Integrand size = 28, antiderivative size = 46 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=-\frac {4 \cos ^5(a+b x)}{5 b}+\frac {8 \cos ^7(a+b x)}{7 b}-\frac {4 \cos ^9(a+b x)}{9 b} \] Output:
-4/5*cos(b*x+a)^5/b+8/7*cos(b*x+a)^7/b-4/9*cos(b*x+a)^9/b
Time = 0.11 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {\cos ^5(a+b x) (-249+220 \cos (2 (a+b x))-35 \cos (4 (a+b x)))}{630 b} \] Input:
Integrate[Cos[a + b*x]^2*Sin[a + b*x]^3*Sin[2*a + 2*b*x]^2,x]
Output:
(Cos[a + b*x]^5*(-249 + 220*Cos[2*(a + b*x)] - 35*Cos[4*(a + b*x)]))/(630* b)
Time = 0.32 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.91, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3042, 4800, 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.
\(\displaystyle \int \sin ^3(a+b x) \sin ^2(2 a+2 b x) \cos ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \sin (a+b x)^3 \sin (2 a+2 b x)^2 \cos (a+b x)^2dx\) |
\(\Big \downarrow \) 4800 |
\(\displaystyle 4 \int \cos ^4(a+b x) \sin ^5(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle 4 \int \cos (a+b x)^4 \sin (a+b x)^5dx\) |
\(\Big \downarrow \) 3045 |
\(\displaystyle -\frac {4 \int \cos ^4(a+b x) \left (1-\cos ^2(a+b x)\right )^2d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 244 |
\(\displaystyle -\frac {4 \int \left (\cos ^8(a+b x)-2 \cos ^6(a+b x)+\cos ^4(a+b x)\right )d\cos (a+b x)}{b}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {4 \left (\frac {1}{9} \cos ^9(a+b x)-\frac {2}{7} \cos ^7(a+b x)+\frac {1}{5} \cos ^5(a+b x)\right )}{b}\) |
Input:
Int[Cos[a + b*x]^2*Sin[a + b*x]^3*Sin[2*a + 2*b*x]^2,x]
Output:
(-4*(Cos[a + b*x]^5/5 - (2*Cos[a + b*x]^7)/7 + Cos[a + b*x]^9/9))/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_.)*((f_.)*sin[(a_.) + (b_.)*(x_)])^( n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Simp[2^p/(e^p*f^p) Int[( e*Cos[a + b*x])^(m + p)*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, e, f, m, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2] && IntegerQ[p]
Time = 10.09 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.30
method | result | size |
parallelrisch | \(\frac {-2048+45 \cos \left (7 b x +7 a \right )-35 \cos \left (9 b x +9 a \right )-420 \cos \left (3 b x +3 a \right )-1890 \cos \left (b x +a \right )+252 \cos \left (5 b x +5 a \right )}{20160 b}\) | \(60\) |
default | \(-\frac {3 \cos \left (b x +a \right )}{32 b}-\frac {\cos \left (3 b x +3 a \right )}{48 b}+\frac {\cos \left (5 b x +5 a \right )}{80 b}+\frac {\cos \left (7 b x +7 a \right )}{448 b}-\frac {\cos \left (9 b x +9 a \right )}{576 b}\) | \(69\) |
risch | \(-\frac {3 \cos \left (b x +a \right )}{32 b}-\frac {\cos \left (3 b x +3 a \right )}{48 b}+\frac {\cos \left (5 b x +5 a \right )}{80 b}+\frac {\cos \left (7 b x +7 a \right )}{448 b}-\frac {\cos \left (9 b x +9 a \right )}{576 b}\) | \(69\) |
orering | \(\text {Expression too large to display}\) | \(1219\) |
Input:
int(cos(b*x+a)^2*sin(b*x+a)^3*sin(2*b*x+2*a)^2,x,method=_RETURNVERBOSE)
Output:
1/20160*(-2048+45*cos(7*b*x+7*a)-35*cos(9*b*x+9*a)-420*cos(3*b*x+3*a)-1890 *cos(b*x+a)+252*cos(5*b*x+5*a))/b
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=-\frac {4 \, {\left (35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}\right )}}{315 \, b} \] Input:
integrate(cos(b*x+a)^2*sin(b*x+a)^3*sin(2*b*x+2*a)^2,x, algorithm="fricas" )
Output:
-4/315*(35*cos(b*x + a)^9 - 90*cos(b*x + a)^7 + 63*cos(b*x + a)^5)/b
Leaf count of result is larger than twice the leaf count of optimal. 318 vs. \(2 (39) = 78\).
