\(\int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx\) [373]

Optimal result

Integrand size = 19, antiderivative size = 60 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cos ^6(c+d x)}{6 d}+\frac {a \sin (c+d x)}{d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d} \] Output:

-1/6*b*cos(d*x+c)^6/d+a*sin(d*x+c)/d-2/3*a*sin(d*x+c)^3/d+1/5*a*sin(d*x+c) 
^5/d
 

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {b \cos ^6(c+d x)}{6 d}+\frac {a \sin (c+d x)}{d}-\frac {2 a \sin ^3(c+d x)}{3 d}+\frac {a \sin ^5(c+d x)}{5 d} \] Input:

Integrate[Cos[c + d*x]^5*(a + b*Sin[c + d*x]),x]
 

Output:

-1/6*(b*Cos[c + d*x]^6)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d 
) + (a*Sin[c + d*x]^5)/(5*d)
 

Rubi [A] (verified)

Time = 0.26 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.23, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {3042, 3147, 455, 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.

\(\displaystyle \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \cos (c+d x)^5 (a+b \sin (c+d x))dx\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {\int (a+b \sin (c+d x)) \left (b^2-b^2 \sin ^2(c+d x)\right )^2d(b \sin (c+d x))}{b^5 d}\)

\(\Big \downarrow \) 455

\(\displaystyle \frac {a \int \left (b^2-b^2 \sin ^2(c+d x)\right )^2d(b \sin (c+d x))-\frac {1}{6} \left (b^2-b^2 \sin ^2(c+d x)\right )^3}{b^5 d}\)

\(\Big \downarrow \) 210

\(\displaystyle \frac {a \int \left (\sin ^4(c+d x) b^4-2 \sin ^2(c+d x) b^4+b^4\right )d(b \sin (c+d x))-\frac {1}{6} \left (b^2-b^2 \sin ^2(c+d x)\right )^3}{b^5 d}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a \left (\frac {1}{5} b^5 \sin ^5(c+d x)-\frac {2}{3} b^5 \sin ^3(c+d x)+b^5 \sin (c+d x)\right )-\frac {1}{6} \left (b^2-b^2 \sin ^2(c+d x)\right )^3}{b^5 d}\)

Input:

Int[Cos[c + d*x]^5*(a + b*Sin[c + d*x]),x]
 

Output:

(-1/6*(b^2 - b^2*Sin[c + d*x]^2)^3 + a*(b^5*Sin[c + d*x] - (2*b^5*Sin[c + 
d*x]^3)/3 + (b^5*Sin[c + d*x]^5)/5))/(b^5*d)
 

Defintions of rubi rules used

rule 210

Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Int[ExpandIntegrand[(a + b*x^2 
)^p, x], x] /; FreeQ[{a, b}, x] && IGtQ[p, 0]
 

rule 455

Int[((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[d*(( 
a + b*x^2)^(p + 1)/(2*b*(p + 1))), x] + Simp[c   Int[(a + b*x^2)^p, x], x] 
/; FreeQ[{a, b, c, d, p}, x] &&  !LeQ[p, -1]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

Maple [A] (verified)

