\(\int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx\) [45]

Optimal result

Integrand size = 20, antiderivative size = 155 \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {5 x}{8}+\frac {5 \cos (a+b x) \sin (a+b x)}{8 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{12 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac {2 \cos ^7(a+b x) \sin (a+b x)}{7 b}-\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b} \]

[Out]

5/8*x+5/8*cos(b*x+a)*sin(b*x+a)/b+5/12*cos(b*x+a)^3*sin(b*x+a)/b+1/3*cos(b*x+a)^5*sin(b*x+a)/b+2/7*cos(b*x+a)^
7*sin(b*x+a)/b-16/7*cos(b*x+a)^9*sin(b*x+a)/b-160/21*cos(b*x+a)^9*sin(b*x+a)^3/b-128/7*cos(b*x+a)^9*sin(b*x+a)
^5/b

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4373, 2648, 2715, 8} \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=-\frac {128 \sin ^5(a+b x) \cos ^9(a+b x)}{7 b}-\frac {160 \sin ^3(a+b x) \cos ^9(a+b x)}{21 b}-\frac {16 \sin (a+b x) \cos ^9(a+b x)}{7 b}+\frac {2 \sin (a+b x) \cos ^7(a+b x)}{7 b}+\frac {\sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac {5 \sin (a+b x) \cos ^3(a+b x)}{12 b}+\frac {5 \sin (a+b x) \cos (a+b x)}{8 b}+\frac {5 x}{8} \]

[In]

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

[Out]

(5*x)/8 + (5*Cos[a + b*x]*Sin[a + b*x])/(8*b) + (5*Cos[a + b*x]^3*Sin[a + b*x])/(12*b) + (Cos[a + b*x]^5*Sin[a
+ b*x])/(3*b) + (2*Cos[a + b*x]^7*Sin[a + b*x])/(7*b) - (16*Cos[a + b*x]^9*Sin[a + b*x])/(7*b) - (160*Cos[a +
b*x]^9*Sin[a + b*x]^3)/(21*b) - (128*Cos[a + b*x]^9*Sin[a + b*x]^5)/(7*b)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2648

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*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}& = 256 \int \cos ^8(a+b x) \sin ^6(a+b x) \, dx \\ & = -\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+\frac {640}{7} \int \cos ^8(a+b x) \sin ^4(a+b x) \, dx \\ & = -\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+\frac {160}{7} \int \cos ^8(a+b x) \sin ^2(a+b x) \, dx \\ & = -\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+\frac {16}{7} \int \cos ^8(a+b x) \, dx \\ & = \frac {2 \cos ^7(a+b x) \sin (a+b x)}{7 b}-\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+2 \int \cos ^6(a+b x) \, dx \\ & = \frac {\cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac {2 \cos ^7(a+b x) \sin (a+b x)}{7 b}-\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+\frac {5}{3} \int \cos ^4(a+b x) \, dx \\ & = \frac {5 \cos ^3(a+b x) \sin (a+b x)}{12 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac {2 \cos ^7(a+b x) \sin (a+b x)}{7 b}-\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+\frac {5}{4} \int \cos ^2(a+b x) \, dx \\ & = \frac {5 \cos (a+b x) \sin (a+b x)}{8 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{12 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac {2 \cos ^7(a+b x) \sin (a+b x)}{7 b}-\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b}+\frac {5 \int 1 \, dx}{8} \\ & = \frac {5 x}{8}+\frac {5 \cos (a+b x) \sin (a+b x)}{8 b}+\frac {5 \cos ^3(a+b x) \sin (a+b x)}{12 b}+\frac {\cos ^5(a+b x) \sin (a+b x)}{3 b}+\frac {2 \cos ^7(a+b x) \sin (a+b x)}{7 b}-\frac {16 \cos ^9(a+b x) \sin (a+b x)}{7 b}-\frac {160 \cos ^9(a+b x) \sin ^3(a+b x)}{21 b}-\frac {128 \cos ^9(a+b x) \sin ^5(a+b x)}{7 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.93 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.55 \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {840 a+840 b x+105 \sin (2 (a+b x))-315 \sin (4 (a+b x))-63 \sin (6 (a+b x))+63 \sin (8 (a+b x))+21 \sin (10 (a+b x))-7 \sin (12 (a+b x))-3 \sin (14 (a+b x))}{1344 b} \]

[In]

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

[Out]

(840*a + 840*b*x + 105*Sin[2*(a + b*x)] - 315*Sin[4*(a + b*x)] - 63*Sin[6*(a + b*x)] + 63*Sin[8*(a + b*x)] + 2
1*Sin[10*(a + b*x)] - 7*Sin[12*(a + b*x)] - 3*Sin[14*(a + b*x)])/(1344*b)

Maple [A] (verified)

Time = 46.86 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.66

method	result	size
risch	\(\frac {5 x}{8}-\frac {\sin \left (14 x b +14 a \right )}{448 b}-\frac {\sin \left (12 x b +12 a \right )}{192 b}+\frac {\sin \left (10 x b +10 a \right )}{64 b}+\frac {3 \sin \left (8 x b +8 a \right )}{64 b}-\frac {3 \sin \left (6 x b +6 a \right )}{64 b}-\frac {15 \sin \left (4 x b +4 a \right )}{64 b}+\frac {5 \sin \left (2 x b +2 a \right )}{64 b}\)	\(103\)
default	\(\frac {-\frac {128 \sin \left (x b +a \right )^{5} \cos \left (x b +a \right )^{9}}{7}-\frac {160 \sin \left (x b +a \right )^{3} \cos \left (x b +a \right )^{9}}{21}-\frac {16 \sin \left (x b +a \right ) \cos \left (x b +a \right )^{9}}{7}+\frac {2 \left (\cos \left (x b +a \right )^{7}+\frac {7 \cos \left (x b +a \right )^{5}}{6}+\frac {35 \cos \left (x b +a \right )^{3}}{24}+\frac {35 \cos \left (x b +a \right )}{16}\right ) \sin \left (x b +a \right )}{7}+\frac {5 x b}{8}+\frac {5 a}{8}}{b}\)	\(111\)

