Integrand size = 19, antiderivative size = 94 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))}{d}+\frac {2 \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^3}{3 d}-\frac {(b \cos (c+d x)-a \sin (c+d x))^5}{5 d} \] Output:
-(a^2+b^2)^2*(b*cos(d*x+c)-a*sin(d*x+c))/d+2/3*(a^2+b^2)*(b*cos(d*x+c)-a*s in(d*x+c))^3/d-1/5*(b*cos(d*x+c)-a*sin(d*x+c))^5/d
Time = 1.22 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.66 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {-150 b \left (a^2+b^2\right )^2 \cos (c+d x)+25 b \left (-3 a^4-2 a^2 b^2+b^4\right ) \cos (3 (c+d x))-3 b \left (5 a^4-10 a^2 b^2+b^4\right ) \cos (5 (c+d x))+150 a \left (a^2+b^2\right )^2 \sin (c+d x)+25 a \left (a^4-2 a^2 b^2-3 b^4\right ) \sin (3 (c+d x))+3 a \left (a^4-10 a^2 b^2+5 b^4\right ) \sin (5 (c+d x))}{240 d} \] Input:
Integrate[(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]
Output:
(-150*b*(a^2 + b^2)^2*Cos[c + d*x] + 25*b*(-3*a^4 - 2*a^2*b^2 + b^4)*Cos[3 *(c + d*x)] - 3*b*(5*a^4 - 10*a^2*b^2 + b^4)*Cos[5*(c + d*x)] + 150*a*(a^2 + b^2)^2*Sin[c + d*x] + 25*a*(a^4 - 2*a^2*b^2 - 3*b^4)*Sin[3*(c + d*x)] + 3*a*(a^4 - 10*a^2*b^2 + 5*b^4)*Sin[5*(c + d*x)])/(240*d)
Time = 0.24 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.95, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {3042, 3551, 210, 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.
| \(\displaystyle \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \cos (c+d x)+b \sin (c+d x))^5dx\) |
| \(\Big \downarrow \) 3551 |
| \(\displaystyle -\frac {\int \left (a^2+b^2-(b \cos (c+d x)-a \sin (c+d x))^2\right )^2d(b \cos (c+d x)-a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 210 |
| \(\displaystyle -\frac {\int \left (\left (\frac {b^4+2 a^2 b^2}{a^4}+1\right ) a^4-2 \left (\frac {b^2}{a^2}+1\right ) (b \cos (c+d x)-a \sin (c+d x))^2 a^2+(b \cos (c+d x)-a \sin (c+d x))^4\right )d(b \cos (c+d x)-a \sin (c+d x))}{d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {2}{3} \left (a^2+b^2\right ) (b \cos (c+d x)-a \sin (c+d x))^3+\left (a^2+b^2\right )^2 (b \cos (c+d x)-a \sin (c+d x))+\frac {1}{5} (b \cos (c+d x)-a \sin (c+d x))^5}{d}\) |
Input:
Int[(a*Cos[c + d*x] + b*Sin[c + d*x])^5,x]
Output:
-(((a^2 + b^2)^2*(b*Cos[c + d*x] - a*Sin[c + d*x]) - (2*(a^2 + b^2)*(b*Cos [c + d*x] - a*Sin[c + d*x])^3)/3 + (b*Cos[c + d*x] - a*Sin[c + d*x])^5/5)/ d)
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 )^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x _Symbol] :> Simp[-d^(-1) Subst[Int[(a^2 + b^2 - x^2)^((n - 1)/2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[(n - 1)/2, 0]
Time = 2.83 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.86
| method | result | size |
| derivativedivides | \(\frac {\frac {a^{5} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}-a^{4} b \cos \left (d x +c \right )^{5}+10 b^{2} a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}{5}+\frac {\left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{15}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+b^{4} a \sin \left (d x +c \right )^{5}-\frac {b^{5} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\) | \(175\) |
| default | \(\frac {\frac {a^{5} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5}-a^{4} b \cos \left (d x +c \right )^{5}+10 b^{2} a^{3} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{4}}{5}+\frac {\left (2+\cos \left (d x +c \right )^{2}\right ) \sin \left (d x +c \right )}{15}\right )+10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )+b^{4} a \sin \left (d x +c \right )^{5}-\frac {b^{5} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5}}{d}\) | \(175\) |
