Integrand size = 20, antiderivative size = 46 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \cos ^9(a+b x)}{9 b}+\frac {64 \cos ^{11}(a+b x)}{11 b}-\frac {32 \cos ^{13}(a+b x)}{13 b} \] Output:
-32/9*cos(b*x+a)^9/b+64/11*cos(b*x+a)^11/b-32/13*cos(b*x+a)^13/b
Time = 0.27 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4 \cos ^9(a+b x) (-505+540 \cos (2 (a+b x))-99 \cos (4 (a+b x)))}{1287 b} \] Input:
Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]
Output:
(4*Cos[a + b*x]^9*(-505 + 540*Cos[2*(a + b*x)] - 99*Cos[4*(a + b*x)]))/(12 87*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, 4775, 3042, 3045, 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.
\(\displaystyle \int \sin ^5(2 a+2 b x) \cos ^3(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (2 a+2 b x)^5 \cos (a+b x)^3dx\) |
\(\Big \downarrow \) 4775 |
\(\displaystyle 32 \int \cos ^8(a+b x) \sin ^5(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 32 \int \cos (a+b x)^8 \sin (a+b x)^5dx\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle -\frac {32 \int \cos ^8(a+b x) \left (1-\cos ^2(a+b x)\right )^2d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {32 \int \left (\cos ^{12}(a+b x)-2 \cos ^{10}(a+b x)+\cos ^8(a+b x)\right )d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {32 \left (\frac {1}{13} \cos ^{13}(a+b x)-\frac {2}{11} \cos ^{11}(a+b x)+\frac {1}{9} \cos ^9(a+b x)\right )}{b}\) |
Input:
Int[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]
Output:
(-32*(Cos[a + b*x]^9/9 - (2*Cos[a + b*x]^11)/11 + Cos[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_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_ Symbol] :> Simp[-(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])
Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_ Symbol] :> Simp[2^p/e^p Int[(e*Cos[a + b*x])^(m + p)*Sin[a + b*x]^p, x], x] /; FreeQ[{a, b, c, d, e, m}, 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 = 23.77 (sec) , antiderivative size = 97, normalized size of antiderivative = 2.11
method | result | size |
default | \(-\frac {5 \cos \left (b x +a \right )}{32 b}-\frac {25 \cos \left (3 b x +3 a \right )}{384 b}+\frac {\cos \left (5 b x +5 a \right )}{128 b}+\frac {\cos \left (7 b x +7 a \right )}{64 b}+\frac {\cos \left (9 b x +9 a \right )}{576 b}-\frac {3 \cos \left (11 b x +11 a \right )}{1408 b}-\frac {\cos \left (13 b x +13 a \right )}{1664 b}\) | \(97\) |
risch | \(-\frac {5 \cos \left (b x +a \right )}{32 b}-\frac {25 \cos \left (3 b x +3 a \right )}{384 b}+\frac {\cos \left (5 b x +5 a \right )}{128 b}+\frac {\cos \left (7 b x +7 a \right )}{64 b}+\frac {\cos \left (9 b x +9 a \right )}{576 b}-\frac {3 \cos \left (11 b x +11 a \right )}{1408 b}-\frac {\cos \left (13 b x +13 a \right )}{1664 b}\) | \(97\) |
parallelrisch | \(\frac {\left (1632 \tan \left (b x +a \right )^{6}+53408 \tan \left (b x +a \right )^{4}+27520 \tan \left (b x +a \right )^{2}+5504\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{6}+\left (-13056 \tan \left (b x +a \right )^{7}-44736 \tan \left (b x +a \right )^{5}-13056 \tan \left (b x +a \right )^{3}\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{5}+\left (39168 \tan \left (b x +a \right )^{8}+137184 \tan \left (b x +a \right )^{6}+27936 \tan \left (b x +a \right )^{4}+43392 \tan \left (b x +a \right )^{2}+16512\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{4}+\left (-52224 \tan \left (b x +a \right )^{9}-178688 \tan \left (b x +a \right )^{7}-214144 \tan \left (b x +a \right )^{5}-178688 \tan \left (b x +a \right )^{3}-52224 \tan \left (b x +a \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{3}+\left (26112 \tan \left (b x +a \right )^{10}+72192 \tan \left (b x +a \right )^{8}+47136 \tan \left (b x +a \right )^{6}+117984 \tan \left (b x +a \right )^{4}+10368 \tan \left (b x +a \right )^{2}-9600\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}+\left (15360 \tan \left (b x +a \right )^{9}+48384 \tan \left (b x +a \right )^{7}+47424 \tan \left (b x +a \right )^{5}+48384 \tan \left (b x +a \right )^{3}+15360 \tan \left (b x +a \right )\right ) \tan \left (\frac {a}{2}+\frac {b x}{2}\right )+3584 \tan \left (b x +a \right )^{10}+21760 \tan \left (b x +a \right )^{8}+49568 \tan \left (b x +a \right )^{6}+5472 \tan \left (b x +a \right )^{4}+5760 \tan \left (b x +a \right )^{2}+1920}{9009 b \left (1+\tan \left (\frac {a}{2}+\frac {b x}{2}\right )^{2}\right )^{3} \left (\tan \left (b x +a \right )^{2}+1\right )^{5}}\) | \(410\) |
orering | \(\text {Expression too large to display}\) | \(2262\) |
Input:
int(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x,method=_RETURNVERBOSE)
Output:
-5/32*cos(b*x+a)/b-25/384*cos(3*b*x+3*a)/b+1/128*cos(5*b*x+5*a)/b+1/64*cos (7*b*x+7*a)/b+1/576*cos(9*b*x+9*a)/b-3/1408*cos(11*b*x+11*a)/b-1/1664*cos( 13*b*x+13*a)/b
Time = 0.09 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, b} \] Input:
integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="fricas")
Output:
-32/1287*(99*cos(b*x + a)^13 - 234*cos(b*x + a)^11 + 143*cos(b*x + a)^9)/b
Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (39) = 78\).
Time = 25.34 (sec) , antiderivative size = 447, normalized size of antiderivative = 9.72 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\begin {cases} - \frac {2234 \sin ^{3}{\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {4544 \sin ^{3}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1001 b} - \frac {1388 \sin ^{2}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {2944 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {512 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1001 b} + \frac {271 \sin {\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{3003 b} + \frac {48 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{143 b} + \frac {640 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {1366 \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {4960 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \cos ^{3}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1287 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (2 a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**3*sin(2*b*x+2*a)**5,x)
Output:
Piecewise((-2234*sin(a + b*x)**3*sin(2*a + 2*b*x)**5/(9009*b) - 4544*sin(a + b*x)**3*sin(2*a + 2*b*x)**3*cos(2*a + 2*b*x)**2/(9009*b) - 256*sin(a + b*x)**3*sin(2*a + 2*b*x)*cos(2*a + 2*b*x)**4/(1001*b) - 1388*sin(a + b*x)* *2*sin(2*a + 2*b*x)**4*cos(a + b*x)*cos(2*a + 2*b*x)/(3003*b) - 2944*sin(a + b*x)**2*sin(2*a + 2*b*x)**2*cos(a + b*x)*cos(2*a + 2*b*x)**3/(3003*b) - 512*sin(a + b*x)**2*cos(a + b*x)*cos(2*a + 2*b*x)**5/(1001*b) + 271*sin(a + b*x)*sin(2*a + 2*b*x)**5*cos(a + b*x)**2/(3003*b) + 48*sin(a + b*x)*sin (2*a + 2*b*x)**3*cos(a + b*x)**2*cos(2*a + 2*b*x)**2/(143*b) + 640*sin(a + b*x)*sin(2*a + 2*b*x)*cos(a + b*x)**2*cos(2*a + 2*b*x)**4/(3003*b) - 1366 *sin(2*a + 2*b*x)**4*cos(a + b*x)**3*cos(2*a + 2*b*x)/(3003*b) - 4960*sin( 2*a + 2*b*x)**2*cos(a + b*x)**3*cos(2*a + 2*b*x)**3/(9009*b) - 256*cos(a + b*x)**3*cos(2*a + 2*b*x)**5/(1287*b), Ne(b, 0)), (x*sin(2*a)**5*cos(a)**3 , True))
