Integrand size = 21, antiderivative size = 158 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3}{16} a \left (2 a^2+b^2\right ) x-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{70 d}+\frac {3 a \left (2 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a \left (2 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{8 d}-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{14 d}-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d} \]
3/16*a*(2*a^2+b^2)*x-1/70*b*(17*a^2+4*b^2)*cos(d*x+c)^5/d+3/16*a*(2*a^2+b^ 2)*cos(d*x+c)*sin(d*x+c)/d+1/8*a*(2*a^2+b^2)*cos(d*x+c)^3*sin(d*x+c)/d-3/1 4*a*b*cos(d*x+c)^5*(a+b*sin(d*x+c))/d-1/7*b*cos(d*x+c)^5*(a+b*sin(d*x+c))^ 2/d
Time = 0.52 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.15 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {840 a^3 c+420 a b^2 c+840 a^3 d x+420 a b^2 d x-105 b \left (8 a^2+b^2\right ) \cos (c+d x)-35 \left (12 a^2 b+b^3\right ) \cos (3 (c+d x))-84 a^2 b \cos (5 (c+d x))+7 b^3 \cos (5 (c+d x))+5 b^3 \cos (7 (c+d x))+560 a^3 \sin (2 (c+d x))+105 a b^2 \sin (2 (c+d x))+70 a^3 \sin (4 (c+d x))-105 a b^2 \sin (4 (c+d x))-35 a b^2 \sin (6 (c+d x))}{2240 d} \]
(840*a^3*c + 420*a*b^2*c + 840*a^3*d*x + 420*a*b^2*d*x - 105*b*(8*a^2 + b^ 2)*Cos[c + d*x] - 35*(12*a^2*b + b^3)*Cos[3*(c + d*x)] - 84*a^2*b*Cos[5*(c + d*x)] + 7*b^3*Cos[5*(c + d*x)] + 5*b^3*Cos[7*(c + d*x)] + 560*a^3*Sin[2 *(c + d*x)] + 105*a*b^2*Sin[2*(c + d*x)] + 70*a^3*Sin[4*(c + d*x)] - 105*a *b^2*Sin[4*(c + d*x)] - 35*a*b^2*Sin[6*(c + d*x)])/(2240*d)
Time = 0.72 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.99, number of steps used = 12, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {3042, 3171, 3042, 3341, 27, 3042, 3148, 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.
| \(\displaystyle \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^4 (a+b \sin (c+d x))^3dx\) |
| \(\Big \downarrow \) 3171 |
| \(\displaystyle \frac {1}{7} \int \cos ^4(c+d x) (a+b \sin (c+d x)) \left (7 a^2+9 b \sin (c+d x) a+2 b^2\right )dx-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \int \cos (c+d x)^4 (a+b \sin (c+d x)) \left (7 a^2+9 b \sin (c+d x) a+2 b^2\right )dx-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3341 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{6} \int 3 \cos ^4(c+d x) \left (7 a \left (2 a^2+b^2\right )+b \left (17 a^2+4 b^2\right ) \sin (c+d x)\right )dx-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \int \cos ^4(c+d x) \left (7 a \left (2 a^2+b^2\right )+b \left (17 a^2+4 b^2\right ) \sin (c+d x)\right )dx-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \int \cos (c+d x)^4 \left (7 a \left (2 a^2+b^2\right )+b \left (17 a^2+4 b^2\right ) \sin (c+d x)\right )dx-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3148 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (7 a \left (2 a^2+b^2\right ) \int \cos ^4(c+d x)dx-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{5 d}\right )-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (7 a \left (2 a^2+b^2\right ) \int \sin \left (c+d x+\frac {\pi }{2}\right )^4dx-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{5 d}\right )-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (7 a \left (2 a^2+b^2\right ) \left (\frac {3}{4} \int \cos ^2(c+d x)dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{5 d}\right )-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (7 a \left (2 a^2+b^2\right ) \left (\frac {3}{4} \int \sin \left (c+d x+\frac {\pi }{2}\right )^2dx+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{5 d}\right )-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 3115 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (7 a \left (2 a^2+b^2\right ) \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (c+d x) \cos (c+d x)}{2 d}\right )+\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}\right )-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{5 d}\right )-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
