\(\int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx\) [151]

Optimal result

Integrand size = 20, antiderivative size = 46 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \cos ^9(a+b x)}{9 b}+\frac {64 \cos ^{11}(a+b x)}{11 b}-\frac {32 \cos ^{13}(a+b x)}{13 b} \]

[Out]

-32/9*cos(b*x+a)^9/b+64/11*cos(b*x+a)^11/b-32/13*cos(b*x+a)^13/b

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4372, 2645, 276} \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \cos ^{13}(a+b x)}{13 b}+\frac {64 \cos ^{11}(a+b x)}{11 b}-\frac {32 \cos ^9(a+b x)}{9 b} \]

[In]

Int[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]

[Out]

(-32*Cos[a + b*x]^9)/(9*b) + (64*Cos[a + b*x]^11)/(11*b) - (32*Cos[a + b*x]^13)/(13*b)

Rule 276

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rule 2645

Int[(cos[(e_.) + (f_.)*(x_)]*(a_.))^(m_.)*sin[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[-(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])

Rule 4372

Int[(cos[(a_.) + (b_.)*(x_)]*(e_.))^(m_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[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]
&& IntegerQ[p]

Rubi steps \begin{align*} \text {integral}& = 32 \int \cos ^8(a+b x) \sin ^5(a+b x) \, dx \\ & = -\frac {32 \text {Subst}\left (\int x^8 \left (1-x^2\right )^2 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {32 \text {Subst}\left (\int \left (x^8-2 x^{10}+x^{12}\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\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} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.40 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.80 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\frac {4 \cos ^9(a+b x) (-505+540 \cos (2 (a+b x))-99 \cos (4 (a+b x)))}{1287 b} \]

[In]

Integrate[Cos[a + b*x]^3*Sin[2*a + 2*b*x]^5,x]

[Out]

(4*Cos[a + b*x]^9*(-505 + 540*Cos[2*(a + b*x)] - 99*Cos[4*(a + b*x)]))/(1287*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(81\) vs. \(2(40)=80\).

Time = 5.41 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.78

method	result	size
parallelrisch	\(\frac {475136-180180 \cos \left (x b +a \right )+18018 \cos \left (7 x b +7 a \right )+9009 \cos \left (5 x b +5 a \right )-2457 \cos \left (11 x b +11 a \right )+2002 \cos \left (9 x b +9 a \right )-693 \cos \left (13 x b +13 a \right )-75075 \cos \left (3 x b +3 a \right )}{1153152 b}\)	\(82\)
default	\(-\frac {5 \cos \left (x b +a \right )}{32 b}-\frac {25 \cos \left (3 x b +3 a \right )}{384 b}+\frac {\cos \left (5 x b +5 a \right )}{128 b}+\frac {\cos \left (7 x b +7 a \right )}{64 b}+\frac {\cos \left (9 x b +9 a \right )}{576 b}-\frac {3 \cos \left (11 x b +11 a \right )}{1408 b}-\frac {\cos \left (13 x b +13 a \right )}{1664 b}\)	\(97\)
risch	\(-\frac {5 \cos \left (x b +a \right )}{32 b}-\frac {25 \cos \left (3 x b +3 a \right )}{384 b}+\frac {\cos \left (5 x b +5 a \right )}{128 b}+\frac {\cos \left (7 x b +7 a \right )}{64 b}+\frac {\cos \left (9 x b +9 a \right )}{576 b}-\frac {3 \cos \left (11 x b +11 a \right )}{1408 b}-\frac {\cos \left (13 x b +13 a \right )}{1664 b}\)	\(97\)

[In]

int(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x,method=_RETURNVERBOSE)

[Out]

1/1153152*(475136-180180*cos(b*x+a)+18018*cos(7*b*x+7*a)+9009*cos(5*b*x+5*a)-2457*cos(11*b*x+11*a)+2002*cos(9*
b*x+9*a)-693*cos(13*b*x+13*a)-75075*cos(3*b*x+3*a))/b

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, b} \]

[In]

integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="fricas")

[Out]

-32/1287*(99*cos(b*x + a)^13 - 234*cos(b*x + a)^11 + 143*cos(b*x + a)^9)/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 447 vs. \(2 (39) = 78\).

Time = 25.73 (sec) , antiderivative size = 447, normalized size of antiderivative = 9.72 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=\begin {cases} - \frac {2234 \sin ^{3}{\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {4544 \sin ^{3}{\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \sin ^{3}{\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1001 b} - \frac {1388 \sin ^{2}{\left (a + b x \right )} \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {2944 \sin ^{2}{\left (a + b x \right )} \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {512 \sin ^{2}{\left (a + b x \right )} \cos {\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1001 b} + \frac {271 \sin {\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )}}{3003 b} + \frac {48 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{143 b} + \frac {640 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{2}{\left (a + b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{3003 b} - \frac {1366 \sin ^{4}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{3003 b} - \frac {4960 \sin ^{2}{\left (2 a + 2 b x \right )} \cos ^{3}{\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{9009 b} - \frac {256 \cos ^{3}{\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{1287 b} & \text {for}\: b \neq 0 \\x \sin ^{5}{\left (2 a \right )} \cos ^{3}{\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(b*x+a)**3*sin(2*b*x+2*a)**5,x)

[Out]

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*s
in(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*c
os(a)**3, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 80, normalized size of antiderivative = 1.74 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {99 \, \cos \left (13 \, b x + 13 \, a\right ) + 351 \, \cos \left (11 \, b x + 11 \, a\right ) - 286 \, \cos \left (9 \, b x + 9 \, a\right ) - 2574 \, \cos \left (7 \, b x + 7 \, a\right ) - 1287 \, \cos \left (5 \, b x + 5 \, a\right ) + 10725 \, \cos \left (3 \, b x + 3 \, a\right ) + 25740 \, \cos \left (b x + a\right )}{164736 \, b} \]

[In]

integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="maxima")

[Out]

-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) - 128
7*cos(5*b*x + 5*a) + 10725*cos(3*b*x + 3*a) + 25740*cos(b*x + a))/b

Giac [A] (verification not implemented)

none

Time = 0.34 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32 \, {\left (99 \, \cos \left (b x + a\right )^{13} - 234 \, \cos \left (b x + a\right )^{11} + 143 \, \cos \left (b x + a\right )^{9}\right )}}{1287 \, b} \]

[In]

integrate(cos(b*x+a)^3*sin(2*b*x+2*a)^5,x, algorithm="giac")

[Out]

-32/1287*(99*cos(b*x + a)^13 - 234*cos(b*x + a)^11 + 143*cos(b*x + a)^9)/b

Mupad [B] (verification not implemented)

Time = 20.85 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.78 \[ \int \cos ^3(a+b x) \sin ^5(2 a+2 b x) \, dx=-\frac {32\,\left (99\,{\cos \left (a+b\,x\right )}^{13}-234\,{\cos \left (a+b\,x\right )}^{11}+143\,{\cos \left (a+b\,x\right )}^9\right )}{1287\,b} \]

[In]

int(cos(a + b*x)^3*sin(2*a + 2*b*x)^5,x)

[Out]

-(32*(143*cos(a + b*x)^9 - 234*cos(a + b*x)^11 + 99*cos(a + b*x)^13))/(1287*b)