Integrand size = 26, antiderivative size = 87 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {5 a x}{16}-\frac {b \cos ^6(c+d x)}{6 d}+\frac {5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac {5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac {a \cos ^5(c+d x) \sin (c+d x)}{6 d} \] Output:
5/16*a*x-1/6*b*cos(d*x+c)^6/d+5/16*a*cos(d*x+c)*sin(d*x+c)/d+5/24*a*cos(d* x+c)^3*sin(d*x+c)/d+1/6*a*cos(d*x+c)^5*sin(d*x+c)/d
Time = 0.11 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.66 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {-32 b \cos ^6(c+d x)+a (60 c+60 d x+45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x)))}{192 d} \] Input:
Integrate[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]
Output:
(-32*b*Cos[c + d*x]^6 + a*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*( c + d*x)] + Sin[6*(c + d*x)]))/(192*d)
Time = 0.29 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, 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.
\(\displaystyle \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \cos (c+d x)^5 (a \cos (c+d x)+b \sin (c+d x))dx\) |
\(\Big \downarrow \) 3569 |
\(\displaystyle \int \left (a \cos ^6(c+d x)+b \sin (c+d x) \cos ^5(c+d x)\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac {5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac {5 a x}{16}-\frac {b \cos ^6(c+d x)}{6 d}\) |
Input:
Int[Cos[c + d*x]^5*(a*Cos[c + d*x] + b*Sin[c + d*x]),x]
Output:
(5*a*x)/16 - (b*Cos[c + d*x]^6)/(6*d) + (5*a*Cos[c + d*x]*Sin[c + d*x])/(1 6*d) + (5*a*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*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 = 2.27 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {b \cos \left (d x +c \right )^{6}}{6}}{d}\) | \(62\) |
default | \(\frac {a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {b \cos \left (d x +c \right )^{6}}{6}}{d}\) | \(62\) |
parts | \(\frac {a \left (\frac {\left (\cos \left (d x +c \right )^{5}+\frac {5 \cos \left (d x +c \right )^{3}}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}-\frac {b \cos \left (d x +c \right )^{6}}{6 d}\) | \(64\) |
parallelrisch | \(\frac {60 a x d -6 b \cos \left (4 d x +4 c \right )-15 b \cos \left (2 d x +2 c \right )-b \cos \left (6 d x +6 c \right )+a \sin \left (6 d x +6 c \right )+9 a \sin \left (4 d x +4 c \right )+45 a \sin \left (2 d x +2 c \right )+22 b}{192 d}\) | \(86\) |
risch | \(\frac {5 a x}{16}-\frac {b \cos \left (6 d x +6 c \right )}{192 d}+\frac {a \sin \left (6 d x +6 c \right )}{192 d}-\frac {b \cos \left (4 d x +4 c \right )}{32 d}+\frac {3 a \sin \left (4 d x +4 c \right )}{64 d}-\frac {5 b \cos \left (2 d x +2 c \right )}{64 d}+\frac {15 a \sin \left (2 d x +2 c \right )}{64 d}\) | \(96\) |
norman | \(\frac {\frac {5 a x}{16}+\frac {11 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{24 d}+\frac {15 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{4 d}-\frac {15 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{4 d}+\frac {5 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{24 d}-\frac {11 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{8 d}+\frac {15 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{8}+\frac {75 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{16}+\frac {25 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{4}+\frac {75 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{16}+\frac {15 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{8}+\frac {5 a x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{12}}{16}+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {20 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\) | \(263\) |
orering | \(\text {Expression too large to display}\) | \(1884\) |
Input:
int(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(a*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/1 6*d*x+5/16*c)-1/6*b*cos(d*x+c)^6)
Time = 0.08 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {8 \, b \cos \left (d x + c\right )^{6} - 15 \, a d x - {\left (8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{48 \, d} \] Input:
integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")
Output:
-1/48*(8*b*cos(d*x + c)^6 - 15*a*d*x - (8*a*cos(d*x + c)^5 + 10*a*cos(d*x + c)^3 + 15*a*cos(d*x + c))*sin(d*x + c))/d
Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (82) = 164\).