| \(\Big \downarrow \) 24 |
| \(\displaystyle \frac {1}{7} \left (\frac {1}{2} \left (7 a \left (2 a^2+b^2\right ) \left (\frac {\sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac {3}{4} \left (\frac {\sin (c+d x) \cos (c+d x)}{2 d}+\frac {x}{2}\right )\right )-\frac {b \left (17 a^2+4 b^2\right ) \cos ^5(c+d x)}{5 d}\right )-\frac {3 a b \cos ^5(c+d x) (a+b \sin (c+d x))}{2 d}\right )-\frac {b \cos ^5(c+d x) (a+b \sin (c+d x))^2}{7 d}\) |
-1/7*(b*Cos[c + d*x]^5*(a + b*Sin[c + d*x])^2)/d + ((-3*a*b*Cos[c + d*x]^5 *(a + b*Sin[c + d*x]))/(2*d) + (-1/5*(b*(17*a^2 + 4*b^2)*Cos[c + d*x]^5)/d + 7*a*(2*a^2 + b^2)*((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))/2)/7
3.5.6.3.1 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[((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])
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 = 1.50 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.92
| method | result | size |
| derivativedivides | \(\frac {a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a \,b^{2} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(145\) |
| default | \(\frac {a^{3} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {3 a^{2} b \left (\cos ^{5}\left (d x +c \right )\right )}{5}+3 a \,b^{2} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+b^{3} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\) | \(145\) |
| parallelrisch | \(\frac {\left (-420 a^{2} b -35 b^{3}\right ) \cos \left (3 d x +3 c \right )+\left (-84 a^{2} b +7 b^{3}\right ) \cos \left (5 d x +5 c \right )+\left (560 a^{3}+105 a \,b^{2}\right ) \sin \left (2 d x +2 c \right )+\left (70 a^{3}-105 a \,b^{2}\right ) \sin \left (4 d x +4 c \right )+5 b^{3} \cos \left (7 d x +7 c \right )-35 a \,b^{2} \sin \left (6 d x +6 c \right )+\left (-840 a^{2} b -105 b^{3}\right ) \cos \left (d x +c \right )+840 a^{3} d x +420 a \,b^{2} d x -1344 a^{2} b -128 b^{3}}{2240 d}\) | \(169\) |
| risch | \(\frac {3 a^{3} x}{8}+\frac {3 a \,b^{2} x}{16}-\frac {3 b \cos \left (d x +c \right ) a^{2}}{8 d}-\frac {3 b^{3} \cos \left (d x +c \right )}{64 d}+\frac {b^{3} \cos \left (7 d x +7 c \right )}{448 d}-\frac {a \,b^{2} \sin \left (6 d x +6 c \right )}{64 d}-\frac {3 b \cos \left (5 d x +5 c \right ) a^{2}}{80 d}+\frac {b^{3} \cos \left (5 d x +5 c \right )}{320 d}+\frac {\sin \left (4 d x +4 c \right ) a^{3}}{32 d}-\frac {3 \sin \left (4 d x +4 c \right ) a \,b^{2}}{64 d}-\frac {3 b \cos \left (3 d x +3 c \right ) a^{2}}{16 d}-\frac {b^{3} \cos \left (3 d x +3 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) a^{3}}{4 d}+\frac {3 \sin \left (2 d x +2 c \right ) a \,b^{2}}{64 d}\) | \(219\) |
| norman | \(\frac {\left (\frac {3}{8} a^{3}+\frac {3}{16} a \,b^{2}\right ) x +\left (\frac {3}{8} a^{3}+\frac {3}{16} a \,b^{2}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{8} a^{3}+\frac {21}{16} a \,b^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {21}{8} a^{3}+\frac {21}{16} a \,b^{2}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{8} a^{3}+\frac {63}{16} a \,b^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {63}{8} a^{3}+\frac {63}{16} a \,b^{2}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{8} a^{3}+\frac {105}{16} a \,b^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {105}{8} a^{3}+\frac {105}{16} a \,b^{2}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {42 a^{2} b +4 b^{3}}{35 d}-\frac {6 a^{2} b \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (12 