3.403 \(\int \cos ^7(c+d x) (a+b \cos (c+d x)) \, dx\)

Optimal. Leaf size=150 \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 b x}{128} \]

[Out]

(35*b*x)/128 + (a*Sin[c + d*x])/d + (35*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*b*Cos[c + d*x]^3*Sin[c + d*
x])/(192*d) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (b*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (a*Sin[c + d*
x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.101661, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2748, 2633, 2635, 8} \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^7(c+d x)}{8 d}+\frac{7 b \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{35 b \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{35 b \sin (c+d x) \cos (c+d x)}{128 d}+\frac{35 b x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + b*Cos[c + d*x]),x]

[Out]

(35*b*x)/128 + (a*Sin[c + d*x])/d + (35*b*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (35*b*Cos[c + d*x]^3*Sin[c + d*
x])/(192*d) + (7*b*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) + (b*Cos[c + d*x]^7*Sin[c + d*x])/(8*d) - (a*Sin[c + d*
x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*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 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 ^7(c+d x) (a+b \cos (c+d x)) \, dx &=a \int \cos ^7(c+d x) \, dx+b \int \cos ^8(c+d x) \, dx\\ &=\frac{b \cos ^7(c+d x) \sin (c+d x)}{8 d}+\frac{1}{8} (7 b) \int \cos ^6(c+d x) \, dx-\frac{a \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=\frac{a \sin (c+d x)}{d}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^7(c+d x)}{7 d}+\frac{1}{48} (35 b) \int \cos ^4(c+d x) \, dx\\ &=\frac{a \sin (c+d x)}{d}+\frac{35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^7(c+d x)}{7 d}+\frac{1}{64} (35 b) \int \cos ^2(c+d x) \, dx\\ &=\frac{a \sin (c+d x)}{d}+\frac{35 b \cos (c+d x) \sin (c+d x)}{128 d}+\frac{35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^7(c+d x)}{7 d}+\frac{1}{128} (35 b) \int 1 \, dx\\ &=\frac{35 b x}{128}+\frac{a \sin (c+d x)}{d}+\frac{35 b \cos (c+d x) \sin (c+d x)}{128 d}+\frac{35 b \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{7 b \cos ^5(c+d x) \sin (c+d x)}{48 d}+\frac{b \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.212083, size = 135, normalized size = 0.9 \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}+\frac{35 b (c+d x)}{128 d}+\frac{7 b \sin (2 (c+d x))}{32 d}+\frac{7 b \sin (4 (c+d x))}{128 d}+\frac{b \sin (6 (c+d x))}{96 d}+\frac{b \sin (8 (c+d x))}{1024 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + b*Cos[c + d*x]),x]

[Out]

(35*b*(c + d*x))/(128*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[c +
d*x]^7)/(7*d) + (7*b*Sin[2*(c + d*x)])/(32*d) + (7*b*Sin[4*(c + d*x)])/(128*d) + (b*Sin[6*(c + d*x)])/(96*d)
+ (b*Sin[8*(c + d*x)])/(1024*d)

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Maple [A]  time = 0.035, size = 100, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\sin \left ( dx+c \right ) }{8} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( dx+c \right ) }{16}} \right ) }+{\frac{35\,dx}{128}}+{\frac{35\,c}{128}} \right ) +{\frac{a\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 ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+b*cos(d*x+c)),x)

[Out]

1/d*(b*(1/8*(cos(d*x+c)^7+7/6*cos(d*x+c)^5+35/24*cos(d*x+c)^3+35/16*cos(d*x+c))*sin(d*x+c)+35/128*d*x+35/128*c
)+1/7*a*(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))

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Maxima [A]  time = 0.984367, size = 142, normalized size = 0.95 \begin{align*} -\frac{3072 \,{\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 )} a + 35 \,{\left (128 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 840 \, d x - 840 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 168 \, \sin \left (4 \, d x + 4 \, c\right ) - 768 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{107520 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

-1/107520*(3072*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a + 35*(128*sin(2
*d*x + 2*c)^3 - 840*d*x - 840*c - 3*sin(8*d*x + 8*c) - 168*sin(4*d*x + 4*c) - 768*sin(2*d*x + 2*c))*b)/d

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Fricas [A]  time = 2.26971, size = 289, normalized size = 1.93 \begin{align*} \frac{3675 \, b d x +{\left (1680 \, b \cos \left (d x + c\right )^{7} + 1920 \, a \cos \left (d x + c\right )^{6} + 1960 \, b \cos \left (d x + c\right )^{5} + 2304 \, a \cos \left (d x + c\right )^{4} + 2450 \, b \cos \left (d x + c\right )^{3} + 3072 \, a \cos \left (d x + c\right )^{2} + 3675 \, b \cos \left (d x + c\right ) + 6144 \, a\right )} \sin \left (d x + c\right )}{13440 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/13440*(3675*b*d*x + (1680*b*cos(d*x + c)^7 + 1920*a*cos(d*x + c)^6 + 1960*b*cos(d*x + c)^5 + 2304*a*cos(d*x
+ c)^4 + 2450*b*cos(d*x + c)^3 + 3072*a*cos(d*x + c)^2 + 3675*b*cos(d*x + c) + 6144*a)*sin(d*x + c))/d

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Sympy [A]  time = 13.7775, size = 286, normalized size = 1.91 \begin{align*} \begin{cases} \frac{16 a \sin ^{7}{\left (c + d x \right )}}{35 d} + \frac{8 a \sin ^{5}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{5 d} + \frac{2 a \sin ^{3}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{d} + \frac{a \sin{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{d} + \frac{35 b x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{35 b x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{105 b x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{35 b x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{35 b x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{35 b \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{385 b \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{511 b \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac{93 b \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right ) \cos ^{7}{\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)**7*(a+b*cos(d*x+c)),x)

[Out]

Piecewise((16*a*sin(c + d*x)**7/(35*d) + 8*a*sin(c + d*x)**5*cos(c + d*x)**2/(5*d) + 2*a*sin(c + d*x)**3*cos(c
+ d*x)**4/d + a*sin(c + d*x)*cos(c + d*x)**6/d + 35*b*x*sin(c + d*x)**8/128 + 35*b*x*sin(c + d*x)**6*cos(c +
d*x)**2/32 + 105*b*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 35*b*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 35*b*x*c
os(c + d*x)**8/128 + 35*b*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 385*b*sin(c + d*x)**5*cos(c + d*x)**3/(384*d)
+ 511*b*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) + 93*b*sin(c + d*x)*cos(c + d*x)**7/(128*d), Ne(d, 0)), (x*(a
+ b*cos(c))*cos(c)**7, True))

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Giac [A]  time = 1.36918, size = 165, normalized size = 1.1 \begin{align*} \frac{35}{128} \, b x + \frac{b \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{a \sin \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{b \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac{7 \, a \sin \left (5 \, d x + 5 \, c\right )}{320 \, d} + \frac{7 \, b \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{7 \, a \sin \left (3 \, d x + 3 \, c\right )}{64 \, d} + \frac{7 \, b \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} + \frac{35 \, a \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)^7*(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

35/128*b*x + 1/1024*b*sin(8*d*x + 8*c)/d + 1/448*a*sin(7*d*x + 7*c)/d + 1/96*b*sin(6*d*x + 6*c)/d + 7/320*a*si
n(5*d*x + 5*c)/d + 7/128*b*sin(4*d*x + 4*c)/d + 7/64*a*sin(3*d*x + 3*c)/d + 7/32*b*sin(2*d*x + 2*c)/d + 35/64*
a*sin(d*x + c)/d