3.60 \(\int \csc ^3(a+b x) \sin ^9(2 a+2 b x) \, dx\)

Optimal. Leaf size=76 \[ \frac{512 \sin ^{15}(a+b x)}{15 b}-\frac{2048 \sin ^{13}(a+b x)}{13 b}+\frac{3072 \sin ^{11}(a+b x)}{11 b}-\frac{2048 \sin ^9(a+b x)}{9 b}+\frac{512 \sin ^7(a+b x)}{7 b} \]

[Out]

(512*Sin[a + b*x]^7)/(7*b) - (2048*Sin[a + b*x]^9)/(9*b) + (3072*Sin[a + b*x]^11)/(11*b) - (2048*Sin[a + b*x]^
13)/(13*b) + (512*Sin[a + b*x]^15)/(15*b)

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Rubi [A]  time = 0.0773139, antiderivative size = 76, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {4288, 2564, 270} \[ \frac{512 \sin ^{15}(a+b x)}{15 b}-\frac{2048 \sin ^{13}(a+b x)}{13 b}+\frac{3072 \sin ^{11}(a+b x)}{11 b}-\frac{2048 \sin ^9(a+b x)}{9 b}+\frac{512 \sin ^7(a+b x)}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^9,x]

[Out]

(512*Sin[a + b*x]^7)/(7*b) - (2048*Sin[a + b*x]^9)/(9*b) + (3072*Sin[a + b*x]^11)/(11*b) - (2048*Sin[a + b*x]^
13)/(13*b) + (512*Sin[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 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 \csc ^3(a+b x) \sin ^9(2 a+2 b x) \, dx &=512 \int \cos ^9(a+b x) \sin ^6(a+b x) \, dx\\ &=\frac{512 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^4 \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{512 \operatorname{Subst}\left (\int \left (x^6-4 x^8+6 x^{10}-4 x^{12}+x^{14}\right ) \, dx,x,\sin (a+b x)\right )}{b}\\ &=\frac{512 \sin ^7(a+b x)}{7 b}-\frac{2048 \sin ^9(a+b x)}{9 b}+\frac{3072 \sin ^{11}(a+b x)}{11 b}-\frac{2048 \sin ^{13}(a+b x)}{13 b}+\frac{512 \sin ^{15}(a+b x)}{15 b}\\ \end{align*}

Mathematica [A]  time = 0.329026, size = 58, normalized size = 0.76 \[ \frac{512 \left (3003 \sin ^{15}(a+b x)-13860 \sin ^{13}(a+b x)+24570 \sin ^{11}(a+b x)-20020 \sin ^9(a+b x)+6435 \sin ^7(a+b x)\right )}{45045 b} \]

Antiderivative was successfully verified.

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^9,x]

[Out]

(512*(6435*Sin[a + b*x]^7 - 20020*Sin[a + b*x]^9 + 24570*Sin[a + b*x]^11 - 13860*Sin[a + b*x]^13 + 3003*Sin[a
+ b*x]^15))/(45045*b)

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Maple [A]  time = 0.058, size = 107, normalized size = 1.4 \begin{align*} 512\,{\frac{1}{b} \left ( -1/15\, \left ( \sin \left ( bx+a \right ) \right ) ^{5} \left ( \cos \left ( bx+a \right ) \right ) ^{10}-1/39\, \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{10}-{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{10}}{143}}+{\frac{\sin \left ( bx+a \right ) }{1287} \left ({\frac{128}{35}}+ \left ( \cos \left ( bx+a \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( bx+a \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}}{35}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(b*x+a)^3*sin(2*b*x+2*a)^9,x)

[Out]

512/b*(-1/15*sin(b*x+a)^5*cos(b*x+a)^10-1/39*sin(b*x+a)^3*cos(b*x+a)^10-1/143*sin(b*x+a)*cos(b*x+a)^10+1/1287*
(128/35+cos(b*x+a)^8+8/7*cos(b*x+a)^6+48/35*cos(b*x+a)^4+64/35*cos(b*x+a)^2)*sin(b*x+a))

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Maxima [A]  time = 1.07465, size = 123, normalized size = 1.62 \begin{align*} -\frac{3003 \, \sin \left (15 \, b x + 15 \, a\right ) + 10395 \, \sin \left (13 \, b x + 13 \, a\right ) - 12285 \, \sin \left (11 \, b x + 11 \, a\right ) - 85085 \, \sin \left (9 \, b x + 9 \, a\right ) - 19305 \, \sin \left (7 \, b x + 7 \, a\right ) + 351351 \, \sin \left (5 \, b x + 5 \, a\right ) + 375375 \, \sin \left (3 \, b x + 3 \, a\right ) - 2027025 \, \sin \left (b x + a\right )}{1441440 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^9,x, algorithm="maxima")

[Out]

-1/1441440*(3003*sin(15*b*x + 15*a) + 10395*sin(13*b*x + 13*a) - 12285*sin(11*b*x + 11*a) - 85085*sin(9*b*x +
9*a) - 19305*sin(7*b*x + 7*a) + 351351*sin(5*b*x + 5*a) + 375375*sin(3*b*x + 3*a) - 2027025*sin(b*x + a))/b

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Fricas [A]  time = 0.529794, size = 246, normalized size = 3.24 \begin{align*} -\frac{512 \,{\left (3003 \, \cos \left (b x + a\right )^{14} - 7161 \, \cos \left (b x + a\right )^{12} + 4473 \, \cos \left (b x + a\right )^{10} - 35 \, \cos \left (b x + a\right )^{8} - 40 \, \cos \left (b x + a\right )^{6} - 48 \, \cos \left (b x + a\right )^{4} - 64 \, \cos \left (b x + a\right )^{2} - 128\right )} \sin \left (b x + a\right )}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^9,x, algorithm="fricas")

[Out]

-512/45045*(3003*cos(b*x + a)^14 - 7161*cos(b*x + a)^12 + 4473*cos(b*x + a)^10 - 35*cos(b*x + a)^8 - 40*cos(b*
x + a)^6 - 48*cos(b*x + a)^4 - 64*cos(b*x + a)^2 - 128)*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(csc(b*x+a)**3*sin(2*b*x+2*a)**9,x)

[Out]

Timed out

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Giac [A]  time = 2.1526, size = 76, normalized size = 1. \begin{align*} \frac{512 \,{\left (3003 \, \sin \left (b x + a\right )^{15} - 13860 \, \sin \left (b x + a\right )^{13} + 24570 \, \sin \left (b x + a\right )^{11} - 20020 \, \sin \left (b x + a\right )^{9} + 6435 \, \sin \left (b x + a\right )^{7}\right )}}{45045 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^9,x, algorithm="giac")

[Out]

512/45045*(3003*sin(b*x + a)^15 - 13860*sin(b*x + a)^13 + 24570*sin(b*x + a)^11 - 20020*sin(b*x + a)^9 + 6435*
sin(b*x + a)^7)/b