| parts | \(\frac {a^{5} \left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{5 d}-\frac {b^{5} \left (\frac {8}{3}+\sin \left (d x +c \right )^{4}+\frac {4 \sin \left (d x +c \right )^{2}}{3}\right ) \cos \left (d x +c \right )}{5 d}+\frac {10 b^{2} a^{3} \left (-\frac {\sin \left (d x +c \right )^{5}}{5}+\frac {\sin \left (d x +c \right )^{3}}{3}\right )}{d}-\frac {a^{4} b \cos \left (d x +c \right )^{5}}{d}+\frac {10 a^{2} b^{3} \left (-\frac {\sin \left (d x +c \right )^{2} \cos \left (d x +c \right )^{3}}{5}-\frac {2 \cos \left (d x +c \right )^{3}}{15}\right )}{d}+\frac {b^{4} a \sin \left (d x +c \right )^{5}}{d}\) | \(175\) |
| parallelrisch | \(\frac {2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}-10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} a^{4} b +\frac {8 \left (a^{5}+10 b^{2} a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3}-40 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6} a^{2} b^{3}+\frac {4 \left (29 a^{5}-40 b^{2} a^{3}+120 b^{4} a \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15}+\frac {4 \left (-15 a^{4} b +10 a^{2} b^{3}-8 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3}+\frac {8 \left (a^{5}+10 b^{2} a^{3}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3}+\frac {8 \left (-5 a^{2} b^{3}-2 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3}+2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 a^{4} b -\frac {8 a^{2} b^{3}}{3}-\frac {16 b^{5}}{15}}{d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(250\) |
| norman | \(\frac {-\frac {30 a^{4} b +40 a^{2} b^{3}+16 b^{5}}{15 d}+\frac {2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {2 a^{5} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{d}-\frac {40 a^{2} b^{3} \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{d}-\frac {10 a^{4} b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{d}-\frac {\left (40 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{3 d}-\frac {2 \left (30 a^{4} b -20 a^{2} b^{3}+16 b^{5}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{3 d}+\frac {4 a \left (29 a^{4}-40 a^{2} b^{2}+120 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{15 d}+\frac {8 a^{3} \left (a^{2}+10 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {8 a^{3} \left (a^{2}+10 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{3 d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{5}}\) | \(279\) |
| risch | \(-\frac {5 b \cos \left (d x +c \right ) a^{4}}{8 d}-\frac {5 b^{3} \cos \left (d x +c \right ) a^{2}}{4 d}-\frac {5 b^{5} \cos \left (d x +c \right )}{8 d}+\frac {5 a^{5} \sin \left (d x +c \right )}{8 d}+\frac {5 a^{3} \sin \left (d x +c \right ) b^{2}}{4 d}+\frac {5 a \sin \left (d x +c \right ) b^{4}}{8 d}-\frac {b \cos \left (5 d x +5 c \right ) a^{4}}{16 d}+\frac {b^{3} \cos \left (5 d x +5 c \right ) a^{2}}{8 d}-\frac {b^{5} \cos \left (5 d x +5 c \right )}{80 d}+\frac {a^{5} \sin \left (5 d x +5 c \right )}{80 d}-\frac {a^{3} \sin \left (5 d x +5 c \right ) b^{2}}{8 d}+\frac {a \sin \left (5 d x +5 c \right ) b^{4}}{16 d}-\frac {5 b \cos \left (3 d x +3 c \right ) a^{4}}{16 d}-\frac {5 b^{3} \cos \left (3 d x +3 c \right ) a^{2}}{24 d}+\frac {5 b^{5} \cos \left (3 d x +3 c \right )}{48 d}+\frac {5 a^{5} \sin \left (3 d x +3 c \right )}{48 d}-\frac {5 a^{3} \sin \left (3 d x +3 c \right ) b^{2}}{24 d}-\frac {5 a \sin \left (3 d x +3 c \right ) b^{4}}{16 d}\) | \(314\) |
