3.1 \(\int \cos ^5(c+d x) (a+a \cos (c+d x)) \, dx\)

Optimal. Leaf size=114 \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16} \]

[Out]

(5*a*x)/16 + (a*Sin[c + d*x])/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*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) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0695691, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2748, 2633, 2635, 8} \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*a*x)/16 + (a*Sin[c + d*x])/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*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) - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)

Rule 2748

Int[((b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[c, Int[(b*S
in[e + f*x])^m, x], x] + Dist[d/b, Int[(b*Sin[e + f*x])^(m + 1), x], x] /; FreeQ[{b, c, d, e, f, m}, x]

Rule 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rule 2635

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] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+a \cos (c+d x)) \, dx &=a \int \cos ^5(c+d x) \, dx+a \int \cos ^6(c+d x) \, dx\\ &=\frac{a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{6} (5 a) \int \cos ^4(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{a \sin (c+d x)}{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}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{1}{8} (5 a) \int \cos ^2(c+d x) \, dx\\ &=\frac{a \sin (c+d x)}{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}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^5(c+d x)}{5 d}+\frac{1}{16} (5 a) \int 1 \, dx\\ &=\frac{5 a x}{16}+\frac{a \sin (c+d x)}{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}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.18795, size = 75, normalized size = 0.66 \[ \frac{a \left (192 \sin ^5(c+d x)-640 \sin ^3(c+d x)+960 \sin (c+d x)+5 (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)\right )}{960 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(960*Sin[c + d*x] - 640*Sin[c + d*x]^3 + 192*Sin[c + d*x]^5 + 5*(60*c + 60*d*x + 45*Sin[2*(c + d*x)] + 9*Si
n[4*(c + d*x)] + Sin[6*(c + d*x)])))/(960*d)

________________________________________________________________________________________

Maple [A]  time = 0.044, size = 80, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( a \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) +{\frac{a\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)+5/16*d*x+5/16*c)+1/5*a*(8/3+cos(d*x+c)^
4+4/3*cos(d*x+c)^2)*sin(d*x+c))

________________________________________________________________________________________

Maxima [A]  time = 1.17699, size = 113, normalized size = 0.99 \begin{align*} \frac{64 \,{\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 \,{\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}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/960*(64*(3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a - 5*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c
- 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a)/d

________________________________________________________________________________________

Fricas [A]  time = 1.94138, size = 204, normalized size = 1.79 \begin{align*} \frac{75 \, a d x +{\left (40 \, a \cos \left (d x + c\right )^{5} + 48 \, a \cos \left (d x + c\right )^{4} + 50 \, a \cos \left (d x + c\right )^{3} + 64 \, a \cos \left (d x + c\right )^{2} + 75 \, a \cos \left (d x + c\right ) + 128 \, a\right )} \sin \left (d x + c\right )}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/240*(75*a*d*x + (40*a*cos(d*x + c)^5 + 48*a*cos(d*x + c)^4 + 50*a*cos(d*x + c)^3 + 64*a*cos(d*x + c)^2 + 75*
a*cos(d*x + c) + 128*a)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 4.76251, size = 216, normalized size = 1.89 \begin{align*} \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{8 a \sin ^{5}{\left (c + d x \right )}}{15 d} + \frac{5 a \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac{4 a \sin ^{3}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{3 d} + \frac{11 a \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac{a \sin{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a \cos{\left (c \right )} + a\right ) \cos ^{5}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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) + 8*a*sin(c + d*x)**5/(15*d
) + 5*a*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) + 4*a*sin(c + d*x)**3*cos(c + d*x)**2/(3*d) + 11*a*sin(c + d*x)*
cos(c + d*x)**5/(16*d) + a*sin(c + d*x)*cos(c + d*x)**4/d, Ne(d, 0)), (x*(a*cos(c) + a)*cos(c)**5, True))

________________________________________________________________________________________

Giac [A]  time = 1.28096, size = 124, normalized size = 1.09 \begin{align*} \frac{5}{16} \, a x + \frac{a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{a \sin \left (5 \, d x + 5 \, c\right )}{80 \, d} + \frac{3 \, a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{5 \, a \sin \left (3 \, d x + 3 \, c\right )}{48 \, d} + \frac{15 \, a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{5 \, a \sin \left (d x + c\right )}{8 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/16*a*x + 1/192*a*sin(6*d*x + 6*c)/d + 1/80*a*sin(5*d*x + 5*c)/d + 3/64*a*sin(4*d*x + 4*c)/d + 5/48*a*sin(3*d
*x + 3*c)/d + 15/64*a*sin(2*d*x + 2*c)/d + 5/8*a*sin(d*x + c)/d