3.129 \(\int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx\)

Optimal. Leaf size=61 \[ \frac{128 \cos ^{15}(a+b x)}{15 b}-\frac{384 \cos ^{13}(a+b x)}{13 b}+\frac{384 \cos ^{11}(a+b x)}{11 b}-\frac{128 \cos ^9(a+b x)}{9 b} \]

[Out]

(-128*Cos[a + b*x]^9)/(9*b) + (384*Cos[a + b*x]^11)/(11*b) - (384*Cos[a + b*x]^13)/(13*b) + (128*Cos[a + b*x]^
15)/(15*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0598614, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {4287, 2565, 270} \[ \frac{128 \cos ^{15}(a+b x)}{15 b}-\frac{384 \cos ^{13}(a+b x)}{13 b}+\frac{384 \cos ^{11}(a+b x)}{11 b}-\frac{128 \cos ^9(a+b x)}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[Cos[a + b*x]*Sin[2*a + 2*b*x]^7,x]

[Out]

(-128*Cos[a + b*x]^9)/(9*b) + (384*Cos[a + b*x]^11)/(11*b) - (384*Cos[a + b*x]^13)/(13*b) + (128*Cos[a + b*x]^
15)/(15*b)

Rule 4287

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/e^p, Int[(e*Cos
[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]

Rule 2565

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^8(a+b x) \sin ^7(a+b x) \, dx\\ &=-\frac{128 \operatorname{Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{128 \operatorname{Subst}\left (\int \left (x^8-3 x^{10}+3 x^{12}-x^{14}\right ) \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{128 \cos ^9(a+b x)}{9 b}+\frac{384 \cos ^{11}(a+b x)}{11 b}-\frac{384 \cos ^{13}(a+b x)}{13 b}+\frac{128 \cos ^{15}(a+b x)}{15 b}\\ \end{align*}

Mathematica [A]  time = 0.454349, size = 47, normalized size = 0.77 \[ \frac{4 \cos ^9(a+b x) (10755 \cos (2 (a+b x))-3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))-8330)}{6435 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^7,x]

[Out]

(4*Cos[a + b*x]^9*(-8330 + 10755*Cos[2*(a + b*x)] - 3366*Cos[4*(a + b*x)] + 429*Cos[6*(a + b*x)]))/(6435*b)

________________________________________________________________________________________

Maple [B]  time = 0.024, size = 111, normalized size = 1.8 \begin{align*} -{\frac{35\,\cos \left ( bx+a \right ) }{128\,b}}-{\frac{35\,\cos \left ( 3\,bx+3\,a \right ) }{384\,b}}+{\frac{21\,\cos \left ( 5\,bx+5\,a \right ) }{640\,b}}+{\frac{3\,\cos \left ( 7\,bx+7\,a \right ) }{128\,b}}-{\frac{7\,\cos \left ( 9\,bx+9\,a \right ) }{1152\,b}}-{\frac{7\,\cos \left ( 11\,bx+11\,a \right ) }{1408\,b}}+{\frac{\cos \left ( 13\,bx+13\,a \right ) }{1664\,b}}+{\frac{\cos \left ( 15\,bx+15\,a \right ) }{1920\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(b*x+a)*sin(2*b*x+2*a)^7,x)

[Out]

-35/128*cos(b*x+a)/b-35/384*cos(3*b*x+3*a)/b+21/640*cos(5*b*x+5*a)/b+3/128*cos(7*b*x+7*a)/b-7/1152*cos(9*b*x+9
*a)/b-7/1408*cos(11*b*x+11*a)/b+1/1664*cos(13*b*x+13*a)/b+1/1920*cos(15*b*x+15*a)/b

________________________________________________________________________________________

Maxima [A]  time = 1.08165, size = 123, normalized size = 2.02 \begin{align*} \frac{429 \, \cos \left (15 \, b x + 15 \, a\right ) + 495 \, \cos \left (13 \, b x + 13 \, a\right ) - 4095 \, \cos \left (11 \, b x + 11 \, a\right ) - 5005 \, \cos \left (9 \, b x + 9 \, a\right ) + 19305 \, \cos \left (7 \, b x + 7 \, a\right ) + 27027 \, \cos \left (5 \, b x + 5 \, a\right ) - 75075 \, \cos \left (3 \, b x + 3 \, a\right ) - 225225 \, \cos \left (b x + a\right )}{823680 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="maxima")

[Out]

1/823680*(429*cos(15*b*x + 15*a) + 495*cos(13*b*x + 13*a) - 4095*cos(11*b*x + 11*a) - 5005*cos(9*b*x + 9*a) +
19305*cos(7*b*x + 7*a) + 27027*cos(5*b*x + 5*a) - 75075*cos(3*b*x + 3*a) - 225225*cos(b*x + a))/b

________________________________________________________________________________________

Fricas [A]  time = 0.538273, size = 136, normalized size = 2.23 \begin{align*} \frac{128 \,{\left (429 \, \cos \left (b x + a\right )^{15} - 1485 \, \cos \left (b x + a\right )^{13} + 1755 \, \cos \left (b x + a\right )^{11} - 715 \, \cos \left (b x + a\right )^{9}\right )}}{6435 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="fricas")

[Out]

128/6435*(429*cos(b*x + a)^15 - 1485*cos(b*x + a)^13 + 1755*cos(b*x + a)^11 - 715*cos(b*x + a)^9)/b

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)**7,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.66449, size = 149, normalized size = 2.44 \begin{align*} \frac{\cos \left (15 \, b x + 15 \, a\right )}{1920 \, b} + \frac{\cos \left (13 \, b x + 13 \, a\right )}{1664 \, b} - \frac{7 \, \cos \left (11 \, b x + 11 \, a\right )}{1408 \, b} - \frac{7 \, \cos \left (9 \, b x + 9 \, a\right )}{1152 \, b} + \frac{3 \, \cos \left (7 \, b x + 7 \, a\right )}{128 \, b} + \frac{21 \, \cos \left (5 \, b x + 5 \, a\right )}{640 \, b} - \frac{35 \, \cos \left (3 \, b x + 3 \, a\right )}{384 \, b} - \frac{35 \, \cos \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)*sin(2*b*x+2*a)^7,x, algorithm="giac")

[Out]

1/1920*cos(15*b*x + 15*a)/b + 1/1664*cos(13*b*x + 13*a)/b - 7/1408*cos(11*b*x + 11*a)/b - 7/1152*cos(9*b*x + 9
*a)/b + 3/128*cos(7*b*x + 7*a)/b + 21/640*cos(5*b*x + 5*a)/b - 35/384*cos(3*b*x + 3*a)/b - 35/128*cos(b*x + a)
/b