| orering | \(-\frac {259 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{4} \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right )}{45 d^{2}}-\frac {7 \left (60 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{2} \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right )^{3}+60 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{3} \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right ) \left (-d^{2} \cos \left (d x +c \right ) a -b \,d^{2} \sin \left (d x +c \right )\right )+5 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{4} \left (d^{3} \sin \left (d x +c \right ) a -b \,d^{3} \cos \left (d x +c \right )\right )\right )}{45 d^{4}}-\frac {120 \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right )^{5}+1200 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right ) \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right )^{3} \left (-d^{2} \cos \left (d x +c \right ) a -b \,d^{2} \sin \left (d x +c \right )\right )+900 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{2} \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right ) \left (-d^{2} \cos \left (d x +c \right ) a -b \,d^{2} \sin \left (d x +c \right )\right )^{2}+600 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{2} \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right )^{2} \left (d^{3} \sin \left (d x +c \right ) a -b \,d^{3} \cos \left (d x +c \right )\right )+200 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{3} \left (-d^{2} \cos \left (d x +c \right ) a -b \,d^{2} \sin \left (d x +c \right )\right ) \left (d^{3} \sin \left (d x +c \right ) a -b \,d^{3} \cos \left (d x +c \right )\right )+100 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{3} \left (-\sin \left (d x +c \right ) d a +b d \cos \left (d x +c \right )\right ) \left (d^{4} \cos \left (d x +c \right ) a +b \,d^{4} \sin \left (d x +c \right )\right )+5 \left (\cos \left (d x +c \right ) a +b \sin \left (d x +c \right )\right )^{4} \left (-d^{5} \sin \left (d x +c \right ) a +b \,d^{5} \cos \left (d x +c \right )\right )}{225 d^{6}}\) | \(616\) |
Input:
int((cos(d*x+c)*a+b*sin(d*x+c))^5,x,method=_RETURNVERBOSE)
Output:
1/d*(1/5*a^5*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c)-a^4*b*cos(d*x+ c)^5+10*b^2*a^3*(-1/5*sin(d*x+c)*cos(d*x+c)^4+1/15*(2+cos(d*x+c)^2)*sin(d* x+c))+10*a^2*b^3*(-1/5*sin(d*x+c)^2*cos(d*x+c)^3-2/15*cos(d*x+c)^3)+b^4*a* sin(d*x+c)^5-1/5*b^5*(8/3+sin(d*x+c)^4+4/3*sin(d*x+c)^2)*cos(d*x+c))
Time = 0.08 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.65 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {15 \, b^{5} \cos \left (d x + c\right ) + 3 \, {\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{5} + 10 \, {\left (5 \, a^{2} b^{3} - b^{5}\right )} \cos \left (d x + c\right )^{3} - {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4} + 3 \, {\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \cos \left (d x + c\right )^{4} + 2 \, {\left (2 \, a^{5} + 5 \, a^{3} b^{2} - 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{15 \, d} \] Input:
integrate((a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="fricas")
Output:
-1/15*(15*b^5*cos(d*x + c) + 3*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(d*x + c)^5 + 10*(5*a^2*b^3 - b^5)*cos(d*x + c)^3 - (8*a^5 + 20*a^3*b^2 + 15*a*b^4 + 3*(a^5 - 10*a^3*b^2 + 5*a*b^4)*cos(d*x + c)^4 + 2*(2*a^5 + 5*a^3*b^2 - 15* a*b^4)*cos(d*x + c)^2)*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (82) = 164\).
