3.33 \(\int \csc (a+b x) \sin ^8(2 a+2 b x) \, dx\)

Optimal. Leaf size=61 \[ \frac{256 \cos ^{15}(a+b x)}{15 b}-\frac{768 \cos ^{13}(a+b x)}{13 b}+\frac{768 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^9(a+b x)}{9 b} \]

[Out]

(-256*Cos[a + b*x]^9)/(9*b) + (768*Cos[a + b*x]^11)/(11*b) - (768*Cos[a + b*x]^13)/(13*b) + (256*Cos[a + b*x]^
15)/(15*b)

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Rubi [A]  time = 0.0613116, 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, 2565, 270} \[ \frac{256 \cos ^{15}(a+b x)}{15 b}-\frac{768 \cos ^{13}(a+b x)}{13 b}+\frac{768 \cos ^{11}(a+b x)}{11 b}-\frac{256 \cos ^9(a+b x)}{9 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]*Sin[2*a + 2*b*x]^8,x]

[Out]

(-256*Cos[a + b*x]^9)/(9*b) + (768*Cos[a + b*x]^11)/(11*b) - (768*Cos[a + b*x]^13)/(13*b) + (256*Cos[a + b*x]^
15)/(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 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 \csc (a+b x) \sin ^8(2 a+2 b x) \, dx &=256 \int \cos ^8(a+b x) \sin ^7(a+b x) \, dx\\ &=-\frac{256 \operatorname{Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b}\\ &=-\frac{256 \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{256 \cos ^9(a+b x)}{9 b}+\frac{768 \cos ^{11}(a+b x)}{11 b}-\frac{768 \cos ^{13}(a+b x)}{13 b}+\frac{256 \cos ^{15}(a+b x)}{15 b}\\ \end{align*}

Mathematica [A]  time = 0.0936457, size = 119, normalized size = 1.95 \[ -\frac{35 \cos (a+b x)}{64 b}-\frac{35 \cos (3 (a+b x))}{192 b}+\frac{21 \cos (5 (a+b x))}{320 b}+\frac{3 \cos (7 (a+b x))}{64 b}-\frac{7 \cos (9 (a+b x))}{576 b}-\frac{7 \cos (11 (a+b x))}{704 b}+\frac{\cos (13 (a+b x))}{832 b}+\frac{\cos (15 (a+b x))}{960 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]*Sin[2*a + 2*b*x]^8,x]

[Out]

(-35*Cos[a + b*x])/(64*b) - (35*Cos[3*(a + b*x)])/(192*b) + (21*Cos[5*(a + b*x)])/(320*b) + (3*Cos[7*(a + b*x)
])/(64*b) - (7*Cos[9*(a + b*x)])/(576*b) - (7*Cos[11*(a + b*x)])/(704*b) + Cos[13*(a + b*x)]/(832*b) + Cos[15*
(a + b*x)]/(960*b)

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Maple [A]  time = 0.036, size = 71, normalized size = 1.2 \begin{align*} 256\,{\frac{1}{b} \left ( -1/15\, \left ( \sin \left ( bx+a \right ) \right ) ^{6} \left ( \cos \left ( bx+a \right ) \right ) ^{9}-{\frac{2\, \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{65}}-{\frac{8\, \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{715}}-{\frac{16\, \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{6435}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)*sin(2*b*x+2*a)^8,x)

[Out]

256/b*(-1/15*sin(b*x+a)^6*cos(b*x+a)^9-2/65*sin(b*x+a)^4*cos(b*x+a)^9-8/715*sin(b*x+a)^2*cos(b*x+a)^9-16/6435*
cos(b*x+a)^9)

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Maxima [A]  time = 1.06837, 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 )}{411840 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(2*b*x+2*a)^8,x, algorithm="maxima")

[Out]

1/411840*(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

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Fricas [A]  time = 0.538362, size = 136, normalized size = 2.23 \begin{align*} \frac{256 \,{\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(csc(b*x+a)*sin(2*b*x+2*a)^8,x, algorithm="fricas")

[Out]

256/6435*(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

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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(csc(b*x+a)*sin(2*b*x+2*a)**8,x)

[Out]

Timed out

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Giac [B]  time = 1.93562, size = 365, normalized size = 5.98 \begin{align*} -\frac{8192 \,{\left (\frac{15 \,{\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac{105 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac{455 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac{5070 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac{30030 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac{70070 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac{115830 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac{109395 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac{75075 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} + \frac{27027 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} + \frac{6435 \,{\left (\cos \left (b x + a\right ) - 1\right )}^{11}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{11}} - 1\right )}}{6435 \, b{\left (\frac{\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)*sin(2*b*x+2*a)^8,x, algorithm="giac")

[Out]

-8192/6435*(15*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 105*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 455*(co
s(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 + 5070*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + 30030*(cos(b*x + a)
- 1)^5/(cos(b*x + a) + 1)^5 + 70070*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 + 115830*(cos(b*x + a) - 1)^7/(c
os(b*x + a) + 1)^7 + 109395*(cos(b*x + a) - 1)^8/(cos(b*x + a) + 1)^8 + 75075*(cos(b*x + a) - 1)^9/(cos(b*x +
a) + 1)^9 + 27027*(cos(b*x + a) - 1)^10/(cos(b*x + a) + 1)^10 + 6435*(cos(b*x + a) - 1)^11/(cos(b*x + a) + 1)^
11 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^15)