Integrand size = 20, antiderivative size = 46 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \sin ^9(a+b x)}{9 b}-\frac {64 \sin ^{11}(a+b x)}{11 b}+\frac {32 \sin ^{13}(a+b x)}{13 b} \] Output:
32/9*sin(b*x+a)^9/b-64/11*sin(b*x+a)^11/b+32/13*sin(b*x+a)^13/b
Time = 0.24 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4 (505+540 \cos (2 (a+b x))+99 \cos (4 (a+b x))) \sin ^9(a+b x)}{1287 b} \] Input:
Integrate[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]
Output:
(4*(505 + 540*Cos[2*(a + b*x)] + 99*Cos[4*(a + b*x)])*Sin[a + b*x]^9)/(128 7*b)
Time = 0.29 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {3042, 4776, 3042, 3044, 244, 2009}
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.
\(\displaystyle \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^3 \sin (2 a+2 b x)^5dx\) |
\(\Big \downarrow \) 4776 |
\(\displaystyle 32 \int \cos ^5(a+b x) \sin ^8(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 32 \int \cos (a+b x)^5 \sin (a+b x)^8dx\) |
\(\Big \downarrow \) 3044 |
\(\displaystyle \frac {32 \int \sin ^8(a+b x) \left (1-\sin ^2(a+b x)\right )^2d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle \frac {32 \int \left (\sin ^{12}(a+b x)-2 \sin ^{10}(a+b x)+\sin ^8(a+b x)\right )d\sin (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {32 \left (\frac {1}{13} \sin ^{13}(a+b x)-\frac {2}{11} \sin ^{11}(a+b x)+\frac {1}{9} \sin ^9(a+b x)\right )}{b}\) |
Input:
Int[Sin[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]
Output:
(32*(Sin[a + b*x]^9/9 - (2*Sin[a + b*x]^11)/11 + Sin[a + b*x]^13/13))/b
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[(c*x)^m*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, c, m}, x] && IGtQ[p , 0]
Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_ Symbol] :> Simp[1/(a*f) Subst[Int[x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a *Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] && !(I ntegerQ[(m - 1)/2] && LtQ[0, m, 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]
Leaf count of result is larger than twice the leaf count of optimal. \(96\) vs. \(2(40)=80\).
Time = 22.87 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.11
method | result | size |
default | \(\frac {5 \sin \left (b x +a \right )}{32 b}-\frac {25 \sin \left (3 b x +3 a \right )}{384 b}-\frac {\sin \left (5 b x +5 a \right )}{128 b}+\frac {\sin \left (7 b x +7 a \right )}{64 b}-\frac {\sin \left (9 b x +9 a \right )}{576 b}-\frac {3 \sin \left (11 b x +11 a \right )}{1408 b}+\frac {\sin \left (13 b x +13 a \right )}{1664 b}\) | \(97\) |
risch | \(\frac {5 \sin \left (b x +a \right )}{32 b}-\frac {25 \sin \left (3 b x +3 a \right )}{384 b}-\frac {\sin \left (5 b x +5 a \right )}{128 b}+\frac {\sin \left (7 b x +7 a \right )}{64 b}-\frac {\sin \left (9 b x +9 a \right )}{576 b}-\frac {3 \sin \left (11 b x +11 a \right )}{1408 b}+\frac {\sin \left (13 b x +13 a \right )}{1664 b}\) | \(97\) |
parallelrisch | \(\frac {\left (512 \tan \left (b x +a \right )^{7}+1216 \tan \left (b x +a \right )^{5}+512 \tan \left (b x +a \right )^{3}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{6}+\left (-3072 \tan \left (b x +a \right )^{8}-4992 \tan \left (b x +a \right )^{6}+4992 \tan \left (b x +a \right )^{4}+3072 \tan \left (b x +a \right )^{2}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{5}+\left (6144 \tan \left (b x +a \right )^{9}-3072 \tan \left (b x +a \right )^{7}-56256 \tan \left (b x +a \right )^{5}-3072 \tan \left (b x +a \right )^{3}+6144 \tan \left (b x +a \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}+\left (-4096 \tan \left (b x +a \right )^{10}+37888 \tan \left (b x +a \right )^{8}+164864 \tan \left (b x +a \right )^{6}-164864 \tan \left (b