3.4 \(\int \cos ^4(a+b x) \, dx\)

Optimal. Leaf size=46 \[ \frac{\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x}{8} \]

[Out]

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

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Rubi [A]  time = 0.0208342, antiderivative size = 46, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2635, 8} \[ \frac{\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x}{8} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4,x]

[Out]

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

Mathematica [A]  time = 0.0380392, size = 33, normalized size = 0.72 \[ \frac{12 (a+b x)+8 \sin (2 (a+b x))+\sin (4 (a+b x))}{32 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4,x]

[Out]

(12*(a + b*x) + 8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)])/(32*b)

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Maple [A]  time = 0.027, size = 38, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ({\frac{\sin \left ( bx+a \right ) }{4} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+{\frac{3\,\cos \left ( bx+a \right ) }{2}} \right ) }+{\frac{3\,bx}{8}}+{\frac{3\,a}{8}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.38624, size = 45, normalized size = 0.98 \begin{align*} \frac{12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \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)^4,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.64788, size = 89, normalized size = 1.93 \begin{align*} \frac{3 \, b x +{\left (2 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{8 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.01157, size = 95, normalized size = 2.07 \begin{align*} \begin{cases} \frac{3 x \sin ^{4}{\left (a + b x \right )}}{8} + \frac{3 x \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{4} + \frac{3 x \cos ^{4}{\left (a + b x \right )}}{8} + \frac{3 \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{8 b} + \frac{5 \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} & \text{for}\: b \neq 0 \\x \cos ^{4}{\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)**4,x)

[Out]

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

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Giac [A]  time = 1.299, size = 43, normalized size = 0.93 \begin{align*} \frac{3}{8} \, x + \frac{\sin \left (4 \, b x + 4 \, a\right )}{32 \, b} + \frac{\sin \left (2 \, b x + 2 \, a\right )}{4 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^4,x, algorithm="giac")

[Out]

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