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

Optimal. Leaf size=80 \[ \frac{\cos ^4(a+b x)}{16 b^2}+\frac{3 \cos ^2(a+b x)}{16 b^2}+\frac{x \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x^2}{16} \]

[Out]

(3*x^2)/16 + (3*Cos[a + b*x]^2)/(16*b^2) + Cos[a + b*x]^4/(16*b^2) + (3*x*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (
x*Cos[a + b*x]^3*Sin[a + b*x])/(4*b)

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Rubi [A]  time = 0.0474994, antiderivative size = 80, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3310, 30} \[ \frac{\cos ^4(a+b x)}{16 b^2}+\frac{3 \cos ^2(a+b x)}{16 b^2}+\frac{x \sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac{3 x \sin (a+b x) \cos (a+b x)}{8 b}+\frac{3 x^2}{16} \]

Antiderivative was successfully verified.

[In]

Int[x*Cos[a + b*x]^4,x]

[Out]

(3*x^2)/16 + (3*Cos[a + b*x]^2)/(16*b^2) + Cos[a + b*x]^4/(16*b^2) + (3*x*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (
x*Cos[a + b*x]^3*Sin[a + b*x])/(4*b)

Rule 3310

Int[((c_.) + (d_.)*(x_))*((b_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(d*(b*Sin[e + f*x])^n)/(f^2*n
^2), x] + (Dist[(b^2*(n - 1))/n, Int[(c + d*x)*(b*Sin[e + f*x])^(n - 2), x], x] - Simp[(b*(c + d*x)*Cos[e + f*
x]*(b*Sin[e + f*x])^(n - 1))/(f*n), x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n, 1]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int x \cos ^4(a+b x) \, dx &=\frac{\cos ^4(a+b x)}{16 b^2}+\frac{x \cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac{3}{4} \int x \cos ^2(a+b x) \, dx\\ &=\frac{3 \cos ^2(a+b x)}{16 b^2}+\frac{\cos ^4(a+b x)}{16 b^2}+\frac{3 x \cos (a+b x) \sin (a+b x)}{8 b}+\frac{x \cos ^3(a+b x) \sin (a+b x)}{4 b}+\frac{3 \int x \, dx}{8}\\ &=\frac{3 x^2}{16}+\frac{3 \cos ^2(a+b x)}{16 b^2}+\frac{\cos ^4(a+b x)}{16 b^2}+\frac{3 x \cos (a+b x) \sin (a+b x)}{8 b}+\frac{x \cos ^3(a+b x) \sin (a+b x)}{4 b}\\ \end{align*}

Mathematica [A]  time = 0.132604, size = 53, normalized size = 0.66 \[ \frac{4 b x (8 \sin (2 (a+b x))+\sin (4 (a+b x))+6 b x)+16 \cos (2 (a+b x))+\cos (4 (a+b x))}{128 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Cos[a + b*x]^4,x]

[Out]

(16*Cos[2*(a + b*x)] + Cos[4*(a + b*x)] + 4*b*x*(6*b*x + 8*Sin[2*(a + b*x)] + Sin[4*(a + b*x)]))/(128*b^2)

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Maple [A]  time = 0.026, size = 110, normalized size = 1.4 \begin{align*}{\frac{1}{{b}^{2}} \left ( \left ( bx+a \right ) \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 ) -{\frac{3\, \left ( bx+a \right ) ^{2}}{16}}+{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{16}}+{\frac{3\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{16}}-a \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 ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b^2*((b*x+a)*(1/4*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)+3/8*b*x+3/8*a)-3/16*(b*x+a)^2+1/16*cos(b*x+a)^4+3
/16*cos(b*x+a)^2-a*(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 = 1.01717, size = 132, normalized size = 1.65 \begin{align*} \frac{24 \,{\left (b x + a\right )}^{2} - 4 \,{\left (12 \, b x + 12 \, a + \sin \left (4 \, b x + 4 \, a\right ) + 8 \, \sin \left (2 \, b x + 2 \, a\right )\right )} a + 4 \,{\left (b x + a\right )} \sin \left (4 \, b x + 4 \, a\right ) + 32 \,{\left (b x + a\right )} \sin \left (2 \, b x + 2 \, a\right ) + \cos \left (4 \, b x + 4 \, a\right ) + 16 \, \cos \left (2 \, b x + 2 \, a\right )}{128 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(24*(b*x + a)^2 - 4*(12*b*x + 12*a + sin(4*b*x + 4*a) + 8*sin(2*b*x + 2*a))*a + 4*(b*x + a)*sin(4*b*x +
4*a) + 32*(b*x + a)*sin(2*b*x + 2*a) + cos(4*b*x + 4*a) + 16*cos(2*b*x + 2*a))/b^2

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Fricas [A]  time = 1.36012, size = 161, normalized size = 2.01 \begin{align*} \frac{3 \, b^{2} x^{2} + \cos \left (b x + a\right )^{4} + 3 \, \cos \left (b x + a\right )^{2} + 2 \,{\left (2 \, b x \cos \left (b x + a\right )^{3} + 3 \, b x \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{16 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(3*b^2*x^2 + cos(b*x + a)^4 + 3*cos(b*x + a)^2 + 2*(2*b*x*cos(b*x + a)^3 + 3*b*x*cos(b*x + a))*sin(b*x +
a))/b^2

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Sympy [A]  time = 3.41895, size = 138, normalized size = 1.72 \begin{align*} \begin{cases} \frac{3 x^{2} \sin ^{4}{\left (a + b x \right )}}{16} + \frac{3 x^{2} \sin ^{2}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{8} + \frac{3 x^{2} \cos ^{4}{\left (a + b x \right )}}{16} + \frac{3 x \sin ^{3}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{8 b} + \frac{5 x \sin{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{8 b} - \frac{3 \sin ^{4}{\left (a + b x \right )}}{32 b^{2}} + \frac{5 \cos ^{4}{\left (a + b x \right )}}{32 b^{2}} & \text{for}\: b \neq 0 \\\frac{x^{2} \cos ^{4}{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x*cos(b*x+a)**4,x)

[Out]

Piecewise((3*x**2*sin(a + b*x)**4/16 + 3*x**2*sin(a + b*x)**2*cos(a + b*x)**2/8 + 3*x**2*cos(a + b*x)**4/16 +
3*x*sin(a + b*x)**3*cos(a + b*x)/(8*b) + 5*x*sin(a + b*x)*cos(a + b*x)**3/(8*b) - 3*sin(a + b*x)**4/(32*b**2)
+ 5*cos(a + b*x)**4/(32*b**2), Ne(b, 0)), (x**2*cos(a)**4/2, True))

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Giac [A]  time = 1.13597, size = 86, normalized size = 1.08 \begin{align*} \frac{3}{16} \, x^{2} + \frac{x \sin \left (4 \, b x + 4 \, a\right )}{32 \, b} + \frac{x \sin \left (2 \, b x + 2 \, a\right )}{4 \, b} + \frac{\cos \left (4 \, b x + 4 \, a\right )}{128 \, b^{2}} + \frac{\cos \left (2 \, b x + 2 \, a\right )}{8 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/16*x^2 + 1/32*x*sin(4*b*x + 4*a)/b + 1/4*x*sin(2*b*x + 2*a)/b + 1/128*cos(4*b*x + 4*a)/b^2 + 1/8*cos(2*b*x +
2*a)/b^2