3.8 \(\int \cos ^8(a+b x) \, dx\)

Optimal. Leaf size=88 \[ \frac{\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac{7 \sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac{35 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac{35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{35 x}{128} \]

[Out]

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

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Rubi [A]  time = 0.0469793, antiderivative size = 88, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac{7 \sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac{35 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac{35 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{35 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^8,x]

[Out]

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

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 ^8(a+b x) \, dx &=\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{7}{8} \int \cos ^6(a+b x) \, dx\\ &=\frac{7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{35}{48} \int \cos ^4(a+b x) \, dx\\ &=\frac{35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{35}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac{35 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{35 \int 1 \, dx}{128}\\ &=\frac{35 x}{128}+\frac{35 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{35 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{7 \cos ^5(a+b x) \sin (a+b x)}{48 b}+\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0545457, size = 55, normalized size = 0.62 \[ \frac{672 \sin (2 (a+b x))+168 \sin (4 (a+b x))+32 \sin (6 (a+b x))+3 \sin (8 (a+b x))+840 a+840 b x}{3072 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^8,x]

[Out]

(840*a + 840*b*x + 672*Sin[2*(a + b*x)] + 168*Sin[4*(a + b*x)] + 32*Sin[6*(a + b*x)] + 3*Sin[8*(a + b*x)])/(30
72*b)

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Maple [A]  time = 0.026, size = 58, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ({\frac{\sin \left ( bx+a \right ) }{8} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{7}+{\frac{7\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{6}}+{\frac{35\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( bx+a \right ) }{16}} \right ) }+{\frac{35\,bx}{128}}+{\frac{35\,a}{128}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^8,x)

[Out]

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

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Maxima [A]  time = 1.63483, size = 80, normalized size = 0.91 \begin{align*} -\frac{128 \, \sin \left (2 \, b x + 2 \, a\right )^{3} - 840 \, b x - 840 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 168 \, \sin \left (4 \, b x + 4 \, a\right ) - 768 \, \sin \left (2 \, b x + 2 \, a\right )}{3072 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^8,x, algorithm="maxima")

[Out]

-1/3072*(128*sin(2*b*x + 2*a)^3 - 840*b*x - 840*a - 3*sin(8*b*x + 8*a) - 168*sin(4*b*x + 4*a) - 768*sin(2*b*x
+ 2*a))/b

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Fricas [A]  time = 1.43896, size = 153, normalized size = 1.74 \begin{align*} \frac{105 \, b x +{\left (48 \, \cos \left (b x + a\right )^{7} + 56 \, \cos \left (b x + a\right )^{5} + 70 \, \cos \left (b x + a\right )^{3} + 105 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{384 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^8,x, algorithm="fricas")

[Out]

1/384*(105*b*x + (48*cos(b*x + a)^7 + 56*cos(b*x + a)^5 + 70*cos(b*x + a)^3 + 105*cos(b*x + a))*sin(b*x + a))/
b

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Sympy [A]  time = 12.0218, size = 184, normalized size = 2.09 \begin{align*} \begin{cases} \frac{35 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac{35 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac{105 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac{35 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac{35 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac{35 \sin ^{7}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{128 b} + \frac{385 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac{511 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} + \frac{93 \sin{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \cos ^{8}{\left (a \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)**8,x)

[Out]

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

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Giac [A]  time = 1.38629, size = 81, normalized size = 0.92 \begin{align*} \frac{35}{128} \, x + \frac{\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} + \frac{\sin \left (6 \, b x + 6 \, a\right )}{96 \, b} + \frac{7 \, \sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac{7 \, \sin \left (2 \, b x + 2 \, a\right )}{32 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^8,x, algorithm="giac")

[Out]

35/128*x + 1/1024*sin(8*b*x + 8*a)/b + 1/96*sin(6*b*x + 6*a)/b + 7/128*sin(4*b*x + 4*a)/b + 7/32*sin(2*b*x + 2
*a)/b