Time = 0.29 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.84 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\begin {cases} \frac {8 a^{5} \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a^{5} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a^{5} \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {a^{4} b \cos ^{5}{\left (c + d x \right )}}{d} + \frac {4 a^{3} b^{2} \sin ^{5}{\left (c + d x \right )}}{3 d} + \frac {10 a^{3} b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} - \frac {10 a^{2} b^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {4 a^{2} b^{3} \cos ^{5}{\left (c + d x \right )}}{3 d} + \frac {a b^{4} \sin ^{5}{\left (c + d x \right )}}{d} - \frac {b^{5} \sin ^{4}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{d} - \frac {4 b^{5} \sin ^{2}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {8 b^{5} \cos ^{5}{\left (c + d x \right )}}{15 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \] Input:
integrate((a*cos(d*x+c)+b*sin(d*x+c))**5,x)
Output:
Piecewise((8*a**5*sin(c + d*x)**5/(15*d) + 4*a**5*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + a**5*sin(c + d*x)*cos(c + d*x)**4/d - a**4*b*cos(c + d*x)* *5/d + 4*a**3*b**2*sin(c + d*x)**5/(3*d) + 10*a**3*b**2*sin(c + d*x)**3*co s(c + d*x)**2/(3*d) - 10*a**2*b**3*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - 4*a**2*b**3*cos(c + d*x)**5/(3*d) + a*b**4*sin(c + d*x)**5/d - b**5*sin(c + d*x)**4*cos(c + d*x)/d - 4*b**5*sin(c + d*x)**2*cos(c + d*x)**3/(3*d) - 8*b**5*cos(c + d*x)**5/(15*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))**5, Tr ue))
Time = 0.04 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.83 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {a^{4} b \cos \left (d x + c\right )^{5}}{d} + \frac {a b^{4} \sin \left (d x + c\right )^{5}}{d} + \frac {{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a^{5}}{15 \, d} - \frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{5} - 5 \, \sin \left (d x + c\right )^{3}\right )} a^{3} b^{2}}{3 \, d} + \frac {2 \, {\left (3 \, \cos \left (d x + c\right )^{5} - 5 \, \cos \left (d x + c\right )^{3}\right )} a^{2} b^{3}}{3 \, d} - \frac {{\left (3 \, \cos \left (d x + c\right )^{5} - 10 \, \cos \left (d x + c\right )^{3} + 15 \, \cos \left (d x + c\right )\right )} b^{5}}{15 \, d} \] Input:
integrate((a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="maxima")
Output:
-a^4*b*cos(d*x + c)^5/d + a*b^4*sin(d*x + c)^5/d + 1/15*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a^5/d - 2/3*(3*sin(d*x + c)^5 - 5*s in(d*x + c)^3)*a^3*b^2/d + 2/3*(3*cos(d*x + c)^5 - 5*cos(d*x + c)^3)*a^2*b ^3/d - 1/15*(3*cos(d*x + c)^5 - 10*cos(d*x + c)^3 + 15*cos(d*x + c))*b^5/d
Leaf count of result is larger than twice the leaf count of optimal. 187 vs. \(2 (90) = 180\).
Time = 0.16 (sec) , antiderivative size = 187, normalized size of antiderivative = 1.99 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=-\frac {{\left (5 \, a^{4} b - 10 \, a^{2} b^{3} + b^{5}\right )} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {5 \, {\left (3 \, a^{4} b + 2 \, a^{2} b^{3} - b^{5}\right )} \cos \left (3 \, d x + 3 \, c\right )}{48 \, d} - \frac {5 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )}{8 \, d} + \frac {{\left (a^{5} - 10 \, a^{3} b^{2} + 5 \, a b^{4}\right )} \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac {5 \, {\left (a^{5} - 2 \, a^{3} b^{2} - 3 \, a b^{4}\right )} \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac {5 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} \sin \left (d x + c\right )}{8 \, d} \] Input:
integrate((a*cos(d*x+c)+b*sin(d*x+c))^5,x, algorithm="giac")
Output:
-1/80*(5*a^4*b - 10*a^2*b^3 + b^5)*cos(5*d*x + 5*c)/d - 5/48*(3*a^4*b + 2* a^2*b^3 - b^5)*cos(3*d*x + 3*c)/d - 5/8*(a^4*b + 2*a^2*b^3 + b^5)*cos(d*x + c)/d + 1/80*(a^5 - 10*a^3*b^2 + 5*a*b^4)*sin(5*d*x + 5*c)/d + 5/48*(a^5 - 2*a^3*b^2 - 3*a*b^4)*sin(3*d*x + 3*c)/d + 5/8*(a^5 + 2*a^3*b^2 + a*b^4)* sin(d*x + c)/d
Time = 16.50 (sec) , antiderivative size = 248, normalized size of antiderivative = 2.64 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {2\,\left (\frac {3\,\sin \left (c+d\,x\right )\,a^5\,{\cos \left (c+d\,x\right )}^4}{2}+2\,\sin \left (c+d\,x\right )\,a^5\,{\cos \left (c+d\,x\right )}^2+4\,\sin \left (c+d\,x\right )\,a^5-\frac {15\,a^4\,b\,{\cos \left (c+d\,x\right )}^5}{2}-15\,\sin \left (c+d\,x\right )\,a^3\,b^2\,{\cos \left (c+d\,x\right )}^4+5\,\sin \left (c+d\,x\right )\,a^3\,b^2\,{\cos \left (c+d\,x\right )}^2+10\,\sin \left (c+d\,x\right )\,a^3\,b^2+15\,a^2\,b^3\,{\cos \left (c+d\,x\right )}^5-25\,a^2\,b^3\,{\cos \left (c+d\,x\right )}^3+\frac {15\,\sin \left (c+d\,x\right )\,a\,b^4\,{\cos \left (c+d\,x\right )}^4}{2}-15\,\sin \left (c+d\,x\right )\,a\,b^4\,{\cos \left (c+d\,x\right )}^2+\frac {15\,\sin \left (c+d\,x\right )\,a\,b^4}{2}-\frac {3\,b^5\,{\cos \left (c+d\,x\right )}^5}{2}+5\,b^5\,{\cos \left (c+d\,x\right )}^3-\frac {15\,b^5\,\cos \left (c+d\,x\right )}{2}\right )}{15\,d} \] Input:
int((a*cos(c + d*x) + b*sin(c + d*x))^5,x)
Output:
(2*(4*a^5*sin(c + d*x) - (15*b^5*cos(c + d*x))/2 + 5*b^5*cos(c + d*x)^3 - (3*b^5*cos(c + d*x)^5)/2 - (15*a^4*b*cos(c + d*x)^5)/2 + 2*a^5*cos(c + d*x )^2*sin(c + d*x) + (3*a^5*cos(c + d*x)^4*sin(c + d*x))/2 + 10*a^3*b^2*sin( c + d*x) - 25*a^2*b^3*cos(c + d*x)^3 + 15*a^2*b^3*cos(c + d*x)^5 + (15*a*b ^4*sin(c + d*x))/2 + 5*a^3*b^2*cos(c + d*x)^2*sin(c + d*x) - 15*a^3*b^2*co s(c + d*x)^4*sin(c + d*x) - 15*a*b^4*cos(c + d*x)^2*sin(c + d*x) + (15*a*b ^4*cos(c + d*x)^4*sin(c + d*x))/2))/(15*d)
Time = 0.16 (sec) , antiderivative size = 220, normalized size of antiderivative = 2.34 \[ \int (a \cos (c+d x)+b \sin (c+d x))^5 \, dx=\frac {-15 \cos \left (d x +c \right )^{5} a^{4} b -20 \cos \left (d x +c \right )^{5} a^{2} b^{3}-8 \cos \left (d x +c \right )^{5} b^{5}+15 \cos \left (d x +c \right )^{4} \sin \left (d x +c \right ) a^{5}-50 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} a^{2} b^{3}-20 \cos \left (d x +c \right )^{3} \sin \left (d x +c \right )^{2} b^{5}+20 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3} a^{5}+50 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{3} a^{3} b^{2}-15 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} b^{5}+8 \sin \left (d x +c \right )^{5} a^{5}+20 \sin \left (d x +c \right )^{5} a^{3} b^{2}+15 \sin \left (d x +c \right )^{5} a \,b^{4}}{15 d} \] Input:
int((a*cos(d*x+c)+b*sin(d*x+c))^5,x)
Output:
( - 15*cos(c + d*x)**5*a**4*b - 20*cos(c + d*x)**5*a**2*b**3 - 8*cos(c + d *x)**5*b**5 + 15*cos(c + d*x)**4*sin(c + d*x)*a**5 - 50*cos(c + d*x)**3*si n(c + d*x)**2*a**2*b**3 - 20*cos(c + d*x)**3*sin(c + d*x)**2*b**5 + 20*cos (c + d*x)**2*sin(c + d*x)**3*a**5 + 50*cos(c + d*x)**2*sin(c + d*x)**3*a** 3*b**2 - 15*cos(c + d*x)*sin(c + d*x)**4*b**5 + 8*sin(c + d*x)**5*a**5 + 2 0*sin(c + d*x)**5*a**3*b**2 + 15*sin(c + d*x)**5*a*b**4)/(15*d)