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

Optimal. Leaf size=90 \[ -\frac{\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}-\frac{\sin (a+b x) \cos ^5(a+b x)}{16 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{64 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{3 x}{128} \]

[Out]

(3*x)/128 + (3*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(64*b) - (Cos[a + b*x]^5*Sin
[a + b*x])/(16*b) - (Cos[a + b*x]^5*Sin[a + b*x]^3)/(8*b)

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Rubi [A]  time = 0.0832022, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.176, Rules used = {2568, 2635, 8} \[ -\frac{\sin ^3(a+b x) \cos ^5(a+b x)}{8 b}-\frac{\sin (a+b x) \cos ^5(a+b x)}{16 b}+\frac{\sin (a+b x) \cos ^3(a+b x)}{64 b}+\frac{3 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{3 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^4*Sin[a + b*x]^4,x]

[Out]

(3*x)/128 + (3*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (Cos[a + b*x]^3*Sin[a + b*x])/(64*b) - (Cos[a + b*x]^5*Sin
[a + b*x])/(16*b) - (Cos[a + b*x]^5*Sin[a + b*x]^3)/(8*b)

Rule 2568

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(a*(b*Cos[e
+ f*x])^(n + 1)*(a*Sin[e + f*x])^(m - 1))/(b*f*(m + n)), x] + Dist[(a^2*(m - 1))/(m + n), Int[(b*Cos[e + f*x])
^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[2*
m, 2*n]

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) \sin ^4(a+b x) \, dx &=-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{3}{8} \int \cos ^4(a+b x) \sin ^2(a+b x) \, dx\\ &=-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{1}{16} \int \cos ^4(a+b x) \, dx\\ &=\frac{\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{3}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac{3 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}+\frac{3 \int 1 \, dx}{128}\\ &=\frac{3 x}{128}+\frac{3 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{\cos ^3(a+b x) \sin (a+b x)}{64 b}-\frac{\cos ^5(a+b x) \sin (a+b x)}{16 b}-\frac{\cos ^5(a+b x) \sin ^3(a+b x)}{8 b}\\ \end{align*}

Mathematica [A]  time = 0.0437347, size = 33, normalized size = 0.37 \[ \frac{24 (a+b x)-8 \sin (4 (a+b x))+\sin (8 (a+b x))}{1024 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^4*Sin[a + b*x]^4,x]

[Out]

(24*(a + b*x) - 8*Sin[4*(a + b*x)] + Sin[8*(a + b*x)])/(1024*b)

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Maple [A]  time = 0.012, size = 72, normalized size = 0.8 \begin{align*}{\frac{1}{b} \left ( -{\frac{ \left ( \cos \left ( bx+a \right ) \right ) ^{5} \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{8}}-{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{5}}{16}}+{\frac{\sin \left ( bx+a \right ) }{64} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{3}+{\frac{3\,\cos \left ( bx+a \right ) }{2}} \right ) }+{\frac{3\,bx}{128}}+{\frac{3\,a}{128}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/b*(-1/8*cos(b*x+a)^5*sin(b*x+a)^3-1/16*sin(b*x+a)*cos(b*x+a)^5+1/64*(cos(b*x+a)^3+3/2*cos(b*x+a))*sin(b*x+a)
+3/128*b*x+3/128*a)

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Maxima [A]  time = 0.997071, size = 45, normalized size = 0.5 \begin{align*} \frac{24 \, b x + 24 \, a + \sin \left (8 \, b x + 8 \, a\right ) - 8 \, \sin \left (4 \, b x + 4 \, a\right )}{1024 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1024*(24*b*x + 24*a + sin(8*b*x + 8*a) - 8*sin(4*b*x + 4*a))/b

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Fricas [A]  time = 1.60475, size = 146, normalized size = 1.62 \begin{align*} \frac{3 \, b x +{\left (16 \, \cos \left (b x + a\right )^{7} - 24 \, \cos \left (b x + a\right )^{5} + 2 \, \cos \left (b x + a\right )^{3} + 3 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{128 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/128*(3*b*x + (16*cos(b*x + a)^7 - 24*cos(b*x + a)^5 + 2*cos(b*x + a)^3 + 3*cos(b*x + a))*sin(b*x + a))/b

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Sympy [A]  time = 12.3905, size = 189, normalized size = 2.1 \begin{align*} \begin{cases} \frac{3 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac{3 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac{9 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac{3 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac{3 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac{3 \sin ^{7}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{128 b} + \frac{11 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{128 b} - \frac{11 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{128 b} - \frac{3 \sin{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sin ^{4}{\left (a \right )} \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*sin(b*x+a)**4,x)

[Out]

Piecewise((3*x*sin(a + b*x)**8/128 + 3*x*sin(a + b*x)**6*cos(a + b*x)**2/32 + 9*x*sin(a + b*x)**4*cos(a + b*x)
**4/64 + 3*x*sin(a + b*x)**2*cos(a + b*x)**6/32 + 3*x*cos(a + b*x)**8/128 + 3*sin(a + b*x)**7*cos(a + b*x)/(12
8*b) + 11*sin(a + b*x)**5*cos(a + b*x)**3/(128*b) - 11*sin(a + b*x)**3*cos(a + b*x)**5/(128*b) - 3*sin(a + b*x
)*cos(a + b*x)**7/(128*b), Ne(b, 0)), (x*sin(a)**4*cos(a)**4, True))

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Giac [A]  time = 1.16124, size = 43, normalized size = 0.48 \begin{align*} \frac{3}{128} \, x + \frac{\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/128*x + 1/1024*sin(8*b*x + 8*a)/b - 1/128*sin(4*b*x + 4*a)/b