a^{2} b +4 b^{3}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (12 a^{2} b +4 b^{3}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (18 a^{2} b -4 b^{3}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (24 a^{2} b +8 b^{3}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (66 a^{2} b -8 b^{3}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {a \left (6 a^{2}+11 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}-\frac {a \left (6 a^{2}+11 b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}+\frac {a \left (10 a^{2}-3 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a \left (10 a^{2}-3 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a \left (18 a^{2}-31 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}-\frac {a \left (18 a^{2}-31 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\) | \(550\) |
1/d*(a^3*(1/4*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/8*d*x+3/8*c)-3/5* a^2*b*cos(d*x+c)^5+3*a*b^2*(-1/6*cos(d*x+c)^5*sin(d*x+c)+1/24*(cos(d*x+c)^ 3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c)+b^3*(-1/7*cos(d*x+c)^5*sin(d *x+c)^2-2/35*cos(d*x+c)^5))
Time = 0.30 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.74 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {80 \, b^{3} \cos \left (d x + c\right )^{7} - 112 \, {\left (3 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )^{5} + 105 \, {\left (2 \, a^{3} + a b^{2}\right )} d x - 35 \, {\left (8 \, a b^{2} \cos \left (d x + c\right )^{5} - 2 \, {\left (2 \, a^{3} + a b^{2}\right )} \cos \left (d x + c\right )^{3} - 3 \, {\left (2 \, a^{3} + a b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{560 \, d} \]
1/560*(80*b^3*cos(d*x + c)^7 - 112*(3*a^2*b + b^3)*cos(d*x + c)^5 + 105*(2 *a^3 + a*b^2)*d*x - 35*(8*a*b^2*cos(d*x + c)^5 - 2*(2*a^3 + a*b^2)*cos(d*x + c)^3 - 3*(2*a^3 + a*b^2)*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 348 vs. \(2 (148) = 296\).
Time = 0.51 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.20 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {3 a^{3} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {3 a^{3} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {3 a^{2} b \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {3 a b^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {9 a b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 a b^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {a b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{2 d} - \frac {3 a b^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {2 b^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{3} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]
Piecewise((3*a**3*x*sin(c + d*x)**4/8 + 3*a**3*x*sin(c + d*x)**2*cos(c + d *x)**2/4 + 3*a**3*x*cos(c + d*x)**4/8 + 3*a**3*sin(c + d*x)**3*cos(c + d*x )/(8*d) + 5*a**3*sin(c + d*x)*cos(c + d*x)**3/(8*d) - 3*a**2*b*cos(c + d*x )**5/(5*d) + 3*a*b**2*x*sin(c + d*x)**6/16 + 9*a*b**2*x*sin(c + d*x)**4*co s(c + d*x)**2/16 + 9*a*b**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*a*b** 2*x*cos(c + d*x)**6/16 + 3*a*b**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + a* b**2*sin(c + d*x)**3*cos(c + d*x)**3/(2*d) - 3*a*b**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) - b**3*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 2*b**3*cos( c + d*x)**7/(35*d), Ne(d, 0)), (x*(a + b*sin(c))**3*cos(c)**4, True))
Time = 0.19 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.74 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=-\frac {1344 \, a^{2} b \cos \left (d x + c\right )^{5} - 70 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{3} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a b^{2} - 64 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} b^{3}}{2240 \, d} \]
