3.407 \(\int \cos ^3(c+d x) (a+b \cos (c+d x)) \, dx\)

Optimal. Leaf size=76 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8} \]

[Out]

(3*b*x)/8 + (a*Sin[c + d*x])/d + (3*b*Cos[c + d*x]*Sin[c + d*x])/(8*d) + (b*Cos[c + d*x]^3*Sin[c + d*x])/(4*d)
- (a*Sin[c + d*x]^3)/(3*d)

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Rubi [A]  time = 0.0586476, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {2748, 2633, 2635, 8} \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{b \sin (c+d x) \cos ^3(c+d x)}{4 d}+\frac{3 b \sin (c+d x) \cos (c+d x)}{8 d}+\frac{3 b x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^3*(a + b*Cos[c + d*x]),x]

[Out]

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

Mathematica [A]  time = 0.0966939, size = 73, normalized size = 0.96 \[ -\frac{a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}+\frac{3 b (c+d x)}{8 d}+\frac{b \sin (2 (c+d x))}{4 d}+\frac{b \sin (4 (c+d x))}{32 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^3*(a + b*Cos[c + d*x]),x]

[Out]

(3*b*(c + d*x))/(8*d) + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/(3*d) + (b*Sin[2*(c + d*x)])/(4*d) + (b*Sin[4*
(c + d*x)])/(32*d)

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Maple [A]  time = 0.033, size = 60, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( b \left ({\frac{\sin \left ( dx+c \right ) }{4} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{8}}+{\frac{3\,c}{8}} \right ) +{\frac{a \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{3}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^3*(a+b*cos(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 0.954641, size = 77, normalized size = 1.01 \begin{align*} -\frac{32 \,{\left (\sin \left (d x + c\right )^{3} - 3 \, \sin \left (d x + c\right )\right )} a - 3 \,{\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} b}{96 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c)),x, algorithm="maxima")

[Out]

-1/96*(32*(sin(d*x + c)^3 - 3*sin(d*x + c))*a - 3*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*b)/d

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Fricas [A]  time = 1.8701, size = 136, normalized size = 1.79 \begin{align*} \frac{9 \, b d x +{\left (6 \, b \cos \left (d x + c\right )^{3} + 8 \, a \cos \left (d x + c\right )^{2} + 9 \, b \cos \left (d x + c\right ) + 16 \, a\right )} \sin \left (d x + c\right )}{24 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c)),x, algorithm="fricas")

[Out]

1/24*(9*b*d*x + (6*b*cos(d*x + c)^3 + 8*a*cos(d*x + c)^2 + 9*b*cos(d*x + c) + 16*a)*sin(d*x + c))/d

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Sympy [A]  time = 1.18761, size = 144, normalized size = 1.89 \begin{align*} \begin{cases} \frac{2 a \sin ^{3}{\left (c + d x \right )}}{3 d} + \frac{a \sin{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{d} + \frac{3 b x \sin ^{4}{\left (c + d x \right )}}{8} + \frac{3 b x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac{3 b x \cos ^{4}{\left (c + d x \right )}}{8} + \frac{3 b \sin ^{3}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{8 d} + \frac{5 b \sin{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} & \text{for}\: d \neq 0 \\x \left (a + b \cos{\left (c \right )}\right ) \cos ^{3}{\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)**3*(a+b*cos(d*x+c)),x)

[Out]

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

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Giac [A]  time = 1.32354, size = 84, normalized size = 1.11 \begin{align*} \frac{3}{8} \, b x + \frac{b \sin \left (4 \, d x + 4 \, c\right )}{32 \, d} + \frac{a \sin \left (3 \, d x + 3 \, c\right )}{12 \, d} + \frac{b \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} + \frac{3 \, a \sin \left (d x + c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^3*(a+b*cos(d*x+c)),x, algorithm="giac")

[Out]

3/8*b*x + 1/32*b*sin(4*d*x + 4*c)/d + 1/12*a*sin(3*d*x + 3*c)/d + 1/4*b*sin(2*d*x + 2*c)/d + 3/4*a*sin(d*x + c
)/d