x +a \right )^{4}-37888 \tan \left (b x +a \right )^{2}+4096\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}+\left (-52224 \tan \left (b x +a \right )^{9}-181248 \tan \left (b x +a \right )^{7}-220224 \tan \left (b x +a \right )^{5}-181248 \tan \left (b x +a \right )^{3}-52224 \tan \left (b x +a \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}+\left (18432 \tan \left (b x +a \right )^{10}+52224 \tan \left (b x +a \right )^{8}+31872 \tan \left (b x +a \right )^{6}-31872 \tan \left (b x +a \right )^{4}-52224 \tan \left (b x +a \right )^{2}-18432\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+9216 \tan \left (b x +a \right )^{9}+36352 \tan \left (b x +a \right )^{7}+54080 \tan \left (b x +a \right )^{5}+36352 \tan \left (b x +a \right )^{3}+9216 \tan \left (b x +a \right )}{9009 b \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )^{3} \left (\tan \left (b x +a \right )^{2}+1\right )^{5}}\) | \(426\) |
orering | \(\text {Expression too large to display}\) | \(2262\) |
Input:
int(sin(b*x+a)^3*sin(2*b*x+2*a)^5,x,method=_RETURNVERBOSE)
Output:
5/32*sin(b*x+a)/b-25/384*sin(3*b*x+3*a)/b-1/128/b*sin(5*b*x+5*a)+1/64/b*si n(7*b*x+7*a)-1/576/b*sin(9*b*x+9*a)-3/1408/b*sin(11*b*x+11*a)+1/1664/b*sin (13*b*x+13*a)
Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.59 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{12} - 360 \, \cos \left (b x + a\right )^{10} + 458 \, \cos \left (b x + a\right )^{8} - 212 \, \cos \left (b x + a\right )^{6} + 3 \, \cos \left (b x + a\right )^{4} + 4 \, \cos \left (b x + a\right )^{2} + 8\right )} \sin \left (b x + a\right )}{1287 \, b} \] Input:
integrate(sin(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="fricas")
Output:
32/1287*(99*cos(b*x + a)^12 - 360*cos(b*x + a)^10 + 458*cos(b*x + a)^8 - 2 12*cos(b*x + a)^6 + 3*cos(b*x + a)^4 + 4*cos(b*x + a)^2 + 8)*sin(b*x + a)/ b
Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (39) = 78\).
Time = 26.38 (sec) , antiderivative size = 447, normalized size of antiderivative = 9.72 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\begin {cases} - \frac {1366 \sin ^{3}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {4960 \sin ^{3}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \sin ^{3}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {271 \sin ^{2}{\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{3003 b} - \frac {48 \sin ^{2}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{143 b} - \frac {640 \sin ^{2}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {1388 \sin {\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {2944 \sin {\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {512 \sin {\left (a + b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1001 b} + \frac {2234 \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{9009 b} + \frac {4544 \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{9009 b} + \frac {256 \sin {\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1001 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{5}{\left (2 a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(sin(b*x+a)**3*sin(2*b*x+2*a)**5,x)
Output:
Piecewise((-1366*sin(a + b*x)**3*sin(2*a + 2*b*x)**4*cos(2*a + 2*b*x)/(300 3*b) - 4960*sin(a + b*x)**3*sin(2*a + 2*b*x)**2*cos(2*a + 2*b*x)**3/(9009* b) - 256*sin(a + b*x)**3*cos(2*a + 2*b*x)**5/(1287*b) - 271*sin(a + b*x)** 2*sin(2*a + 2*b*x)**5*cos(a + b*x)/(3003*b) - 48*sin(a + b*x)**2*sin(2*a + 2*b*x)**3*cos(a + b*x)*cos(2*a + 2*b*x)**2/(143*b) - 640*sin(a + b*x)**2* sin(2*a + 2*b*x)*cos(a + b*x)*cos(2*a + 2*b*x)**4/(3003*b) - 1388*sin(a + b*x)*sin(2*a + 2*b*x)**4*cos(a + b*x)**2*cos(2*a + 2*b*x)/(3003*b) - 2944* sin(a + b*x)*sin(2*a + 2*b*x)**2*cos(a + b*x)**2*cos(2*a + 2*b*x)**3/(3003 *b) - 512*sin(a + b*x)*cos(a + b*x)**2*cos(2*a + 2*b*x)**5/(1001*b) + 2234 *sin(2*a + 2*b*x)**5*cos(a + b*x)**3/(9009*b) + 4544*sin(2*a + 2*b*x)**3*c os(a + b*x)**3*cos(2*a + 2*b*x)**2/(9009*b) + 256*sin(2*a + 2*b*x)*cos(a + b*x)**3*cos(2*a + 2*b*x)**4/(1001*b), Ne(b, 0)), (x*sin(a)**3*sin(2*a)**5 , True))