[In]

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

[Out]

5/8*x-1/448/b*sin(14*b*x+14*a)-1/192/b*sin(12*b*x+12*a)+1/64/b*sin(10*b*x+10*a)+3/64/b*sin(8*b*x+8*a)-3/64/b*s
in(6*b*x+6*a)-15/64/b*sin(4*b*x+4*a)+5/64*sin(2*b*x+2*a)/b

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.56 \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {105 \, b x - {\left (3072 \, \cos \left (b x + a\right )^{13} - 7424 \, \cos \left (b x + a\right )^{11} + 4736 \, \cos \left (b x + a\right )^{9} - 48 \, \cos \left (b x + a\right )^{7} - 56 \, \cos \left (b x + a\right )^{5} - 70 \, \cos \left (b x + a\right )^{3} - 105 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{168 \, b} \]

[In]

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

[Out]

1/168*(105*b*x - (3072*cos(b*x + a)^13 - 7424*cos(b*x + a)^11 + 4736*cos(b*x + a)^9 - 48*cos(b*x + a)^7 - 56*c
os(b*x + a)^5 - 70*cos(b*x + a)^3 - 105*cos(b*x + a))*sin(b*x + a))/b

Sympy [F(-1)]

Timed out. \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.56 \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {840 \, b x - 3 \, \sin \left (14 \, b x + 14 \, a\right ) - 7 \, \sin \left (12 \, b x + 12 \, a\right ) + 21 \, \sin \left (10 \, b x + 10 \, a\right ) + 63 \, \sin \left (8 \, b x + 8 \, a\right ) - 63 \, \sin \left (6 \, b x + 6 \, a\right ) - 315 \, \sin \left (4 \, b x + 4 \, a\right ) + 105 \, \sin \left (2 \, b x + 2 \, a\right )}{1344 \, b} \]

[In]

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

[Out]

1/1344*(840*b*x - 3*sin(14*b*x + 14*a) - 7*sin(12*b*x + 12*a) + 21*sin(10*b*x + 10*a) + 63*sin(8*b*x + 8*a) -
63*sin(6*b*x + 6*a) - 315*sin(4*b*x + 4*a) + 105*sin(2*b*x + 2*a))/b

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.61 \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {105 \, b x + 105 \, a + \frac {105 \, \tan \left (b x + a\right )^{13} + 700 \, \tan \left (b x + a\right )^{11} + 1981 \, \tan \left (b x + a\right )^{9} + 3072 \, \tan \left (b x + a\right )^{7} - 1981 \, \tan \left (b x + a\right )^{5} - 700 \, \tan \left (b x + a\right )^{3} - 105 \, \tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{7}}}{168 \, b} \]

[In]

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

[Out]

1/168*(105*b*x + 105*a + (105*tan(b*x + a)^13 + 700*tan(b*x + a)^11 + 1981*tan(b*x + a)^9 + 3072*tan(b*x + a)^
7 - 1981*tan(b*x + a)^5 - 700*tan(b*x + a)^3 - 105*tan(b*x + a))/(tan(b*x + a)^2 + 1)^7)/b

Mupad [B] (verification not implemented)

Time = 21.91 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.96 \[ \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx=\frac {5\,x}{8}+\frac {\frac {5\,{\mathrm {tan}\left (a+b\,x\right )}^{13}}{8}+\frac {25\,{\mathrm {tan}\left (a+b\,x\right )}^{11}}{6}+\frac {283\,{\mathrm {tan}\left (a+b\,x\right )}^9}{24}+\frac {128\,{\mathrm {tan}\left (a+b\,x\right )}^7}{7}-\frac {283\,{\mathrm {tan}\left (a+b\,x\right )}^5}{24}-\frac {25\,{\mathrm {tan}\left (a+b\,x\right )}^3}{6}-\frac {5\,\mathrm {tan}\left (a+b\,x\right )}{8}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^{14}+7\,{\mathrm {tan}\left (a+b\,x\right )}^{12}+21\,{\mathrm {tan}\left (a+b\,x\right )}^{10}+35\,{\mathrm {tan}\left (a+b\,x\right )}^8+35\,{\mathrm {tan}\left (a+b\,x\right )}^6+21\,{\mathrm {tan}\left (a+b\,x\right )}^4+7\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \]

[In]

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

[Out]

(5*x)/8 + ((128*tan(a + b*x)^7)/7 - (25*tan(a + b*x)^3)/6 - (283*tan(a + b*x)^5)/24 - (5*tan(a + b*x))/8 + (28
3*tan(a + b*x)^9)/24 + (25*tan(a + b*x)^11)/6 + (5*tan(a + b*x)^13)/8)/(b*(7*tan(a + b*x)^2 + 21*tan(a + b*x)^
4 + 35*tan(a + b*x)^6 + 35*tan(a + b*x)^8 + 21*tan(a + b*x)^10 + 7*tan(a + b*x)^12 + tan(a + b*x)^14 + 1))