Time = 4.70 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.15

method	result	size
derivativedivides	\(\frac {\frac {b \sin \left (d x +c \right )^{6}}{6}+\frac {a \sin \left (d x +c \right )^{5}}{5}-\frac {b \sin \left (d x +c \right )^{4}}{2}-\frac {2 a \sin \left (d x +c \right )^{3}}{3}+\frac {\sin \left (d x +c \right )^{2} b}{2}+a \sin \left (d x +c \right )}{d}\)	\(69\)
default	\(\frac {\frac {b \sin \left (d x +c \right )^{6}}{6}+\frac {a \sin \left (d x +c \right )^{5}}{5}-\frac {b \sin \left (d x +c \right )^{4}}{2}-\frac {2 a \sin \left (d x +c \right )^{3}}{3}+\frac {\sin \left (d x +c \right )^{2} b}{2}+a \sin \left (d x +c \right )}{d}\)	\(69\)
risch	\(\frac {5 a \sin \left (d x +c \right )}{8 d}-\frac {b \cos \left (6 d x +6 c \right )}{192 d}+\frac {a \sin \left (5 d x +5 c \right )}{80 d}-\frac {b \cos \left (4 d x +4 c \right )}{32 d}+\frac {5 a \sin \left (3 d x +3 c \right )}{48 d}-\frac {5 b \cos \left (2 d x +2 c \right )}{64 d}\)	\(89\)
parallelrisch	\(\frac {2 \tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9} b +\frac {7 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8}}{3}+\frac {26 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{5}+\frac {10 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5} b}{3}+\frac {26 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{4}}{5}+\frac {7 \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{3}+b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )}{d \left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\)	\(139\)
norman	\(\frac {\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{d}+\frac {14 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{3 d}+\frac {52 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{5}}{5 d}+\frac {52 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{7}}{5 d}+\frac {14 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{9}}{3 d}+\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{11}}{d}+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{d}+\frac {20 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{6}}{3 d}+\frac {2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{10}}{d}}{\left (1+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}\right )^{6}}\)	\(169\)
orering	\(-\frac {5369 \left (-5 \cos \left (d x +c \right )^{4} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right ) d +\cos \left (d x +c \right )^{6} b d \right )}{3600 d^{2}}-\frac {37037 \left (-60 \cos \left (d x +c \right )^{2} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )^{3} d^{3}+75 \cos \left (d x +c \right )^{4} b \,d^{3} \sin \left (d x +c \right )^{2}+65 \cos \left (d x +c \right )^{4} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right ) d^{3}-16 \cos \left (d x +c \right )^{6} b \,d^{3}\right )}{64800 d^{4}}-\frac {44473 \left (-120 \sin \left (d x +c \right )^{5} d^{5} \left (a +b \sin \left (d x +c \right )\right )+1200 \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )^{4} b \,d^{5}+1800 \cos \left (d x +c \right )^{2} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )^{3} d^{5}-3075 \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2} b \,d^{5}-1205 \cos \left (d x +c \right )^{4} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right ) d^{5}+376 \cos \left (d x +c \right )^{6} b \,d^{5}\right )}{518400 d^{6}}-\frac {1001 \left (-47460 \sin \left (d x +c \right )^{3} d^{7} \left (a +b \sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{2}-73500 \sin \left (d x +c \right )^{4} d^{7} b \cos \left (d x +c \right )^{2}+4200 \sin \left (d x +c \right )^{5} d^{7} \left (a +b \sin \left (d x +c \right )\right )+2520 \sin \left (d x +c \right )^{6} d^{7} b +114975 \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )^{2} b \,d^{7}+26465 \cos \left (d x +c \right )^{4} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right ) d^{7}-10816 \cos \left (d x +c \right )^{6} d^{7} b \right )}{172800 d^{8}}-\frac {91 \left (-628805 \sin \left (d x +c \right ) d^{9} \left (a +b \sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{4}-4258875 \sin \left (d x +c \right )^{2} d^{9} b \cos \left (d x +c \right )^{4}+1208400 \sin \left (d x +c \right )^{3} d^{9} \left (a +b \sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{2}+3351600 \sin \left (d x +c \right )^{4} d^{9} b \cos \left (d x +c \right )^{2}-115920 \sin \left (d x +c \right )^{5} d^{9} \left (a +b \sin \left (d x +c \right )\right )-166320 \sin \left (d x +c \right )^{6} d^{9} b +347776 \cos \left (d x +c \right )^{6} d^{9} b \right )}{518400 d^{10}}-\frac {15424865 d^{11} \cos \left (d x +c \right )^{4} \left (a +b \sin \left (d x +c \right )\right ) \sin \left (d x +c \right )-11862016 d^{11} \cos \left (d x +c \right )^{6} b [phone] \sin \left (d x +c \right )^{2} d^{11} b \cos \left (d x +c \right )^{4}-30406860 \sin \left (d x +c \right )^{3} d^{11} \left (a +b \sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{2}-137032500 \sin \left (d x +c \right )^{4} d^{11} b \cos \left (d x +c \right )^{2}+2996400 \sin \left (d x +c \right )^{5} d^{11} \left (a +b \sin \left (d x +c \right )\right )+7817040 \sin \left (d x +c \right )^{6} d^{11} b}{518400 d^{12}}\)	\(778\)

Input:

int(cos(d*x+c)^5*(a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)
 

Output:

1/d*(1/6*b*sin(d*x+c)^6+1/5*a*sin(d*x+c)^5-1/2*b*sin(d*x+c)^4-2/3*a*sin(d* 
x+c)^3+1/2*sin(d*x+c)^2*b+a*sin(d*x+c))
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.85 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=-\frac {5 \, b \cos \left (d x + c\right )^{6} - 2 \, {\left (3 \, a \cos \left (d x + c\right )^{4} + 4 \, a \cos \left (d x + c\right )^{2} + 8 \, a\right )} \sin \left (d x + c\right )}{30 \, d} \] Input:

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="fricas")
 

Output:

-1/30*(5*b*cos(d*x + c)^6 - 2*(3*a*cos(d*x + c)^4 + 4*a*cos(d*x + c)^2 + 8 
*a)*sin(d*x + c))/d
 

Sympy [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 83, normalized size of antiderivative = 1.38 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=\begin {cases} \frac {8 a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac {4 a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac {a \sin {\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} - \frac {b \cos ^{6}{\left (c + d x \right )}}{6 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right ) \cos ^{5}{\left (c \right )} & \text {otherwise} \end {cases} \] Input:

integrate(cos(d*x+c)**5*(a+b*sin(d*x+c)),x)
 

Output:

Piecewise((8*a*sin(c + d*x)**5/(15*d) + 4*a*sin(c + d*x)**3*cos(c + d*x)** 
2/(3*d) + a*sin(c + d*x)*cos(c + d*x)**4/d - b*cos(c + d*x)**6/(6*d), Ne(d 
, 0)), (x*(a + b*sin(c))*cos(c)**5, True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {5 \, b \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 15 \, b \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} + 15 \, b \sin \left (d x + c\right )^{2} + 30 \, a \sin \left (d x + c\right )}{30 \, d} \] Input:

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="maxima")
 

Output:

1/30*(5*b*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 - 15*b*sin(d*x + c)^4 - 20*a 
*sin(d*x + c)^3 + 15*b*sin(d*x + c)^2 + 30*a*sin(d*x + c))/d
                                                                                    
                                                                                    
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.17 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {5 \, b \sin \left (d x + c\right )^{6} + 6 \, a \sin \left (d x + c\right )^{5} - 15 \, b \sin \left (d x + c\right )^{4} - 20 \, a \sin \left (d x + c\right )^{3} + 15 \, b \sin \left (d x + c\right )^{2} + 30 \, a \sin \left (d x + c\right )}{30 \, d} \] Input:

integrate(cos(d*x+c)^5*(a+b*sin(d*x+c)),x, algorithm="giac")
 

Output:

1/30*(5*b*sin(d*x + c)^6 + 6*a*sin(d*x + c)^5 - 15*b*sin(d*x + c)^4 - 20*a 
*sin(d*x + c)^3 + 15*b*sin(d*x + c)^2 + 30*a*sin(d*x + c))/d
 

Mupad [B] (verification not implemented)

Time = 15.29 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {\frac {b\,{\sin \left (c+d\,x\right )}^6}{6}+\frac {a\,{\sin \left (c+d\,x\right )}^5}{5}-\frac {b\,{\sin \left (c+d\,x\right )}^4}{2}-\frac {2\,a\,{\sin \left (c+d\,x\right )}^3}{3}+\frac {b\,{\sin \left (c+d\,x\right )}^2}{2}+a\,\sin \left (c+d\,x\right )}{d} \] Input:

int(cos(c + d*x)^5*(a + b*sin(c + d*x)),x)
 

Output:

(a*sin(c + d*x) - (2*a*sin(c + d*x)^3)/3 + (a*sin(c + d*x)^5)/5 + (b*sin(c 
 + d*x)^2)/2 - (b*sin(c + d*x)^4)/2 + (b*sin(c + d*x)^6)/6)/d
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13 \[ \int \cos ^5(c+d x) (a+b \sin (c+d x)) \, dx=\frac {\sin \left (d x +c \right ) \left (5 \sin \left (d x +c \right )^{5} b +6 \sin \left (d x +c \right )^{4} a -15 \sin \left (d x +c \right )^{3} b -20 \sin \left (d x +c \right )^{2} a +15 \sin \left (d x +c \right ) b +30 a \right )}{30 d} \] Input:

int(cos(d*x+c)^5*(a+b*sin(d*x+c)),x)
 

Output:

(sin(c + d*x)*(5*sin(c + d*x)**5*b + 6*sin(c + d*x)**4*a - 15*sin(c + d*x) 
**3*b - 20*sin(c + d*x)**2*a + 15*sin(c + d*x)*b + 30*a))/(30*d)