3.404 \(\int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx\)

Optimal. Leaf size=128 \[ \frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{d}+\frac{b \sin (c+d x)}{d} \]

[Out]

(5*a*x)/16 + (b*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) - (b*Sin[c + d*x]^3)/d + (3*b*Sin[c + d*x]^5)/(5*d) - (b*Sin[c +
d*x]^7)/(7*d)

________________________________________________________________________________________

Rubi [A]  time = 0.0857035, antiderivative size = 128, 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, 2635, 8, 2633} \[ \frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^3(c+d x)}{d}+\frac{b \sin (c+d x)}{d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + b*Cos[c + d*x]),x]

[Out]

(5*a*x)/16 + (b*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) - (b*Sin[c + d*x]^3)/d + (3*b*Sin[c + d*x]^5)/(5*d) - (b*Sin[c +
d*x]^7)/(7*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 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]

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]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) (a+b \cos (c+d x)) \, dx &=a \int \cos ^6(c+d x) \, dx+b \int \cos ^7(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{b \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{b \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{b \sin ^3(c+d x)}{d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{1}{8} (5 a) \int \cos ^2(c+d x) \, dx\\ &=\frac{b \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{b \sin ^3(c+d x)}{d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^7(c+d x)}{7 d}+\frac{1}{16} (5 a) \int 1 \, dx\\ &=\frac{5 a x}{16}+\frac{b \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{b \sin ^3(c+d x)}{d}+\frac{3 b \sin ^5(c+d x)}{5 d}-\frac{b \sin ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.181262, size = 89, normalized size = 0.7 \[ \frac{35 a (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))+60 c+60 d x)-960 b \sin ^7(c+d x)+4032 b \sin ^5(c+d x)-6720 b \sin ^3(c+d x)+6720 b \sin (c+d x)}{6720 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + b*Cos[c + d*x]),x]

[Out]

(6720*b*Sin[c + d*x] - 6720*b*Sin[c + d*x]^3 + 4032*b*Sin[c + d*x]^5 - 960*b*Sin[c + d*x]^7 + 35*a*(60*c + 60*
d*x + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(6720*d)

________________________________________________________________________________________

Maple [A]  time = 0.033, size = 90, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ({\frac{b\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) }+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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+b*cos(d*x+c)),x)

[Out]

1/d*(1/7*b*(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)+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))

________________________________________________________________________________________

Maxima [A]  time = 0.965041, size = 127, normalized size = 0.99 \begin{align*} -\frac{35 \,{\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 \,{\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 )} b}{6720 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

-1/6720*(35*(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 + 192*(5*sin(d
*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*b)/d

________________________________________________________________________________________

Fricas [A]  time = 2.28918, size = 244, normalized size = 1.91 \begin{align*} \frac{525 \, a d x +{\left (240 \, b \cos \left (d x + c\right )^{6} + 280 \, a \cos \left (d x + c\right )^{5} + 288 \, b \cos \left (d x + c\right )^{4} + 350 \, a \cos \left (d x + c\right )^{3} + 384 \, b \cos \left (d x + c\right )^{2} + 525 \, a \cos \left (d x + c\right ) + 768 \, b\right )} \sin \left (d x + c\right )}{1680 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/1680*(525*a*d*x + (240*b*cos(d*x + c)^6 + 280*a*cos(d*x + c)^5 + 288*b*cos(d*x + c)^4 + 350*a*cos(d*x + c)^3
+ 384*b*cos(d*x + c)^2 + 525*a*cos(d*x + c) + 768*b)*sin(d*x + c))/d

________________________________________________________________________________________

Sympy [A]  time = 8.76348, size = 238, normalized size = 1.86 \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{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{16 b \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 b \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 b \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{b \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right ) \cos ^{6}{\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)**6*(a+b*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) + 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) + 16*b*sin(c + d*x)**7/(35*d) + 8*b*sin(c + d*x)**
5*cos(c + d*x)**2/(5*d) + 2*b*sin(c + d*x)**3*cos(c + d*x)**4/d + b*sin(c + d*x)*cos(c + d*x)**6/d, Ne(d, 0)),
(x*(a + b*cos(c))*cos(c)**6, True))

________________________________________________________________________________________

Giac [A]  time = 1.39491, size = 144, normalized size = 1.12 \begin{align*} \frac{5}{16} \, a x + \frac{b \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{a \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{7 \, b \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{3 \, a \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac{7 \, b \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac{15 \, a \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} + \frac{35 \, b \sin \left (d x + c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

5/16*a*x + 1/448*b*sin(7*d*x + 7*c)/d + 1/192*a*sin(6*d*x + 6*c)/d + 7/320*b*sin(5*d*x + 5*c)/d + 3/64*a*sin(4
*d*x + 4*c)/d + 7/64*b*sin(3*d*x + 3*c)/d + 15/64*a*sin(2*d*x + 2*c)/d + 35/64*b*sin(d*x + c)/d