Time = 0.06 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {99 \, \sin \left (13 \, b x + 13 \, a\right ) - 351 \, \sin \left (11 \, b x + 11 \, a\right ) - 286 \, \sin \left (9 \, b x + 9 \, a\right ) + 2574 \, \sin \left (7 \, b x + 7 \, a\right ) - 1287 \, \sin \left (5 \, b x + 5 \, a\right ) - 10725 \, \sin \left (3 \, b x + 3 \, a\right ) + 25740 \, \sin \left (b x + a\right )}{164736 \, b} \] Input:
integrate(sin(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="maxima")
Output:
1/164736*(99*sin(13*b*x + 13*a) - 351*sin(11*b*x + 11*a) - 286*sin(9*b*x + 9*a) + 2574*sin(7*b*x + 7*a) - 1287*sin(5*b*x + 5*a) - 10725*sin(3*b*x + 3*a) + 25740*sin(b*x + a))/b
Time = 0.15 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32 \, {\left (99 \, \sin \left (b x + a\right )^{13} - 234 \, \sin \left (b x + a\right )^{11} + 143 \, \sin \left (b x + a\right )^{9}\right )}}{1287 \, b} \] Input:
integrate(sin(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="giac")
Output:
32/1287*(99*sin(b*x + a)^13 - 234*sin(b*x + a)^11 + 143*sin(b*x + a)^9)/b
Time = 18.05 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {32\,\left (99\,{\sin \left (a+b\,x\right )}^{13}-234\,{\sin \left (a+b\,x\right )}^{11}+143\,{\sin \left (a+b\,x\right )}^9\right )}{1287\,b} \] Input:
int(sin(a + b*x)^3*sin(2*a + 2*b*x)^5,x)
Output:
(32*(143*sin(a + b*x)^9 - 234*sin(a + b*x)^11 + 99*sin(a + b*x)^13))/(1287 *b)
Time = 0.19 (sec) , antiderivative size = 345, normalized size of antiderivative = 7.50 \[ \int \sin ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {-4096 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )-990 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{4} \sin \left (b x +a \right )^{3}+60 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{4} \sin \left (b x +a \right )-1760 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{3}+384 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )+2816 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{3}-4608 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )+297 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{5} \sin \left (b x +a \right )^{2}-6 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{5}+880 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )^{2}-64 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{3}-4224 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+2304 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )-4096 \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+2048 \sin \left (2 b x +2 a \right )}{9009 b} \] Input:
int(sin(b*x+a)^3*sin(2*b*x+2*a)^5,x)
Output:
( - 4096*cos(2*a + 2*b*x)*cos(a + b*x)*sin(a + b*x) - 990*cos(2*a + 2*b*x) *sin(2*a + 2*b*x)**4*sin(a + b*x)**3 + 60*cos(2*a + 2*b*x)*sin(2*a + 2*b*x )**4*sin(a + b*x) - 1760*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)**2*sin(a + b*x) **3 + 384*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)**2*sin(a + b*x) + 2816*cos(2*a + 2*b*x)*sin(a + b*x)**3 - 4608*cos(2*a + 2*b*x)*sin(a + b*x) + 297*cos(a + b*x)*sin(2*a + 2*b*x)**5*sin(a + b*x)**2 - 6*cos(a + b*x)*sin(2*a + 2*b *x)**5 + 880*cos(a + b*x)*sin(2*a + 2*b*x)**3*sin(a + b*x)**2 - 64*cos(a + b*x)*sin(2*a + 2*b*x)**3 - 4224*cos(a + b*x)*sin(2*a + 2*b*x)*sin(a + b*x )**2 + 2304*cos(a + b*x)*sin(2*a + 2*b*x) - 4096*sin(2*a + 2*b*x)*sin(a + b*x)**2 + 2048*sin(2*a + 2*b*x))/(9009*b)