\(\int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx\) [59]

Optimal result

Integrand size = 20, antiderivative size = 61 \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=-\frac {1024 \cos ^{11}(a+b x)}{11 b}+\frac {3072 \cos ^{13}(a+b x)}{13 b}-\frac {1024 \cos ^{15}(a+b x)}{5 b}+\frac {1024 \cos ^{17}(a+b x)}{17 b} \]

[Out]

-1024/11*cos(b*x+a)^11/b+3072/13*cos(b*x+a)^13/b-1024/5*cos(b*x+a)^15/b+1024/17*cos(b*x+a)^17/b

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 61, 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, 276} \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=\frac {1024 \cos ^{17}(a+b x)}{17 b}-\frac {1024 \cos ^{15}(a+b x)}{5 b}+\frac {3072 \cos ^{13}(a+b x)}{13 b}-\frac {1024 \cos ^{11}(a+b x)}{11 b} \]

[In]

Int[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^10,x]

[Out]

(-1024*Cos[a + b*x]^11)/(11*b) + (3072*Cos[a + b*x]^13)/(13*b) - (1024*Cos[a + b*x]^15)/(5*b) + (1024*Cos[a +
b*x]^17)/(17*b)

Rule 276

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]

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}& = 1024 \int \cos ^{10}(a+b x) \sin ^7(a+b x) \, dx \\ & = -\frac {1024 \text {Subst}\left (\int x^{10} \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {1024 \text {Subst}\left (\int \left (x^{10}-3 x^{12}+3 x^{14}-x^{16}\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {1024 \cos ^{11}(a+b x)}{11 b}+\frac {3072 \cos ^{13}(a+b x)}{13 b}-\frac {1024 \cos ^{15}(a+b x)}{5 b}+\frac {1024 \cos ^{17}(a+b x)}{17 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.38 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.95 \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=-\frac {35 \cos (a+b x)}{32 b}-\frac {7 \cos (3 (a+b x))}{16 b}+\frac {7 \cos (5 (a+b x))}{80 b}+\frac {\cos (7 (a+b x))}{8 b}-\frac {5 \cos (11 (a+b x))}{176 b}-\frac {\cos (13 (a+b x))}{208 b}+\frac {\cos (15 (a+b x))}{320 b}+\frac {\cos (17 (a+b x))}{1088 b} \]

[In]

Integrate[Csc[a + b*x]^3*Sin[2*a + 2*b*x]^10,x]

[Out]

(-35*Cos[a + b*x])/(32*b) - (7*Cos[3*(a + b*x)])/(16*b) + (7*Cos[5*(a + b*x)])/(80*b) + Cos[7*(a + b*x)]/(8*b)
- (5*Cos[11*(a + b*x)])/(176*b) - Cos[13*(a + b*x)]/(208*b) + Cos[15*(a + b*x)]/(320*b) + Cos[17*(a + b*x)]/(
1088*b)

Maple [A] (verified)

Time = 168.67 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77

method	result	size
default	\(\frac {\frac {1024 \cos \left (x b +a \right )^{17}}{17}-\frac {1024 \cos \left (x b +a \right )^{15}}{5}+\frac {3072 \cos \left (x b +a \right )^{13}}{13}-\frac {1024 \cos \left (x b +a \right )^{11}}{11}}{b}\)	\(47\)
risch	\(-\frac {35 \cos \left (x b +a \right )}{32 b}+\frac {\cos \left (17 x b +17 a \right )}{1088 b}+\frac {\cos \left (15 x b +15 a \right )}{320 b}-\frac {\cos \left (13 x b +13 a \right )}{208 b}-\frac {5 \cos \left (11 x b +11 a \right )}{176 b}+\frac {\cos \left (7 x b +7 a \right )}{8 b}+\frac {7 \cos \left (5 x b +5 a \right )}{80 b}-\frac {7 \cos \left (3 x b +3 a \right )}{16 b}\)	\(111\)

[In]

int(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x,method=_RETURNVERBOSE)

[Out]

1024/b*(1/17*cos(b*x+a)^17-1/5*cos(b*x+a)^15+3/13*cos(b*x+a)^13-1/11*cos(b*x+a)^11)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=\frac {1024 \, {\left (715 \, \cos \left (b x + a\right )^{17} - 2431 \, \cos \left (b x + a\right )^{15} + 2805 \, \cos \left (b x + a\right )^{13} - 1105 \, \cos \left (b x + a\right )^{11}\right )}}{12155 \, b} \]

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x, algorithm="fricas")

[Out]

1024/12155*(715*cos(b*x + a)^17 - 2431*cos(b*x + a)^15 + 2805*cos(b*x + a)^13 - 1105*cos(b*x + a)^11)/b

