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

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

[Out]

(128*Sin[a + b*x]^9)/(9*b) - (384*Sin[a + b*x]^11)/(11*b) + (384*Sin[a + b*x]^13)/(13*b) - (128*Sin[a + b*x]^1
5)/(15*b)

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Rubi [A]  time = 0.0574345, 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 = {4288, 2564, 270} \[ -\frac{128 \sin ^{15}(a+b x)}{15 b}+\frac{384 \sin ^{13}(a+b x)}{13 b}-\frac{384 \sin ^{11}(a+b x)}{11 b}+\frac{128 \sin ^9(a+b x)}{9 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(128*Sin[a + b*x]^9)/(9*b) - (384*Sin[a + b*x]^11)/(11*b) + (384*Sin[a + b*x]^13)/(13*b) - (128*Sin[a + b*x]^1
5)/(15*b)

Rule 4288

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

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, 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 \sin (a+b x) \sin ^7(2 a+2 b x) \, dx &=128 \int \cos ^7(a+b x) \sin ^8(a+b x) \, dx\\ &=\frac{128 \operatorname{Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\sin (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,\sin (a+b x)\right )}{b}\\ &=\frac{128 \sin ^9(a+b x)}{9 b}-\frac{384 \sin ^{11}(a+b x)}{11 b}+\frac{384 \sin ^{13}(a+b x)}{13 b}-\frac{128 \sin ^{15}(a+b x)}{15 b}\\ \end{align*}

Mathematica [A]  time = 0.46577, size = 47, normalized size = 0.77 \[ \frac{4 \sin ^9(a+b x) (10755 \cos (2 (a+b x))+3366 \cos (4 (a+b x))+429 \cos (6 (a+b x))+8330)}{6435 b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.082, size = 111, normalized size = 1.8 \begin{align*}{\frac{35\,\sin \left ( bx+a \right ) }{128\,b}}-{\frac{35\,\sin \left ( 3\,bx+3\,a \right ) }{384\,b}}-{\frac{21\,\sin \left ( 5\,bx+5\,a \right ) }{640\,b}}+{\frac{3\,\sin \left ( 7\,bx+7\,a \right ) }{128\,b}}+{\frac{7\,\sin \left ( 9\,bx+9\,a \right ) }{1152\,b}}-{\frac{7\,\sin \left ( 11\,bx+11\,a \right ) }{1408\,b}}-{\frac{\sin \left ( 13\,bx+13\,a \right ) }{1664\,b}}+{\frac{\sin \left ( 15\,bx+15\,a \right ) }{1920\,b}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.23679, size = 123, normalized size = 2.02 \begin{align*} \frac{429 \, \sin \left (15 \, b x + 15 \, a\right ) - 495 \, \sin \left (13 \, b x + 13 \, a\right ) - 4095 \, \sin \left (11 \, b x + 11 \, a\right ) + 5005 \, \sin \left (9 \, b x + 9 \, a\right ) + 19305 \, \sin \left (7 \, b x + 7 \, a\right ) - 27027 \, \sin \left (5 \, b x + 5 \, a\right ) - 75075 \, \sin \left (3 \, b x + 3 \, a\right ) + 225225 \, \sin \left (b x + a\right )}{823680 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0.52975, size = 238, normalized size = 3.9 \begin{align*} \frac{128 \,{\left (429 \, \cos \left (b x + a\right )^{14} - 1518 \, \cos \left (b x + a\right )^{12} + 1854 \, \cos \left (b x + a\right )^{10} - 800 \, \cos \left (b x + a\right )^{8} + 5 \, \cos \left (b x + a\right )^{6} + 6 \, \cos \left (b x + a\right )^{4} + 8 \, \cos \left (b x + a\right )^{2} + 16\right )} \sin \left (b x + a\right )}{6435 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

128/6435*(429*cos(b*x + a)^14 - 1518*cos(b*x + a)^12 + 1854*cos(b*x + a)^10 - 800*cos(b*x + a)^8 + 5*cos(b*x +
a)^6 + 6*cos(b*x + a)^4 + 8*cos(b*x + a)^2 + 16)*sin(b*x + a)/b

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

[Out]

Timed out

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Giac [B]  time = 1.52236, size = 149, normalized size = 2.44 \begin{align*} \frac{\sin \left (15 \, b x + 15 \, a\right )}{1920 \, b} - \frac{\sin \left (13 \, b x + 13 \, a\right )}{1664 \, b} - \frac{7 \, \sin \left (11 \, b x + 11 \, a\right )}{1408 \, b} + \frac{7 \, \sin \left (9 \, b x + 9 \, a\right )}{1152 \, b} + \frac{3 \, \sin \left (7 \, b x + 7 \, a\right )}{128 \, b} - \frac{21 \, \sin \left (5 \, b x + 5 \, a\right )}{640 \, b} - \frac{35 \, \sin \left (3 \, b x + 3 \, a\right )}{384 \, b} + \frac{35 \, \sin \left (b x + a\right )}{128 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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