\(\int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx\) [129]

Optimal result

Integrand size = 18, antiderivative size = 61 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {128 \cos ^9(a+b x)}{9 b}+\frac {384 \cos ^{11}(a+b x)}{11 b}-\frac {384 \cos ^{13}(a+b x)}{13 b}+\frac {128 \cos ^{15}(a+b x)}{15 b} \]

[Out]

-128/9*cos(b*x+a)^9/b+384/11*cos(b*x+a)^11/b-384/13*cos(b*x+a)^13/b+128/15*cos(b*x+a)^15/b

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {4372, 2645, 276} \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \cos ^{15}(a+b x)}{15 b}-\frac {384 \cos ^{13}(a+b x)}{13 b}+\frac {384 \cos ^{11}(a+b x)}{11 b}-\frac {128 \cos ^9(a+b x)}{9 b} \]

[In]

Int[Cos[a + b*x]*Sin[2*a + 2*b*x]^7,x]

[Out]

(-128*Cos[a + b*x]^9)/(9*b) + (384*Cos[a + b*x]^11)/(11*b) - (384*Cos[a + b*x]^13)/(13*b) + (128*Cos[a + b*x]^
15)/(15*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}& = 128 \int \cos ^8(a+b x) \sin ^7(a+b x) \, dx \\ & = -\frac {128 \text {Subst}\left (\int x^8 \left (1-x^2\right )^3 \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {128 \text {Subst}\left (\int \left (x^8-3 x^{10}+3 x^{12}-x^{14}\right ) \, dx,x,\cos (a+b x)\right )}{b} \\ & = -\frac {128 \cos ^9(a+b x)}{9 b}+\frac {384 \cos ^{11}(a+b x)}{11 b}-\frac {384 \cos ^{13}(a+b x)}{13 b}+\frac {128 \cos ^{15}(a+b x)}{15 b} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.81 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.77 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {4 \cos ^9(a+b x) (-8330+10755 \cos (2 (a+b x))-3366 \cos (4 (a+b x))+429 \cos (6 (a+b x)))}{6435 b} \]

[In]

Integrate[Cos[a + b*x]*Sin[2*a + 2*b*x]^7,x]

[Out]

(4*Cos[a + b*x]^9*(-8330 + 10755*Cos[2*(a + b*x)] - 3366*Cos[4*(a + b*x)] + 429*Cos[6*(a + b*x)]))/(6435*b)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(110\) vs. \(2(53)=106\).

Time = 4.97 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.82

method	result	size
default	\(-\frac {35 \cos \left (x b +a \right )}{128 b}-\frac {35 \cos \left (3 x b +3 a \right )}{384 b}+\frac {21 \cos \left (5 x b +5 a \right )}{640 b}+\frac {3 \cos \left (7 x b +7 a \right )}{128 b}-\frac {7 \cos \left (9 x b +9 a \right )}{1152 b}-\frac {7 \cos \left (11 x b +11 a \right )}{1408 b}+\frac {\cos \left (13 x b +13 a \right )}{1664 b}+\frac {\cos \left (15 x b +15 a \right )}{1920 b}\)	\(111\)
risch	\(-\frac {35 \cos \left (x b +a \right )}{128 b}-\frac {35 \cos \left (3 x b +3 a \right )}{384 b}+\frac {21 \cos \left (5 x b +5 a \right )}{640 b}+\frac {3 \cos \left (7 x b +7 a \right )}{128 b}-\frac {7 \cos \left (9 x b +9 a \right )}{1152 b}-\frac {7 \cos \left (11 x b +11 a \right )}{1408 b}+\frac {\cos \left (13 x b +13 a \right )}{1664 b}+\frac {\cos \left (15 x b +15 a \right )}{1920 b}\)	\(111\)
parallelrisch	\(\frac {\left (1024 \tan \left (x b +a \right )^{12}+6400 \tan \left (x b +a \right )^{10}+16768 \tan \left (x b +a \right )^{8}+126592 \tan \left (x b +a \right )^{6}+79616 \tan \left (x b +a \right )^{4}+27648 \tan \left (x b +a \right )^{2}+4096\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}+\left (-4096 \tan \left (x b +a \right )^{13}-26624 \tan \left (x b +a \right )^{11}-73216 \tan \left (x b +a \right )^{9}-109824 \tan \left (x b +a \right )^{7}-73216 \tan \left (x b +a \right )^{5}-26624 \tan \left (x b +a \right )^{3}-4096 \tan \left (x b +a \right )\right ) \tan \left (\frac {a}{2}+\frac {x b}{2}\right )+4096 \tan \left (x b +a \right )^{14}+27648 \tan \left (x b +a \right )^{12}+79616 \tan \left (x b +a \right )^{10}+126592 \tan \left (x b +a \right )^{8}+16768 \tan \left (x b +a \right )^{6}+6400 \tan \left (x b +a \right )^{4}+1024 \tan \left (x b +a \right )^{2}}{6435 b \left (1+\tan \left (\frac {a}{2}+\frac {x b}{2}\right )^{2}\right ) \left (1+\tan \left (x b +a \right )^{2}\right )^{7}}\)	\(257\)

[In]

int(cos(b*x+a)*sin(2*b*x+2*a)^7,x,method=_RETURNVERBOSE)

[Out]

-35/128*cos(b*x+a)/b-35/384*cos(3*b*x+3*a)/b+21/640*cos(5*b*x+5*a)/b+3/128*cos(7*b*x+7*a)/b-7/1152*cos(9*b*x+9
*a)/b-7/1408*cos(11*b*x+11*a)/b+1/1664*cos(13*b*x+13*a)/b+1/1920*cos(15*b*x+15*a)/b

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \, {\left (429 \, \cos \left (b x + a\right )^{15} - 1485 \, \cos \left (b x + a\right )^{13} + 1755 \, \cos \left (b x + a\right )^{11} - 715 \, \cos \left (b x + a\right )^{9}\right )}}{6435 \, b} \]

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="fricas")

[Out]

128/6435*(429*cos(b*x + a)^15 - 1485*cos(b*x + a)^13 + 1755*cos(b*x + a)^11 - 715*cos(b*x + a)^9)/b

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 270 vs. \(2 (53) = 106\).