Sympy [F(-1)]

Timed out. \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=\text {Timed out} \]

[In]

integrate(csc(b*x+a)**3*sin(2*b*x+2*a)**10,x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=\frac {715 \, \cos \left (17 \, b x + 17 \, a\right ) + 2431 \, \cos \left (15 \, b x + 15 \, a\right ) - 3740 \, \cos \left (13 \, b x + 13 \, a\right ) - 22100 \, \cos \left (11 \, b x + 11 \, a\right ) + 97240 \, \cos \left (7 \, b x + 7 \, a\right ) + 68068 \, \cos \left (5 \, b x + 5 \, a\right ) - 340340 \, \cos \left (3 \, b x + 3 \, a\right ) - 850850 \, \cos \left (b x + a\right )}{777920 \, b} \]

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x, algorithm="maxima")

[Out]

1/777920*(715*cos(17*b*x + 17*a) + 2431*cos(15*b*x + 15*a) - 3740*cos(13*b*x + 13*a) - 22100*cos(11*b*x + 11*a
) + 97240*cos(7*b*x + 7*a) + 68068*cos(5*b*x + 5*a) - 340340*cos(3*b*x + 3*a) - 850850*cos(b*x + a))/b

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 314 vs. \(2 (53) = 106\).

Time = 0.37 (sec) , antiderivative size = 314, normalized size of antiderivative = 5.15 \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=-\frac {32768 \, {\left (\frac {17 \, {\left (\cos \left (b x + a\right ) - 1\right )}}{\cos \left (b x + a\right ) + 1} - \frac {136 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{2}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{2}} + \frac {680 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{3}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{3}} + \frac {9775 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{4}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{4}} + \frac {71825 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{5}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{5}} + \frac {221000 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{6}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{6}} + \frac {486200 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{7}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{7}} + \frac {668525 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{8}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{8}} + \frac {692835 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{9}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{9}} + \frac {466752 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{10}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{10}} + \frac {233376 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{11}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{11}} + \frac {65637 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{12}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{12}} + \frac {12155 \, {\left (\cos \left (b x + a\right ) - 1\right )}^{13}}{{\left (\cos \left (b x + a\right ) + 1\right )}^{13}} - 1\right )}}{12155 \, b {\left (\frac {\cos \left (b x + a\right ) - 1}{\cos \left (b x + a\right ) + 1} - 1\right )}^{17}} \]

[In]

integrate(csc(b*x+a)^3*sin(2*b*x+2*a)^10,x, algorithm="giac")

[Out]

-32768/12155*(17*(cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 136*(cos(b*x + a) - 1)^2/(cos(b*x + a) + 1)^2 + 680*(
cos(b*x + a) - 1)^3/(cos(b*x + a) + 1)^3 + 9775*(cos(b*x + a) - 1)^4/(cos(b*x + a) + 1)^4 + 71825*(cos(b*x + a
) - 1)^5/(cos(b*x + a) + 1)^5 + 221000*(cos(b*x + a) - 1)^6/(cos(b*x + a) + 1)^6 + 486200*(cos(b*x + a) - 1)^7
/(cos(b*x + a) + 1)^7 + 668525*(cos(b*x + a) - 1)^8/(cos(b*x + a) + 1)^8 + 692835*(cos(b*x + a) - 1)^9/(cos(b*
x + a) + 1)^9 + 466752*(cos(b*x + a) - 1)^10/(cos(b*x + a) + 1)^10 + 233376*(cos(b*x + a) - 1)^11/(cos(b*x + a
) + 1)^11 + 65637*(cos(b*x + a) - 1)^12/(cos(b*x + a) + 1)^12 + 12155*(cos(b*x + a) - 1)^13/(cos(b*x + a) + 1)
^13 - 1)/(b*((cos(b*x + a) - 1)/(cos(b*x + a) + 1) - 1)^17)

Mupad [B] (verification not implemented)

Time = 19.63 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \csc ^3(a+b x) \sin ^{10}(2 a+2 b x) \, dx=-\frac {-\frac {1024\,{\cos \left (a+b\,x\right )}^{17}}{17}+\frac {1024\,{\cos \left (a+b\,x\right )}^{15}}{5}-\frac {3072\,{\cos \left (a+b\,x\right )}^{13}}{13}+\frac {1024\,{\cos \left (a+b\,x\right )}^{11}}{11}}{b} \]

[In]

int(sin(2*a + 2*b*x)^10/sin(a + b*x)^3,x)

[Out]

-((1024*cos(a + b*x)^11)/11 - (3072*cos(a + b*x)^13)/13 + (1024*cos(a + b*x)^15)/5 - (1024*cos(a + b*x)^17)/17
)/b