\(\int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx\) [63]

Optimal result

Integrand size = 20, antiderivative size = 31 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64 \cos ^7(a+b x)}{7 b}+\frac {64 \cos ^9(a+b x)}{9 b} \]

[Out]

-64/7*cos(b*x+a)^7/b+64/9*cos(b*x+a)^9/b

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4373, 2645, 14} \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {64 \cos ^9(a+b x)}{9 b}-\frac {64 \cos ^7(a+b x)}{7 b} \]

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^6,x]

[Out]

(-64*Cos[a + b*x]^7)/(7*b) + (64*Cos[a + b*x]^9)/(9*b)

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2645

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 4373

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]

Rubi steps \begin{align*} \text {integral}& = 64 \int \cos ^6(a+b x) \sin ^3(a+b x) \, dx \\ & = -\frac {64 \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {64 \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {64 \cos ^7(a+b x)}{7 b}+\frac {64 \cos ^9(a+b x)}{9 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {32 \cos ^7(a+b x) (-11+7 \cos (2 (a+b x)))}{63 b} \]

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^6,x]

[Out]

(32*Cos[a + b*x]^7*(-11 + 7*Cos[2*(a + b*x)]))/(63*b)

Maple [A] (verified)

Time = 23.73 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.87

method	result	size
default	\(\frac {\frac {64 \cos \left (x b +a \right )^{9}}{9}-\frac {64 \cos \left (x b +a \right )^{7}}{7}}{b}\)	\(27\)
risch	\(-\frac {3 \cos \left (x b +a \right )}{2 b}+\frac {\cos \left (9 x b +9 a \right )}{36 b}+\frac {3 \cos \left (7 x b +7 a \right )}{28 b}-\frac {2 \cos \left (3 x b +3 a \right )}{3 b}\)	\(55\)

[In]

int(csc(b*x+a)^3*sin(2*b*x+2*a)^6,x,method=_RETURNVERBOSE)

[Out]

64/b*(1/9*cos(b*x+a)^9-1/7*cos(b*x+a)^7)

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {64 \, {\left (7 \, \cos \left (b x + a\right )^{9} - 9 \, \cos \left (b x + a\right )^{7}\right )}}{63 \, b} \]

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^6,x, algorithm="fricas")

[Out]

64/63*(7*cos(b*x + a)^9 - 9*cos(b*x + a)^7)/b

Sympy [F(-1)]

Timed out. \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\text {Timed out} \]

[In]

integrate(csc(b*x+a)**3*sin(2*b*x+2*a)**6,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.52 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {7 \, \cos \left (9 \, b x + 9 \, a\right ) + 27 \, \cos \left (7 \, b x + 7 \, a\right ) - 168 \, \cos \left (3 \, b x + 3 \, a\right ) - 378 \, \cos \left (b x + a\right )}{252 \, b} \]

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^6,x, algorithm="maxima")

[Out]

1/252*(7*cos(9*b*x + 9*a) + 27*cos(7*b*x + 7*a) - 168*cos(3*b*x + 3*a) - 378*cos(b*x + a))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (27) = 54\).

Time = 0.31 (sec) , antiderivative size = 182, normalized size of antiderivative = 5.87 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {256 \, {\left (\frac {9 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} + \frac {27 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {189 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {189 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {315 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {105 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {63 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} - 1\right )}}{63 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{9}} \]

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^6,x, algorithm="giac")

[Out]

-256/63*(9*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) + 27*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 189*(cos(b*x
+ a) - 1)^3/(cos(b*x + a) + 1)^3 + 189*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + 315*(cos(b*x + a) - 1)^5/(
cos(b*x + a) + 1)^5 + 105*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 + 63*(cos(b*x + a) - 1)^7/(cos(b*x + a) +
1)^7 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^9)

Mupad [B] (verification not implemented)

Time = 20.31 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.84 \[ \int \csc ^3(a+b x) \sin ^6(2 a+2 b x) \, dx=-\frac {64\,\left (9\,{\cos \left (a+b\,x\right )}^7-7\,{\cos \left (a+b\,x\right )}^9\right )}{63\,b} \]

[In]

int(sin(2*a + 2*b*x)^6/sin(a + b*x)^3,x)

[Out]

-(64*(9*cos(a + b*x)^7 - 7*cos(a + b*x)^9))/(63*b)