Integrand size = 20, antiderivative size = 111 \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {3 x}{4}+\frac {3 \cos (a+b x) \sin (a+b x)}{4 b}+\frac {\cos ^3(a+b x) \sin (a+b x)}{2 b}+\frac {2 \cos ^5(a+b x) \sin (a+b x)}{5 b}-\frac {12 \cos ^7(a+b x) \sin (a+b x)}{5 b}-\frac {32 \cos ^7(a+b x) \sin ^3(a+b x)}{5 b} \] Output:
3/4*x+3/4*cos(b*x+a)*sin(b*x+a)/b+1/2*cos(b*x+a)^3*sin(b*x+a)/b+2/5*cos(b* x+a)^5*sin(b*x+a)/b-12/5*cos(b*x+a)^7*sin(b*x+a)/b-32/5*cos(b*x+a)^7*sin(b *x+a)^3/b
Time = 0.13 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.56 \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {120 b x+20 \sin (2 (a+b x))-40 \sin (4 (a+b x))-10 \sin (6 (a+b x))+5 \sin (8 (a+b x))+2 \sin (10 (a+b x))}{160 b} \] Input:
Integrate[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^6,x]
Output:
(120*b*x + 20*Sin[2*(a + b*x)] - 40*Sin[4*(a + b*x)] - 10*Sin[6*(a + b*x)] + 5*Sin[8*(a + b*x)] + 2*Sin[10*(a + b*x)])/(160*b)
Time = 0.60 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.20, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.650, Rules used = {3042, 4776, 3042, 3048, 3042, 3048, 3042, 3115, 3042, 3115, 3042, 3115, 24}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \sin ^6(2 a+2 b x) \csc ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\sin (2 a+2 b x)^6}{\sin (a+b x)^2}dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle 64 \int \cos ^6(a+b x) \sin ^4(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 64 \int \cos (a+b x)^6 \sin (a+b x)^4dx\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle 64 \left (\frac {3}{10} \int \cos ^6(a+b x) \sin ^2(a+b x)dx-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 64 \left (\frac {3}{10} \int \cos (a+b x)^6 \sin (a+b x)^2dx-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3048 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \int \cos ^6(a+b x)dx-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \int \sin \left (a+b x+\frac {\pi }{2}\right )^6dx-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \cos ^4(a+b x)dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \int \sin \left (a+b x+\frac {\pi }{2}\right )^4dx+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \cos ^2(a+b x)dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \int \sin \left (a+b x+\frac {\pi }{2}\right )^2dx+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 3115 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {5}{6} \left (\frac {3}{4} \left (\frac {\int 1dx}{2}+\frac {\sin (a+b x) \cos (a+b x)}{2 b}\right )+\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}\right )+\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
\(\Big \downarrow \) 24 |
\(\displaystyle 64 \left (\frac {3}{10} \left (\frac {1}{8} \left (\frac {\sin (a+b x) \cos ^5(a+b x)}{6 b}+\frac {5}{6} \left (\frac {\sin (a+b x) \cos ^3(a+b x)}{4 b}+\frac {3}{4} \left (\frac {\sin (a+b x) \cos (a+b x)}{2 b}+\frac {x}{2}\right )\right )\right )-\frac {\sin (a+b x) \cos ^7(a+b x)}{8 b}\right )-\frac {\sin ^3(a+b x) \cos ^7(a+b x)}{10 b}\right )\) |
Input:
Int[Csc[a + b*x]^2*Sin[2*a + 2*b*x]^6,x]
Output:
64*(-1/10*(Cos[a + b*x]^7*Sin[a + b*x]^3)/b + (3*(-1/8*(Cos[a + b*x]^7*Sin [a + b*x])/b + ((Cos[a + b*x]^5*Sin[a + b*x])/(6*b) + (5*((Cos[a + b*x]^3* Sin[a + b*x])/(4*b) + (3*(x/2 + (Cos[a + b*x]*Sin[a + b*x])/(2*b)))/4))/6) /8))/10)
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] + Simp[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]
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] + Simp[b^2*((n - 1)/n) Int[(b*Sin [c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && IntegerQ[ 2*n]
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_ Symbol] :> Simp[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] && I ntegerQ[p]
Time = 10.25 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68
method | result | size |
