Integrand size = 28, antiderivative size = 220 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {4 a b^3 \cos ^5(c+d x)}{5 d}-\frac {4 a^3 b \cos ^7(c+d x)}{7 d}+\frac {4 a b^3 \cos ^7(c+d x)}{7 d}+\frac {a^4 \sin (c+d x)}{d}-\frac {a^4 \sin ^3(c+d x)}{d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {3 a^4 \sin ^5(c+d x)}{5 d}-\frac {12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {b^4 \sin ^7(c+d x)}{7 d} \] Output:
-4/5*a*b^3*cos(d*x+c)^5/d-4/7*a^3*b*cos(d*x+c)^7/d+4/7*a*b^3*cos(d*x+c)^7/ d+a^4*sin(d*x+c)/d-a^4*sin(d*x+c)^3/d+2*a^2*b^2*sin(d*x+c)^3/d+3/5*a^4*sin (d*x+c)^5/d-12/5*a^2*b^2*sin(d*x+c)^5/d+1/5*b^4*sin(d*x+c)^5/d-1/7*a^4*sin (d*x+c)^7/d+6/7*a^2*b^2*sin(d*x+c)^7/d-1/7*b^4*sin(d*x+c)^7/d
Time = 1.90 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.75 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {-20 a^3 b \cos ^7(c+d x)+35 a^4 \sin (c+d x)-35 a^2 \left (a^2-2 b^2\right ) \sin ^3(c+d x)+7 \left (3 a^4-12 a^2 b^2+b^4\right ) \sin ^5(c+d x)-5 \left (a^4-6 a^2 b^2+b^4\right ) \sin ^7(c+d x)+4 a b^3 \cos (c+d x) \left (-2+\frac {2}{\sqrt {\cos ^2(c+d x)}}-\sin ^2(c+d x)+8 \sin ^4(c+d x)-5 \sin ^6(c+d x)\right )}{35 d} \] Input:
Integrate[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]
Output:
(-20*a^3*b*Cos[c + d*x]^7 + 35*a^4*Sin[c + d*x] - 35*a^2*(a^2 - 2*b^2)*Sin [c + d*x]^3 + 7*(3*a^4 - 12*a^2*b^2 + b^4)*Sin[c + d*x]^5 - 5*(a^4 - 6*a^2 *b^2 + b^4)*Sin[c + d*x]^7 + 4*a*b^3*Cos[c + d*x]*(-2 + 2/Sqrt[Cos[c + d*x ]^2] - Sin[c + d*x]^2 + 8*Sin[c + d*x]^4 - 5*Sin[c + d*x]^6))/(35*d)
Time = 0.49 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {3042, 3569, 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.
| \(\displaystyle \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \cos (c+d x)^3 (a \cos (c+d x)+b \sin (c+d x))^4dx\) |
| \(\Big \downarrow \) 3569 |
| \(\displaystyle \int \left (a^4 \cos ^7(c+d x)+4 a^3 b \sin (c+d x) \cos ^6(c+d x)+6 a^2 b^2 \sin ^2(c+d x) \cos ^5(c+d x)+4 a b^3 \sin ^3(c+d x) \cos ^4(c+d x)+b^4 \sin ^4(c+d x) \cos ^3(c+d x)\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a^4 \sin ^7(c+d x)}{7 d}+\frac {3 a^4 \sin ^5(c+d x)}{5 d}-\frac {a^4 \sin ^3(c+d x)}{d}+\frac {a^4 \sin (c+d x)}{d}-\frac {4 a^3 b \cos ^7(c+d x)}{7 d}+\frac {6 a^2 b^2 \sin ^7(c+d x)}{7 d}-\frac {12 a^2 b^2 \sin ^5(c+d x)}{5 d}+\frac {2 a^2 b^2 \sin ^3(c+d x)}{d}+\frac {4 a b^3 \cos ^7(c+d x)}{7 d}-\frac {4 a b^3 \cos ^5(c+d x)}{5 d}-\frac {b^4 \sin ^7(c+d x)}{7 d}+\frac {b^4 \sin ^5(c+d x)}{5 d}\) |
Input:
Int[Cos[c + d*x]^3*(a*Cos[c + d*x] + b*Sin[c + d*x])^4,x]
Output:
(-4*a*b^3*Cos[c + d*x]^5)/(5*d) - (4*a^3*b*Cos[c + d*x]^7)/(7*d) + (4*a*b^ 3*Cos[c + d*x]^7)/(7*d) + (a^4*Sin[c + d*x])/d - (a^4*Sin[c + d*x]^3)/d + (2*a^2*b^2*Sin[c + d*x]^3)/d + (3*a^4*Sin[c + d*x]^5)/(5*d) - (12*a^2*b^2* Sin[c + d*x]^5)/(5*d) + (b^4*Sin[c + d*x]^5)/(5*d) - (a^4*Sin[c + d*x]^7)/ (7*d) + (6*a^2*b^2*Sin[c + d*x]^7)/(7*d) - (b^4*Sin[c + d*x]^7)/(7*d)
Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*si n[(c_.) + (d_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandTrig[cos[c + d*x]^m*(a *cos[c + d*x] + b*sin[c + d*x])^n, x], x] /; FreeQ[{a, b, c, d}, x] && Inte gerQ[m] && IGtQ[n, 0]
Time = 4.68 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.74
| method | result | size |