Time = 0.44 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.01 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\begin {cases} \frac {5 a x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {b \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a \cos {\left (c \right )} + b \sin {\left (c \right )}\right ) \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**5*(a*cos(d*x+c)+b*sin(d*x+c)),x)
Output:
Piecewise((5*a*x*sin(c + d*x)**6/16 + 15*a*x*sin(c + d*x)**4*cos(c + d*x)* *2/16 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*a*x*cos(c + d*x)**6/ 16 + 5*a*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 5*a*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 11*a*sin(c + d*x)*cos(c + d*x)**5/(16*d) - b*cos(c + d*x) **6/(6*d), Ne(d, 0)), (x*(a*cos(c) + b*sin(c))*cos(c)**5, True))
Time = 0.03 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.71 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=-\frac {32 \, b \cos \left (d x + c\right )^{6} + {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a}{192 \, d} \] Input:
integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")
Output:
-1/192*(32*b*cos(d*x + c)^6 + (4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*si n(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a)/d
Time = 0.14 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.09 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {5}{16} \, a x - \frac {b \cos \left (6 \, d x + 6 \, c\right )}{192 \, d} - \frac {b \cos \left (4 \, d x + 4 \, c\right )}{32 \, d} - \frac {5 \, b \cos \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac {a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {3 \, a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {15 \, a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \] Input:
integrate(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")
Output:
5/16*a*x - 1/192*b*cos(6*d*x + 6*c)/d - 1/32*b*cos(4*d*x + 4*c)/d - 5/64*b *cos(2*d*x + 2*c)/d + 1/192*a*sin(6*d*x + 6*c)/d + 3/64*a*sin(4*d*x + 4*c) /d + 15/64*a*sin(2*d*x + 2*c)/d
Time = 21.68 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.71 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {5\,a\,x}{16}+\frac {-\frac {11\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{8}+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{24}-\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{4}+\frac {20\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{3}+\frac {15\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{4}-\frac {5\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{24}+2\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\frac {11\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{8}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6} \] Input:
int(cos(c + d*x)^5*(a*cos(c + d*x) + b*sin(c + d*x)),x)
Output:
(5*a*x)/16 + ((11*a*tan(c/2 + (d*x)/2))/8 - (5*a*tan(c/2 + (d*x)/2)^3)/24 + (15*a*tan(c/2 + (d*x)/2)^5)/4 - (15*a*tan(c/2 + (d*x)/2)^7)/4 + (5*a*tan (c/2 + (d*x)/2)^9)/24 - (11*a*tan(c/2 + (d*x)/2)^11)/8 + 2*b*tan(c/2 + (d* x)/2)^2 + (20*b*tan(c/2 + (d*x)/2)^6)/3 + 2*b*tan(c/2 + (d*x)/2)^10)/(d*(t an(c/2 + (d*x)/2)^2 + 1)^6)
Time = 0.15 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.07 \[ \int \cos ^5(c+d x) (a \cos (c+d x)+b \sin (c+d x)) \, dx=\frac {8 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{5} a -26 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{3} a +33 \cos \left (d x +c \right ) \sin \left (d x +c \right ) a +8 \sin \left (d x +c \right )^{6} b -24 \sin \left (d x +c \right )^{4} b +24 \sin \left (d x +c \right )^{2} b +15 a d x}{48 d} \] Input:
int(cos(d*x+c)^5*(a*cos(d*x+c)+b*sin(d*x+c)),x)
Output:
(8*cos(c + d*x)*sin(c + d*x)**5*a - 26*cos(c + d*x)*sin(c + d*x)**3*a + 33 *cos(c + d*x)*sin(c + d*x)*a + 8*sin(c + d*x)**6*b - 24*sin(c + d*x)**4*b + 24*sin(c + d*x)**2*b + 15*a*d*x)/(48*d)