risch | \(\frac {3 x}{4}+\frac {\sin \left (10 b x +10 a \right )}{80 b}+\frac {\sin \left (8 b x +8 a \right )}{32 b}-\frac {\sin \left (6 b x +6 a \right )}{16 b}-\frac {\sin \left (4 b x +4 a \right )}{4 b}+\frac {\sin \left (2 b x +2 a \right )}{8 b}\) | \(75\) |
default | \(\frac {-\frac {32 \sin \left (b x +a \right )^{3} \cos \left (b x +a \right )^{7}}{5}-\frac {12 \sin \left (b x +a \right ) \cos \left (b x +a \right )^{7}}{5}+\frac {2 \left (\cos \left (b x +a \right )^{5}+\frac {5 \cos \left (b x +a \right )^{3}}{4}+\frac {15 \cos \left (b x +a \right )}{8}\right ) \sin \left (b x +a \right )}{5}+\frac {3 b x}{4}+\frac {3 a}{4}}{b}\) | \(83\) |
Input:
int(csc(b*x+a)^2*sin(2*b*x+2*a)^6,x,method=_RETURNVERBOSE)
Output:
3/4*x+1/80/b*sin(10*b*x+10*a)+1/32/b*sin(8*b*x+8*a)-1/16/b*sin(6*b*x+6*a)- 1/4/b*sin(4*b*x+4*a)+1/8*sin(2*b*x+2*a)/b
Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.59 \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {15 \, b x + {\left (128 \, \cos \left (b x + a\right )^{9} - 176 \, \cos \left (b x + a\right )^{7} + 8 \, \cos \left (b x + a\right )^{5} + 10 \, \cos \left (b x + a\right )^{3} + 15 \, \cos \left (b x + a\right )\right )} \sin \left (b x + a\right )}{20 \, b} \] Input:
integrate(csc(b*x+a)^2*sin(2*b*x+2*a)^6,x, algorithm="fricas")
Output:
1/20*(15*b*x + (128*cos(b*x + a)^9 - 176*cos(b*x + a)^7 + 8*cos(b*x + a)^5 + 10*cos(b*x + a)^3 + 15*cos(b*x + a))*sin(b*x + a))/b
Timed out. \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\text {Timed out} \] Input:
integrate(csc(b*x+a)**2*sin(2*b*x+2*a)**6,x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.59 \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {120 \, b x + 2 \, \sin \left (10 \, b x + 10 \, a\right ) + 5 \, \sin \left (8 \, b x + 8 \, a\right ) - 10 \, \sin \left (6 \, b x + 6 \, a\right ) - 40 \, \sin \left (4 \, b x + 4 \, a\right ) + 20 \, \sin \left (2 \, b x + 2 \, a\right )}{160 \, b} \] Input:
integrate(csc(b*x+a)^2*sin(2*b*x+2*a)^6,x, algorithm="maxima")
Output:
1/160*(120*b*x + 2*sin(10*b*x + 10*a) + 5*sin(8*b*x + 8*a) - 10*sin(6*b*x + 6*a) - 40*sin(4*b*x + 4*a) + 20*sin(2*b*x + 2*a))/b
Time = 0.16 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.68 \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {15 \, b x + 15 \, a + \frac {15 \, \tan \left (b x + a\right )^{9} + 70 \, \tan \left (b x + a\right )^{7} + 128 \, \tan \left (b x + a\right )^{5} - 70 \, \tan \left (b x + a\right )^{3} - 15 \, \tan \left (b x + a\right )}{{\left (\tan \left (b x + a\right )^{2} + 1\right )}^{5}}}{20 \, b} \] Input:
integrate(csc(b*x+a)^2*sin(2*b*x+2*a)^6,x, algorithm="giac")
Output:
1/20*(15*b*x + 15*a + (15*tan(b*x + a)^9 + 70*tan(b*x + a)^7 + 128*tan(b*x + a)^5 - 70*tan(b*x + a)^3 - 15*tan(b*x + a))/(tan(b*x + a)^2 + 1)^5)/b
Time = 20.17 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.98 \[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\frac {3\,x}{4}+\frac {\frac {3\,{\mathrm {tan}\left (a+b\,x\right )}^9}{4}+\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^7}{2}+\frac {32\,{\mathrm {tan}\left (a+b\,x\right )}^5}{5}-\frac {7\,{\mathrm {tan}\left (a+b\,x\right )}^3}{2}-\frac {3\,\mathrm {tan}\left (a+b\,x\right )}{4}}{b\,\left ({\mathrm {tan}\left (a+b\,x\right )}^{10}+5\,{\mathrm {tan}\left (a+b\,x\right )}^8+10\,{\mathrm {tan}\left (a+b\,x\right )}^6+10\,{\mathrm {tan}\left (a+b\,x\right )}^4+5\,{\mathrm {tan}\left (a+b\,x\right )}^2+1\right )} \] Input:
int(sin(2*a + 2*b*x)^6/sin(a + b*x)^2,x)
Output:
(3*x)/4 + ((32*tan(a + b*x)^5)/5 - (7*tan(a + b*x)^3)/2 - (3*tan(a + b*x)) /4 + (7*tan(a + b*x)^7)/2 + (3*tan(a + b*x)^9)/4)/(b*(5*tan(a + b*x)^2 + 1 0*tan(a + b*x)^4 + 10*tan(a + b*x)^6 + 5*tan(a + b*x)^8 + tan(a + b*x)^10 + 1))
\[ \int \csc ^2(a+b x) \sin ^6(2 a+2 b x) \, dx=\int \csc \left (b x +a \right )^{2} \sin \left (2 b x +2 a \right )^{6}d x \] Input:
int(csc(b*x+a)^2*sin(2*b*x+2*a)^6,x)
Output:
int(csc(a + b*x)**2*sin(2*a + 2*b*x)**6,x)