-1/2240*(1344*a^2*b*cos(d*x + c)^5 - 70*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*a^3 - 35*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*s in(4*d*x + 4*c))*a*b^2 - 64*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*b^3)/d
Time = 0.61 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {b^{3} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} - \frac {a b^{2} \sin \left (6 \, d x + 6 \, c\right )}{64 \, d} + \frac {3}{16} \, {\left (2 \, a^{3} + a b^{2}\right )} x - \frac {{\left (12 \, a^{2} b - b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (12 \, a^{2} b + b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {3 \, {\left (8 \, a^{2} b + b^{3}\right )} \cos \left (d x + c\right )}{64 \, d} + \frac {{\left (2 \, a^{3} - 3 \, a b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (16 \, a^{3} + 3 \, a b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]
1/448*b^3*cos(7*d*x + 7*c)/d - 1/64*a*b^2*sin(6*d*x + 6*c)/d + 3/16*(2*a^3 + a*b^2)*x - 1/320*(12*a^2*b - b^3)*cos(5*d*x + 5*c)/d - 1/64*(12*a^2*b + b^3)*cos(3*d*x + 3*c)/d - 3/64*(8*a^2*b + b^3)*cos(d*x + c)/d + 1/64*(2*a ^3 - 3*a*b^2)*sin(4*d*x + 4*c)/d + 1/64*(16*a^3 + 3*a*b^2)*sin(2*d*x + 2*c )/d
Time = 5.90 (sec) , antiderivative size = 474, normalized size of antiderivative = 3.00 \[ \int \cos ^4(c+d x) (a+b \sin (c+d x))^3 \, dx=\frac {3\,a\,\mathrm {atan}\left (\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (2\,a^2+b^2\right )}{8\,\left (\frac {3\,a^3}{4}+\frac {3\,a\,b^2}{8}\right )}\right )\,\left (2\,a^2+b^2\right )}{8\,d}-\frac {\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {3\,a\,b^2}{8}-\frac {5\,a^3}{4}\right )+\frac {6\,a^2\,b}{5}-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (3\,a^3+\frac {11\,a\,b^2}{2}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (3\,a^3+\frac {11\,a\,b^2}{2}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {3\,a\,b^2}{8}-\frac {5\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {31\,a\,b^2}{8}-\frac {9\,a^3}{4}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {31\,a\,b^2}{8}-\frac {9\,a^3}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (12\,a^2\,b+4\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {12\,a^2\,b}{5}+\frac {4\,b^3}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (18\,a^2\,b-4\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (24\,a^2\,b+8\,b^3\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {66\,a^2\,b}{5}-\frac {8\,b^3}{5}\right )+\frac {4\,b^3}{35}+6\,a^2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}-\frac {3\,a\,\left (2\,a^2+b^2\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{8\,d} \]
(3*a*atan((3*a*tan(c/2 + (d*x)/2)*(2*a^2 + b^2))/(8*((3*a*b^2)/8 + (3*a^3) /4)))*(2*a^2 + b^2))/(8*d) - (tan(c/2 + (d*x)/2)*((3*a*b^2)/8 - (5*a^3)/4) + (6*a^2*b)/5 - tan(c/2 + (d*x)/2)^3*((11*a*b^2)/2 + 3*a^3) + tan(c/2 + ( d*x)/2)^11*((11*a*b^2)/2 + 3*a^3) - tan(c/2 + (d*x)/2)^13*((3*a*b^2)/8 - ( 5*a^3)/4) + tan(c/2 + (d*x)/2)^5*((31*a*b^2)/8 - (9*a^3)/4) - tan(c/2 + (d *x)/2)^9*((31*a*b^2)/8 - (9*a^3)/4) + tan(c/2 + (d*x)/2)^10*(12*a^2*b + 4* b^3) + tan(c/2 + (d*x)/2)^2*((12*a^2*b)/5 + (4*b^3)/5) + tan(c/2 + (d*x)/2 )^8*(18*a^2*b - 4*b^3) + tan(c/2 + (d*x)/2)^6*(24*a^2*b + 8*b^3) + tan(c/2 + (d*x)/2)^4*((66*a^2*b)/5 - (8*b^3)/5) + (4*b^3)/35 + 6*a^2*b*tan(c/2 + (d*x)/2)^12)/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan (c/2 + (d*x)/2)^6 + 35*tan(c/2 + (d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7 *tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1)) - (3*a*(2*a^2 + b^2)* (atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d)