**2 + 768*cos(c + d*x)*sin(c + d*x)**6* a*b + 448*cos(c + d*x)*sin(c + d*x)**5*a**2 - 952*cos(c + d*x)*sin(c + d*x )**5*b**2 - 2304*cos(c + d*x)*sin(c + d*x)**4*a*b - 1456*cos(c + d*x)*sin( c + d*x)**3*a**2 + 826*cos(c + d*x)*sin(c + d*x)**3*b**2 + 2304*cos(c + d* x)*sin(c + d*x)**2*a*b + 1848*cos(c + d*x)*sin(c + d*x)*a**2 - 105*cos(c + d*x)*sin(c + d*x)*b**2 - 768*cos(c + d*x)*a*b + 840*a**2*d*x + 768*a*b + 105*b**2*d*x)/(2688*d)