Time = 0.04 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {99 \, \cos \left (13 \, b x + 13 \, a\right ) + 351 \, \cos \left (11 \, b x + 11 \, a\right ) - 286 \, \cos \left (9 \, b x + 9 \, a\right ) - 2574 \, \cos \left (7 \, b x + 7 \, a\right ) - 1287 \, \cos \left (5 \, b x + 5 \, a\right ) + 10725 \, \cos \left (3 \, b x + 3 \, a\right ) + 25740 \, \cos \left (b x + a\right )}{164736 \, b} \] Input:
integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="maxima")
Output:
-1/164736*(99*cos(13*b*x + 13*a) + 351*cos(11*b*x + 11*a) - 286*cos(9*b*x + 9*a) - 2574*cos(7*b*x + 7*a) - 1287*cos(5*b*x + 5*a) + 10725*cos(3*b*x + 3*a) + 25740*cos(b*x + a))/b
Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, b} \] Input:
integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="giac")
Output:
-32/1287*(99*cos(b*x + a)^13 - 234*cos(b*x + a)^11 + 143*cos(b*x + a)^9)/b
Time = 19.42 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32\,\left (99\,{\cos \left (a+b\,x\right )}^{13}-234\,{\cos \left (a+b\,x\right )}^{11}+143\,{\cos \left (a+b\,x\right )}^9\right )}{1287\,b} \] Input:
int(cos(a + b*x)^3*sin(2*a + 2*b*x)^5,x)
Output:
-(32*(143*cos(a + b*x)^9 - 234*cos(a + b*x)^11 + 99*cos(a + b*x)^13))/(128 7*b)
Time = 0.18 (sec) , antiderivative size = 346, normalized size of antiderivative = 7.52 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4950 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{4} \sin \left (b x +a \right )^{2}-4650 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{4}+8800 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{2}-6880 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2}-14080 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}-8960 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right )-9600 \cos \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{2}+4800 \cos \left (2 b x +2 a \right )+9600 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )+1485 \sin \left (2 b x +2 a \right )^{5} \sin \left (b x +a \right )^{3}-1455 \sin \left (2 b x +2 a \right )^{5} \sin \left (b x +a \right )+4400 \sin \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )^{3}-4080 \sin \left (2 b x +2 a \right )^{3} \sin \left (b x +a \right )-21120 \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{3}+9600 \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )-12944}{45045 b} \] Input:
int(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x)
Output:
(4950*cos(2*a + 2*b*x)*cos(a + b*x)*sin(2*a + 2*b*x)**4*sin(a + b*x)**2 - 4650*cos(2*a + 2*b*x)*cos(a + b*x)*sin(2*a + 2*b*x)**4 + 8800*cos(2*a + 2* b*x)*cos(a + b*x)*sin(2*a + 2*b*x)**2*sin(a + b*x)**2 - 6880*cos(2*a + 2*b *x)*cos(a + b*x)*sin(2*a + 2*b*x)**2 - 14080*cos(2*a + 2*b*x)*cos(a + b*x) *sin(a + b*x)**2 - 8960*cos(2*a + 2*b*x)*cos(a + b*x) - 9600*cos(2*a + 2*b *x)*sin(a + b*x)**2 + 4800*cos(2*a + 2*b*x) + 9600*cos(a + b*x)*sin(2*a + 2*b*x)*sin(a + b*x) + 1485*sin(2*a + 2*b*x)**5*sin(a + b*x)**3 - 1455*sin( 2*a + 2*b*x)**5*sin(a + b*x) + 4400*sin(2*a + 2*b*x)**3*sin(a + b*x)**3 - 4080*sin(2*a + 2*b*x)**3*sin(a + b*x) - 21120*sin(2*a + 2*b*x)*sin(a + b*x )**3 + 9600*sin(2*a + 2*b*x)*sin(a + b*x) - 12944)/(45045*b)