| parts | \(\frac {a^{4} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7 d}+\frac {b^{4} \left (-\frac {\sin \left (d x +c \right )^{7}}{7}+\frac {\sin \left (d x +c \right )^{5}}{5}\right )}{d}+\frac {6 a^{2} b^{2} \left (\frac {\sin \left (d x +c \right )^{7}}{7}-\frac {2 \sin \left (d x +c \right )^{5}}{5}+\frac {\sin \left (d x +c \right )^{3}}{3}\right )}{d}+\frac {4 b^{3} a \left (\frac {\cos \left (d x +c \right )^{7}}{7}-\frac {\cos \left (d x +c \right )^{5}}{5}\right )}{d}-\frac {4 a^{3} b \cos \left (d x +c \right )^{7}}{7 d}\) | \(163\) |
| derivativedivides | \(\frac {\frac {a^{4} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}-\frac {4 a^{3} b \cos \left (d x +c \right )^{7}}{7}+6 a^{2} b^{2} \left (-\frac {\cos \left (d x +c \right )^{6} \sin \left (d x +c \right )}{7}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )+4 b^{3} a \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{7}-\frac {2 \cos \left (d x +c \right )^{5}}{35}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}{7}-\frac {3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}{35}+\frac {\left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{35}\right )}{d}\) | \(206\) |
| default | \(\frac {\frac {a^{4} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}-\frac {4 a^{3} b \cos \left (d x +c \right )^{7}}{7}+6 a^{2} b^{2} \left (-\frac {\cos \left (d x +c \right )^{6} \sin \left (d x +c \right )}{7}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )+4 b^{3} a \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{5}}{7}-\frac {2 \cos \left (d x +c \right )^{5}}{35}\right )+b^{4} \left (-\frac {\sin \left (d x +c \right )^{3} \cos \left (d x +c \right )^{4}}{7}-\frac {3 \sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}{35}+\frac {\left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{35}\right )}{d}\) | \(206\) |
| parallelrisch | \(\frac {70 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13} a^{4}-280 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12} a^{3} b +\left (140 a^{4}+560 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}-560 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10} a \,b^{3}+\left (602 a^{4}-448 a^{2} b^{2}+224 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}+\left (-1400 a^{3} b +560 b^{3} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}+\left (424 a^{4}+1824 a^{2} b^{2}-192 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}-1120 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a \,b^{3}+\left (602 a^{4}-448 a^{2} b^{2}+224 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}+\left (-840 a^{3} b +224 b^{3} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}+\left (140 a^{4}+560 a^{2} b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}-112 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a \,b^{3}+70 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-40 a^{3} b -16 b^{3} a}{35 d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(327\) |
| risch | \(-\frac {5 a^{3} b \cos \left (d x +c \right )}{16 d}-\frac {3 a \,b^{3} \cos \left (d x +c \right )}{16 d}+\frac {35 a^{4} \sin \left (d x +c \right )}{64 d}+\frac {15 a^{2} b^{2} \sin \left (d x +c \right )}{32 d}+\frac {3 b^{4} \sin \left (d x +c \right )}{64 d}-\frac {a^{3} b \cos \left (7 d x +7 c \right )}{112 d}+\frac {a \,b^{3} \cos \left (7 d x +7 c \right )}{112 d}+\frac {\sin \left (7 d x +7 c \right ) a^{4}}{448 d}-\frac {3 \sin \left (7 d x +7 c \right ) a^{2} b^{2}}{224 d}+\frac {\sin \left (7 d x +7 c \right ) b^{4}}{448 d}-\frac {a^{3} b \cos \left (5 d x +5 c \right )}{16 d}+\frac {a \,b^{3} \cos \left (5 d x +5 c \right )}{80 d}+\frac {7 \sin \left (5 d x +5 c \right ) a^{4}}{320 d}-\frac {9 \sin \left (5 d x +5 c \right ) a^{2} b^{2}}{160 d}-\frac {\sin \left (5 d x +5 c \right ) b^{4}}{320 d}-\frac {3 a^{3} b \cos \left (3 d x +3 c \right )}{16 d}-\frac {a \,b^{3} \cos \left (3 d x +3 c \right )}{16 d}+\frac {7 \sin \left (3 d x +3 c \right ) a^{4}}{64 d}-\frac {\sin \left (3 d x +3 c \right ) a^{2} b^{2}}{32 d}-\frac {\sin \left (3 d x +3 c \right ) b^{4}}{64 d}\) | \(347\) |
| norman | \(\frac {-\frac {40 a^{3} b +16 b^{3} a}{35 d}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{4} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{13}}{d}-\frac {16 b^{3} a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{5 d}-\frac {16 b^{3} a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}-\frac {32 b^{3} a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {8 a^{3} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{d}+\frac {2 \left (43 a^{4}-32 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {2 \left (43 a^{4}-32 a^{2} b^{2}+16 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{5 d}+\frac {8 \left (53 a^{4}+228 a^{2} b^{2}-24 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{35 d}-\frac {\left (40 a^{3} b -16 b^{3} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {\left (120 a^{3} b -32 b^{3} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5 d}+\frac {4 a^{2} \left (a^{2}+4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{d}+\frac {4 a^{2} \left (a^{2}+4 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{7}}\) | \(371\) |
| orering | \(\text {Expression too large to display}\) | \(5891\) |
Input:
int(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^4,x,method=_RETURNVERBOSE)
Output:
1/7*a^4/d*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c) +b^4/d*(-1/7*sin(d*x+c)^7+1/5*sin(d*x+c)^5)+6*a^2*b^2/d*(1/7*sin(d*x+c)^7- 2/5*sin(d*x+c)^5+1/3*sin(d*x+c)^3)+4*b^3*a/d*(1/7*cos(d*x+c)^7-1/5*cos(d*x +c)^5)-4/7*a^3*b*cos(d*x+c)^7/d
Time = 0.08 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.68 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {28 \, a b^{3} \cos \left (d x + c\right )^{5} + 20 \, {\left (a^{3} b - a b^{3}\right )} \cos \left (d x + c\right )^{7} - {\left (5 \, {\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{6} + 2 \, {\left (3 \, a^{4} + 3 \, a^{2} b^{2} - 4 \, b^{4}\right )} \cos \left (d x + c\right )^{4} + 16 \, a^{4} + 16 \, a^{2} b^{2} + 2 \, b^{4} + {\left (8 \, a^{4} + 8 \, a^{2} b^{2} + b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{35 \, d} \] Input:
integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="fricas" )
Output:
-1/35*(28*a*b^3*cos(d*x + c)^5 + 20*(a^3*b - a*b^3)*cos(d*x + c)^7 - (5*(a ^4 - 6*a^2*b^2 + b^4)*cos(d*x + c)^6 + 2*(3*a^4 + 3*a^2*b^2 - 4*b^4)*cos(d *x + c)^4 + 16*a^4 + 16*a^2*b^2 + 2*b^4 + (8*a^4 + 8*a^2*b^2 + b^4)*cos(d* x + c)^2)*sin(d*x + c))/d
