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

Optimal. Leaf size=155 \[ -\frac{128 \sin ^5(a+b x) \cos ^9(a+b x)}{7 b}-\frac{160 \sin ^3(a+b x) \cos ^9(a+b x)}{21 b}-\frac{16 \sin (a+b x) \cos ^9(a+b x)}{7 b}+\frac{2 \sin (a+b x) \cos ^7(a+b x)}{7 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{12 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{5 x}{8} \]

[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)

________________________________________________________________________________________

Rubi [A]  time = 0.171443, antiderivative size = 155, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4288, 2568, 2635, 8} \[ -\frac{128 \sin ^5(a+b x) \cos ^9(a+b x)}{7 b}-\frac{160 \sin ^3(a+b x) \cos ^9(a+b x)}{21 b}-\frac{16 \sin (a+b x) \cos ^9(a+b x)}{7 b}+\frac{2 \sin (a+b x) \cos ^7(a+b x)}{7 b}+\frac{\sin (a+b x) \cos ^5(a+b x)}{3 b}+\frac{5 \sin (a+b x) \cos ^3(a+b x)}{12 b}+\frac{5 \sin (a+b x) \cos (a+b x)}{8 b}+\frac{5 x}{8} \]

Antiderivative was successfully verified.

[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 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 2568

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rubi steps

\begin{align*} \int \csc ^2(a+b x) \sin ^8(2 a+2 b x) \, dx &=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]  time = 0.248023, size = 85, normalized size = 0.55 \[ \frac{105 \sin (2 (a+b x))-315 \sin (4 (a+b x))-63 \sin (6 (a+b x))+63 \sin (8 (a+b x))+21 \sin (10 (a+b x))-7 \sin (12 (a+b x))-3 \sin (14 (a+b x))+840 a+840 b x}{1344 b} \]

Antiderivative was successfully verified.

[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]  time = 0.06, size = 111, normalized size = 0.7 \begin{align*} 256\,{\frac{1}{b} \left ( -1/14\, \left ( \sin \left ( bx+a \right ) \right ) ^{5} \left ( \cos \left ( bx+a \right ) \right ) ^{9}-{\frac{5\, \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{168}}-{\frac{\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{9}}{112}}+{\frac{\sin \left ( bx+a \right ) }{896} \left ( \left ( \cos \left ( bx+a \right ) \right ) ^{7}+7/6\, \left ( \cos \left ( bx+a \right ) \right ) ^{5}+{\frac{35\, \left ( \cos \left ( bx+a \right ) \right ) ^{3}}{24}}+{\frac{35\,\cos \left ( bx+a \right ) }{16}} \right ) }+{\frac{5\,bx}{2048}}+{\frac{5\,a}{2048}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

256/b*(-1/14*sin(b*x+a)^5*cos(b*x+a)^9-5/168*sin(b*x+a)^3*cos(b*x+a)^9-1/112*sin(b*x+a)*cos(b*x+a)^9+1/896*(co
s(b*x+a)^7+7/6*cos(b*x+a)^5+35/24*cos(b*x+a)^3+35/16*cos(b*x+a))*sin(b*x+a)+5/2048*b*x+5/2048*a)

________________________________________________________________________________________

Maxima [A]  time = 1.11489, size = 117, normalized size = 0.75 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

Fricas [A]  time = 0.545882, size = 244, normalized size = 1.57 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]  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)**2*sin(2*b*x+2*a)**8,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.89029, size = 128, normalized size = 0.83 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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