Time = 25.61 (sec) , antiderivative size = 270, normalized size of antiderivative = 4.43 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=\begin {cases} - \frac {1241 \sin {\left (a + b x \right )} \sin ^{7}{\left (2 a + 2 b x \right )}}{6435 b} - \frac {376 \sin {\left (a + b x \right )} \sin ^{5}{\left (2 a + 2 b x \right )} \cos ^{2}{\left (2 a + 2 b x \right )}}{715 b} - \frac {640 \sin {\left (a + b x \right )} \sin ^{3}{\left (2 a + 2 b x \right )} \cos ^{4}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {1024 \sin {\left (a + b x \right )} \sin {\left (2 a + 2 b x \right )} \cos ^{6}{\left (2 a + 2 b x \right )}}{6435 b} - \frac {3838 \sin ^{6}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos {\left (2 a + 2 b x \right )}}{6435 b} - \frac {1648 \sin ^{4}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{3}{\left (2 a + 2 b x \right )}}{1287 b} - \frac {768 \sin ^{2}{\left (2 a + 2 b x \right )} \cos {\left (a + b x \right )} \cos ^{5}{\left (2 a + 2 b x \right )}}{715 b} - \frac {2048 \cos {\left (a + b x \right )} \cos ^{7}{\left (2 a + 2 b x \right )}}{6435 b} & \text {for}\: b \neq 0 \\x \sin ^{7}{\left (2 a \right )} \cos {\left (a \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)**7,x)

[Out]

Piecewise((-1241*sin(a + b*x)*sin(2*a + 2*b*x)**7/(6435*b) - 376*sin(a + b*x)*sin(2*a + 2*b*x)**5*cos(2*a + 2*
b*x)**2/(715*b) - 640*sin(a + b*x)*sin(2*a + 2*b*x)**3*cos(2*a + 2*b*x)**4/(1287*b) - 1024*sin(a + b*x)*sin(2*
a + 2*b*x)*cos(2*a + 2*b*x)**6/(6435*b) - 3838*sin(2*a + 2*b*x)**6*cos(a + b*x)*cos(2*a + 2*b*x)/(6435*b) - 16
48*sin(2*a + 2*b*x)**4*cos(a + b*x)*cos(2*a + 2*b*x)**3/(1287*b) - 768*sin(2*a + 2*b*x)**2*cos(a + b*x)*cos(2*
a + 2*b*x)**5/(715*b) - 2048*cos(a + b*x)*cos(2*a + 2*b*x)**7/(6435*b), Ne(b, 0)), (x*sin(2*a)**7*cos(a), True
))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.49 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {429 \, \cos \left (15 \, b x + 15 \, a\right ) + 495 \, \cos \left (13 \, b x + 13 \, a\right ) - 4095 \, \cos \left (11 \, b x + 11 \, a\right ) - 5005 \, \cos \left (9 \, b x + 9 \, a\right ) + 19305 \, \cos \left (7 \, b x + 7 \, a\right ) + 27027 \, \cos \left (5 \, b x + 5 \, a\right ) - 75075 \, \cos \left (3 \, b x + 3 \, a\right ) - 225225 \, \cos \left (b x + a\right )}{823680 \, b} \]

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="maxima")

[Out]

1/823680*(429*cos(15*b*x + 15*a) + 495*cos(13*b*x + 13*a) - 4095*cos(11*b*x + 11*a) - 5005*cos(9*b*x + 9*a) +
19305*cos(7*b*x + 7*a) + 27027*cos(5*b*x + 5*a) - 75075*cos(3*b*x + 3*a) - 225225*cos(b*x + a))/b

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=\frac {128 \, {\left (429 \, \cos \left (b x + a\right )^{15} - 1485 \, \cos \left (b x + a\right )^{13} + 1755 \, \cos \left (b x + a\right )^{11} - 715 \, \cos \left (b x + a\right )^{9}\right )}}{6435 \, b} \]

[In]

integrate(cos(b*x+a)*sin(2*b*x+2*a)^7,x, algorithm="giac")

[Out]

128/6435*(429*cos(b*x + a)^15 - 1485*cos(b*x + a)^13 + 1755*cos(b*x + a)^11 - 715*cos(b*x + a)^9)/b

Mupad [B] (verification not implemented)

Time = 19.65 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.75 \[ \int \cos (a+b x) \sin ^7(2 a+2 b x) \, dx=-\frac {-\frac {128\,{\cos \left (a+b\,x\right )}^{15}}{15}+\frac {384\,{\cos \left (a+b\,x\right )}^{13}}{13}-\frac {384\,{\cos \left (a+b\,x\right )}^{11}}{11}+\frac {128\,{\cos \left (a+b\,x\right )}^9}{9}}{b} \]

[In]

int(cos(a + b*x)*sin(2*a + 2*b*x)^7,x)

[Out]

-((128*cos(a + b*x)^9)/9 - (384*cos(a + b*x)^11)/11 + (384*cos(a + b*x)^13)/13 - (128*cos(a + b*x)^15)/15)/b