Time = 0.55 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.30 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\begin {cases} \frac {16 a^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a^{4} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac {a^{4} \sin {\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} - \frac {4 a^{3} b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {16 a^{2} b^{2} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {8 a^{2} b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac {2 a^{2} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {4 a b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {8 a b^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} + \frac {2 b^{4} \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{4} \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{4} \cos ^{3}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**3*(a*cos(d*x+c)+b*sin(d*x+c))**4,x)
Output:
Piecewise((16*a**4*sin(c + d*x)**7/(35*d) + 8*a**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*a**4*sin(c + d*x)**3*cos(c + d*x)**4/d + a**4*sin(c + d *x)*cos(c + d*x)**6/d - 4*a**3*b*cos(c + d*x)**7/(7*d) + 16*a**2*b**2*sin( c + d*x)**7/(35*d) + 8*a**2*b**2*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2 *a**2*b**2*sin(c + d*x)**3*cos(c + d*x)**4/d - 4*a*b**3*sin(c + d*x)**2*co s(c + d*x)**5/(5*d) - 8*a*b**3*cos(c + d*x)**7/(35*d) + 2*b**4*sin(c + d*x )**7/(35*d) + b**4*sin(c + d*x)**5*cos(c + d*x)**2/(5*d), Ne(d, 0)), (x*(a *cos(c) + b*sin(c))**4*cos(c)**3, True))
Time = 0.03 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.70 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {20 \, a^{3} b \cos \left (d x + c\right )^{7} + {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{4} - 2 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{2} b^{2} - 4 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a b^{3} + {\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} b^{4}}{35 \, d} \] Input:
integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="maxima" )
Output:
-1/35*(20*a^3*b*cos(d*x + c)^7 + (5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 3 5*sin(d*x + c)^3 - 35*sin(d*x + c))*a^4 - 2*(15*sin(d*x + c)^7 - 42*sin(d* x + c)^5 + 35*sin(d*x + c)^3)*a^2*b^2 - 4*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a*b^3 + (5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*b^4)/d
Time = 0.23 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.04 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {{\left (a^{3} b - a b^{3}\right )} \cos \left (7 \, d x + 7 \, c\right )}{112 \, d} - \frac {{\left (5 \, a^{3} b - a b^{3}\right )} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {{\left (3 \, a^{3} b + a b^{3}\right )} \cos \left (3 \, d x + 3 \, c\right )}{16 \, d} - \frac {{\left (5 \, a^{3} b + 3 \, a b^{3}\right )} \cos \left (d x + c\right )}{16 \, d} + \frac {{\left (a^{4} - 6 \, a^{2} b^{2} + b^{4}\right )} \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {{\left (7 \, a^{4} - 18 \, a^{2} b^{2} - b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac {{\left (7 \, a^{4} - 2 \, a^{2} b^{2} - b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac {{\left (35 \, a^{4} + 30 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^4,x, algorithm="giac")
Output:
-1/112*(a^3*b - a*b^3)*cos(7*d*x + 7*c)/d - 1/80*(5*a^3*b - a*b^3)*cos(5*d *x + 5*c)/d - 1/16*(3*a^3*b + a*b^3)*cos(3*d*x + 3*c)/d - 1/16*(5*a^3*b + 3*a*b^3)*cos(d*x + c)/d + 1/448*(a^4 - 6*a^2*b^2 + b^4)*sin(7*d*x + 7*c)/d + 1/320*(7*a^4 - 18*a^2*b^2 - b^4)*sin(5*d*x + 5*c)/d + 1/64*(7*a^4 - 2*a ^2*b^2 - b^4)*sin(3*d*x + 3*c)/d + 1/64*(35*a^4 + 30*a^2*b^2 + 3*b^4)*sin( d*x + c)/d