Time = 11.17 (sec) , antiderivative size = 318, normalized size of antiderivative = 6.91 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=\begin {cases} - \frac {8 \sin ^{5}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} + \frac {16 \sin ^{4}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )}}{315 b} - \frac {16 \sin ^{4}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{315 b} + \frac {44 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} - \frac {113 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )}}{315 b} + \frac {8 \sin ^{2}{\left (a + b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{315 b} - \frac {88 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{4}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{315 b} - \frac {2 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{5}{\left (a + b x \right )}}{63 b} - \frac {32 \cos ^{5}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{315 b} & \text {for}\: b \neq 0 \\x \sin ^{3}{\left (a \right )} \sin ^{2}{\left (2 a \right )} \cos ^{2}{\left (a \right )} & \text {otherwise} \end {cases} \] Input:
integrate(cos(b*x+a)**2*sin(b*x+a)**3*sin(2*b*x+2*a)**2,x)
Output:
Piecewise((-8*sin(a + b*x)**5*sin(2*a + 2*b*x)*cos(2*a + 2*b*x)/(315*b) + 16*sin(a + b*x)**4*sin(2*a + 2*b*x)**2*cos(a + b*x)/(315*b) - 16*sin(a + b *x)**4*cos(a + b*x)*cos(2*a + 2*b*x)**2/(315*b) + 44*sin(a + b*x)**3*sin(2 *a + 2*b*x)*cos(a + b*x)**2*cos(2*a + 2*b*x)/(315*b) - 113*sin(a + b*x)**2 *sin(2*a + 2*b*x)**2*cos(a + b*x)**3/(315*b) + 8*sin(a + b*x)**2*cos(a + b *x)**3*cos(2*a + 2*b*x)**2/(315*b) - 88*sin(a + b*x)*sin(2*a + 2*b*x)*cos( a + b*x)**4*cos(2*a + 2*b*x)/(315*b) - 2*sin(2*a + 2*b*x)**2*cos(a + b*x)* *5/(63*b) - 32*cos(a + b*x)**5*cos(2*a + 2*b*x)**2/(315*b), Ne(b, 0)), (x* sin(a)**3*sin(2*a)**2*cos(a)**2, True))
Time = 0.05 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.26 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=-\frac {35 \, \cos \left (9 \, b x + 9 \, a\right ) - 45 \, \cos \left (7 \, b x + 7 \, a\right ) - 252 \, \cos \left (5 \, b x + 5 \, a\right ) + 420 \, \cos \left (3 \, b x + 3 \, a\right ) + 1890 \, \cos \left (b x + a\right )}{20160 \, b} \] Input:
integrate(cos(b*x+a)^2*sin(b*x+a)^3*sin(2*b*x+2*a)^2,x, algorithm="maxima" )
Output:
-1/20160*(35*cos(9*b*x + 9*a) - 45*cos(7*b*x + 7*a) - 252*cos(5*b*x + 5*a) + 420*cos(3*b*x + 3*a) + 1890*cos(b*x + a))/b
Time = 0.14 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=-\frac {4 \, {\left (35 \, \cos \left (b x + a\right )^{9} - 90 \, \cos \left (b x + a\right )^{7} + 63 \, \cos \left (b x + a\right )^{5}\right )}}{315 \, b} \] Input:
integrate(cos(b*x+a)^2*sin(b*x+a)^3*sin(2*b*x+2*a)^2,x, algorithm="giac")
Output:
-4/315*(35*cos(b*x + a)^9 - 90*cos(b*x + a)^7 + 63*cos(b*x + a)^5)/b
Time = 0.08 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=-\frac {4\,\left (35\,{\cos \left (a+b\,x\right )}^9-90\,{\cos \left (a+b\,x\right )}^7+63\,{\cos \left (a+b\,x\right )}^5\right )}{315\,b} \] Input:
int(cos(a + b*x)^2*sin(a + b*x)^3*sin(2*a + 2*b*x)^2,x)
Output:
-(4*(63*cos(a + b*x)^5 - 90*cos(a + b*x)^7 + 35*cos(a + b*x)^9))/(315*b)
Time = 0.14 (sec) , antiderivative size = 343, normalized size of antiderivative = 7.46 \[ \int \cos ^2(a+b x) \sin ^3(a+b x) \sin ^2(2 a+2 b x) \, dx=\frac {880 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{3}-440 \cos \left (2 b x +2 a \right ) \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )-700 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{5}+1100 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )^{3}-440 \cos \left (2 b x +2 a \right ) \sin \left (2 b x +2 a \right ) \sin \left (b x +a \right )+875 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{4}-825 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{2}+110 \cos \left (b x +a \right ) \sin \left (2 b x +2 a \right )^{2}-280 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{4}+360 \cos \left (b x +a \right ) \sin \left (b x +a \right )^{2}-160 \cos \left (b x +a \right )+880 \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{4}-880 \sin \left (2 b x +2 a \right )^{2} \sin \left (b x +a \right )^{2}+110 \sin \left (2 b x +2 a \right )^{2}-440 \sin \left (b x +a \right )^{4}+440 \sin \left (b x +a \right )^{2}+512}{1575 b} \] Input:
int(cos(b*x+a)^2*sin(b*x+a)^3*sin(2*b*x+2*a)^2,x)
Output:
(880*cos(2*a + 2*b*x)*cos(a + b*x)*sin(2*a + 2*b*x)*sin(a + b*x)**3 - 440* cos(2*a + 2*b*x)*cos(a + b*x)*sin(2*a + 2*b*x)*sin(a + b*x) - 700*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)*sin(a + b*x)**5 + 1100*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)*sin(a + b*x)**3 - 440*cos(2*a + 2*b*x)*sin(2*a + 2*b*x)*sin(a + b *x) + 875*cos(a + b*x)*sin(2*a + 2*b*x)**2*sin(a + b*x)**4 - 825*cos(a + b *x)*sin(2*a + 2*b*x)**2*sin(a + b*x)**2 + 110*cos(a + b*x)*sin(2*a + 2*b*x )**2 - 280*cos(a + b*x)*sin(a + b*x)**4 + 360*cos(a + b*x)*sin(a + b*x)**2 - 160*cos(a + b*x) + 880*sin(2*a + 2*b*x)**2*sin(a + b*x)**4 - 880*sin(2* a + 2*b*x)**2*sin(a + b*x)**2 + 110*sin(2*a + 2*b*x)**2 - 440*sin(a + b*x) **4 + 440*sin(a + b*x)**2 + 512)/(1575*b)