3.58 \(\int \cos ^6(a+b x) \sin ^2(a+b x) \, dx\)

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

[Out]

(5*x)/128 + (5*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(192*b) + (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 = 0.0658742, antiderivative size = 88, 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} \[ -\frac{\sin (a+b x) \cos ^7(a+b x)}{8 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{48 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{192 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{128 b}+\frac{5 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]^6*Sin[a + b*x]^2,x]

[Out]

(5*x)/128 + (5*Cos[a + b*x]*Sin[a + b*x])/(128*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(192*b) + (Cos[a + b*x]^5*
Sin[a + b*x])/(48*b) - (Cos[a + b*x]^7*Sin[a + b*x])/(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 ^6(a+b x) \sin ^2(a+b x) \, dx &=-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{1}{8} \int \cos ^6(a+b x) \, dx\\ &=\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{5}{48} \int \cos ^4(a+b x) \, dx\\ &=\frac{5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{5}{64} \int \cos ^2(a+b x) \, dx\\ &=\frac{5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{\cos ^5(a+b x) \sin (a+b x)}{48 b}-\frac{\cos ^7(a+b x) \sin (a+b x)}{8 b}+\frac{5 \int 1 \, dx}{128}\\ &=\frac{5 x}{128}+\frac{5 \cos (a+b x) \sin (a+b x)}{128 b}+\frac{5 \cos ^3(a+b x) \sin (a+b x)}{192 b}+\frac{\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 = 0.144025, size = 52, normalized size = 0.59 \[ \frac{48 \sin (2 (a+b x))-24 \sin (4 (a+b x))-16 \sin (6 (a+b x))-3 \sin (8 (a+b x))+120 b x}{3072 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]^6*Sin[a + b*x]^2,x]

[Out]

(120*b*x + 48*Sin[2*(a + b*x)] - 24*Sin[4*(a + b*x)] - 16*Sin[6*(a + b*x)] - 3*Sin[8*(a + b*x)])/(3072*b)

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Maple [A]  time = 0.039, size = 64, normalized size = 0.7 \begin{align*}{\frac{1}{b} \left ( -{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( bx+a \right ) }{48} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( bx+a \right ) }{8}} \right ) }+{\frac{5\,bx}{128}}+{\frac{5\,a}{128}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)^6*sin(b*x+a)^2,x)

[Out]

1/b*(-1/8*sin(b*x+a)*cos(b*x+a)^7+1/48*(cos(b*x+a)^5+5/4*cos(b*x+a)^3+15/8*cos(b*x+a))*sin(b*x+a)+5/128*b*x+5/
128*a)

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Maxima [A]  time = 0.997626, size = 65, normalized size = 0.74 \begin{align*} \frac{64 \, \sin \left (2 \, b x + 2 \, a\right )^{3} + 120 \, b x + 120 \, a - 3 \, \sin \left (8 \, b x + 8 \, a\right ) - 24 \, \sin \left (4 \, b x + 4 \, a\right )}{3072 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^2,x, algorithm="maxima")

[Out]

1/3072*(64*sin(2*b*x + 2*a)^3 + 120*b*x + 120*a - 3*sin(8*b*x + 8*a) - 24*sin(4*b*x + 4*a))/b

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Fricas [A]  time = 1.62223, size = 149, normalized size = 1.69 \begin{align*} \frac{15 \, b x -{\left (48 \, \cos \left (b x + a\right )^{7} - 8 \, \cos \left (b x + a\right )^{5} - 10 \, \cos \left (b x + a\right )^{3} - 15 \, \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)^6*sin(b*x+a)^2,x, algorithm="fricas")

[Out]

1/384*(15*b*x - (48*cos(b*x + a)^7 - 8*cos(b*x + a)^5 - 10*cos(b*x + a)^3 - 15*cos(b*x + a))*sin(b*x + a))/b

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Sympy [A]  time = 12.0965, size = 189, normalized size = 2.15 \begin{align*} \begin{cases} \frac{5 x \sin ^{8}{\left (a + b x \right )}}{128} + \frac{5 x \sin ^{6}{\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )}}{32} + \frac{15 x \sin ^{4}{\left (a + b x \right )} \cos ^{4}{\left (a + b x \right )}}{64} + \frac{5 x \sin ^{2}{\left (a + b x \right )} \cos ^{6}{\left (a + b x \right )}}{32} + \frac{5 x \cos ^{8}{\left (a + b x \right )}}{128} + \frac{5 \sin ^{7}{\left (a + b x \right )} \cos{\left (a + b x \right )}}{128 b} + \frac{55 \sin ^{5}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )}}{384 b} + \frac{73 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (a + b x \right )}}{384 b} - \frac{5 \sin{\left (a + b x \right )} \cos ^{7}{\left (a + b x \right )}}{128 b} & \text{for}\: b \neq 0 \\x \sin ^{2}{\left (a \right )} \cos ^{6}{\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)**6*sin(b*x+a)**2,x)

[Out]

Piecewise((5*x*sin(a + b*x)**8/128 + 5*x*sin(a + b*x)**6*cos(a + b*x)**2/32 + 15*x*sin(a + b*x)**4*cos(a + b*x
)**4/64 + 5*x*sin(a + b*x)**2*cos(a + b*x)**6/32 + 5*x*cos(a + b*x)**8/128 + 5*sin(a + b*x)**7*cos(a + b*x)/(1
28*b) + 55*sin(a + b*x)**5*cos(a + b*x)**3/(384*b) + 73*sin(a + b*x)**3*cos(a + b*x)**5/(384*b) - 5*sin(a + b*
x)*cos(a + b*x)**7/(128*b), Ne(b, 0)), (x*sin(a)**2*cos(a)**6, True))

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Giac [A]  time = 1.17535, size = 81, normalized size = 0.92 \begin{align*} \frac{5}{128} \, x - \frac{\sin \left (8 \, b x + 8 \, a\right )}{1024 \, b} - \frac{\sin \left (6 \, b x + 6 \, a\right )}{192 \, b} - \frac{\sin \left (4 \, b x + 4 \, a\right )}{128 \, b} + \frac{\sin \left (2 \, b x + 2 \, a\right )}{64 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)^6*sin(b*x+a)^2,x, algorithm="giac")

[Out]

5/128*x - 1/1024*sin(8*b*x + 8*a)/b - 1/192*sin(6*b*x + 6*a)/b - 1/128*sin(4*b*x + 4*a)/b + 1/64*sin(2*b*x + 2
*a)/b