Time = 17.17 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.32 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=-\frac {\frac {b^4\,\sin \left (3\,c+3\,d\,x\right )}{64}-\frac {3\,b^4\,\sin \left (c+d\,x\right )}{64}-\frac {7\,a^4\,\sin \left (3\,c+3\,d\,x\right )}{64}-\frac {7\,a^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {a^4\,\sin \left (7\,c+7\,d\,x\right )}{448}-\frac {35\,a^4\,\sin \left (c+d\,x\right )}{64}+\frac {b^4\,\sin \left (5\,c+5\,d\,x\right )}{320}-\frac {b^4\,\sin \left (7\,c+7\,d\,x\right )}{448}+\frac {a\,b^3\,\cos \left (3\,c+3\,d\,x\right )}{16}+\frac {3\,a^3\,b\,\cos \left (3\,c+3\,d\,x\right )}{16}-\frac {a\,b^3\,\cos \left (5\,c+5\,d\,x\right )}{80}+\frac {a^3\,b\,\cos \left (5\,c+5\,d\,x\right )}{16}-\frac {a\,b^3\,\cos \left (7\,c+7\,d\,x\right )}{112}+\frac {a^3\,b\,\cos \left (7\,c+7\,d\,x\right )}{112}-\frac {15\,a^2\,b^2\,\sin \left (c+d\,x\right )}{32}+\frac {a^2\,b^2\,\sin \left (3\,c+3\,d\,x\right )}{32}+\frac {9\,a^2\,b^2\,\sin \left (5\,c+5\,d\,x\right )}{160}+\frac {3\,a^2\,b^2\,\sin \left (7\,c+7\,d\,x\right )}{224}+\frac {3\,a\,b^3\,\cos \left (c+d\,x\right )}{16}+\frac {5\,a^3\,b\,\cos \left (c+d\,x\right )}{16}}{d} \] Input:
int(cos(c + d*x)^3*(a*cos(c + d*x) + b*sin(c + d*x))^4,x)
Output:
-((b^4*sin(3*c + 3*d*x))/64 - (3*b^4*sin(c + d*x))/64 - (7*a^4*sin(3*c + 3 *d*x))/64 - (7*a^4*sin(5*c + 5*d*x))/320 - (a^4*sin(7*c + 7*d*x))/448 - (3 5*a^4*sin(c + d*x))/64 + (b^4*sin(5*c + 5*d*x))/320 - (b^4*sin(7*c + 7*d*x ))/448 + (a*b^3*cos(3*c + 3*d*x))/16 + (3*a^3*b*cos(3*c + 3*d*x))/16 - (a* b^3*cos(5*c + 5*d*x))/80 + (a^3*b*cos(5*c + 5*d*x))/16 - (a*b^3*cos(7*c + 7*d*x))/112 + (a^3*b*cos(7*c + 7*d*x))/112 - (15*a^2*b^2*sin(c + d*x))/32 + (a^2*b^2*sin(3*c + 3*d*x))/32 + (9*a^2*b^2*sin(5*c + 5*d*x))/160 + (3*a^ 2*b^2*sin(7*c + 7*d*x))/224 + (3*a*b^3*cos(c + d*x))/16 + (5*a^3*b*cos(c + d*x))/16)/d
Time = 0.15 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.30 \[ \int \cos ^3(c+d x) (a \cos (c+d x)+b \sin (c+d x))^4 \, dx=\frac {20 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a^{3} b -20 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} a \,b^{3}-60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a^{3} b +32 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} a \,b^{3}+60 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a^{3} b -4 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} a \,b^{3}-20 \cos \left (d x +c \right ) a^{3} b -8 \cos \left (d x +c \right ) a \,b^{3}-5 \sin \left (d x +c \right )^{7} a^{4}+30 \sin \left (d x +c \right )^{7} a^{2} b^{2}-5 \sin \left (d x +c \right )^{7} b^{4}+21 \sin \left (d x +c \right )^{5} a^{4}-84 \sin \left (d x +c \right )^{5} a^{2} b^{2}+7 \sin \left (d x +c \right )^{5} b^{4}-35 \sin \left (d x +c \right )^{3} a^{4}+70 \sin \left (d x +c \right )^{3} a^{2} b^{2}+35 \sin \left (d x +c \right ) a^{4}+20 a^{3} b +8 a \,b^{3}}{35 d} \] Input:
int(cos(d*x+c)^3*(a*cos(d*x+c)+b*sin(d*x+c))^4,x)
Output:
(20*cos(c + d*x)*sin(c + d*x)**6*a**3*b - 20*cos(c + d*x)*sin(c + d*x)**6* a*b**3 - 60*cos(c + d*x)*sin(c + d*x)**4*a**3*b + 32*cos(c + d*x)*sin(c + d*x)**4*a*b**3 + 60*cos(c + d*x)*sin(c + d*x)**2*a**3*b - 4*cos(c + d*x)*s in(c + d*x)**2*a*b**3 - 20*cos(c + d*x)*a**3*b - 8*cos(c + d*x)*a*b**3 - 5 *sin(c + d*x)**7*a**4 + 30*sin(c + d*x)**7*a**2*b**2 - 5*sin(c + d*x)**7*b **4 + 21*sin(c + d*x)**5*a**4 - 84*sin(c + d*x)**5*a**2*b**2 + 7*sin(c + d *x)**5*b**4 - 35*sin(c + d*x)**3*a**4 + 70*sin(c + d*x)**3*a**2*b**2 + 35* sin(c + d*x)*a**4 + 20*a**3*b + 8*a*b**3)/(35*d)