\(\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\) [378]

Optimal result

Integrand size = 29, antiderivative size = 185 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {9 a^2 x}{256}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \]

[Out]

9/256*a^2*x-2/5*a^2*cos(d*x+c)^5/d+4/7*a^2*cos(d*x+c)^7/d-2/9*a^2*cos(d*x+c)^9/d+9/256*a^2*cos(d*x+c)*sin(d*x+
c)/d+3/128*a^2*cos(d*x+c)^3*sin(d*x+c)/d-3/32*a^2*cos(d*x+c)^5*sin(d*x+c)/d-3/16*a^2*cos(d*x+c)^5*sin(d*x+c)^3
/d-1/10*a^2*cos(d*x+c)^5*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2648, 2715, 8, 2645, 276} \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}-\frac {a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {9 a^2 x}{256} \]

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(9*a^2*x)/256 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (4*a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Cos[c + d*x]^9)/(9*d) + (
9*a^2*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (3*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (3*a^2*Cos[c + d*x]^5
*Sin[c + d*x])/(32*d) - (3*a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10
*d)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

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 2648

Int[(cos[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[(-a)*(b*Cos[e
+ f*x])^(n + 1)*((a*Sin[e + f*x])^(m - 1)/(b*f*(m + n))), x] + Dist[a^2*((m - 1)/(m + n)), Int[(b*Cos[e + f*x
])^n*(a*Sin[e + f*x])^(m - 2), x], x] /; FreeQ[{a, b, e, f, n}, x] && GtQ[m, 1] && NeQ[m + n, 0] && IntegersQ[
2*m, 2*n]

Rule 2715

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Sin[c + d*x])^(n - 1)/(d*n))
, x] + Dist[b^2*((n - 1)/n), Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integ
erQ[2*n]

Rule 2952

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*
(x_)])^(m_), x_Symbol] :> Int[ExpandTrig[(g*cos[e + f*x])^p, (d*sin[e + f*x])^n*(a + b*sin[e + f*x])^m, x], x]
/; FreeQ[{a, b, d, e, f, g, n, p}, x] && EqQ[a^2 - b^2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (a^2 \cos ^4(c+d x) \sin ^4(c+d x)+2 a^2 \cos ^4(c+d x) \sin ^5(c+d x)+a^2 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+a^2 \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{8} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{2} a^2 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{16} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (3 a^2\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{32} a^2 \int \cos ^4(c+d x) \, dx+\frac {1}{64} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{128} \left (3 a^2\right ) \int 1 \, dx+\frac {1}{128} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {3 a^2 x}{128}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{256} \left (3 a^2\right ) \int 1 \, dx \\ & = \frac {9 a^2 x}{256}-\frac {2 a^2 \cos ^5(c+d x)}{5 d}+\frac {4 a^2 \cos ^7(c+d x)}{7 d}-\frac {2 a^2 \cos ^9(c+d x)}{9 d}+\frac {9 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {3 a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {3 a^2 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {3 a^2 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^2 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.91 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.63 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (22680 c+22680 d x-30240 \cos (c+d x)-6720 \cos (3 (c+d x))+4032 \cos (5 (c+d x))+720 \cos (7 (c+d x))-560 \cos (9 (c+d x))-1260 \sin (2 (c+d x))-7560 \sin (4 (c+d x))+630 \sin (6 (c+d x))+945 \sin (8 (c+d x))-126 \sin (10 (c+d x)))}{645120 d} \]

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(22680*c + 22680*d*x - 30240*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 4032*Cos[5*(c + d*x)] + 720*Cos[7*(c
+ d*x)] - 560*Cos[9*(c + d*x)] - 1260*Sin[2*(c + d*x)] - 7560*Sin[4*(c + d*x)] + 630*Sin[6*(c + d*x)] + 945*Si
n[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)

Maple [A] (verified)

Time = 0.74 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(-\frac {\left (-18 d x +\sin \left (2 d x +2 c \right )+6 \sin \left (4 d x +4 c \right )-\frac {\sin \left (6 d x +6 c \right )}{2}-\frac {3 \sin \left (8 d x +8 c \right )}{4}+\frac {\sin \left (10 d x +10 c \right )}{10}+24 \cos \left (d x +c \right )+\frac {16 \cos \left (3 d x +3 c \right )}{3}-\frac {16 \cos \left (5 d x +5 c \right )}{5}-\frac {4 \cos \left (7 d x +7 c \right )}{7}+\frac {4 \cos \left (9 d x +9 c \right )}{9}+\frac {8192}{315}\right ) a^{2}}{512 d}\)	\(120\)
risch	\(\frac {9 a^{2} x}{256}-\frac {3 a^{2} \cos \left (d x +c \right )}{64 d}-\frac {a^{2} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a^{2} \cos \left (9 d x +9 c \right )}{1152 d}+\frac {3 a^{2} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {a^{2} \cos \left (7 d x +7 c \right )}{896 d}+\frac {a^{2} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{160 d}-\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{256 d}-\frac {a^{2} \cos \left (3 d x +3 c \right )}{96 d}-\frac {a^{2} \sin \left (2 d x +2 c \right )}{512 d}\)	\(175\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\)	\(218\)
default	\(\frac {a^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )}{d}\)	\(218\)

[In]

int(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/512*(-18*d*x+sin(2*d*x+2*c)+6*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)-3/4*sin(8*d*x+8*c)+1/10*sin(10*d*x+10*c)+24
*cos(d*x+c)+16/3*cos(3*d*x+3*c)-16/5*cos(5*d*x+5*c)-4/7*cos(7*d*x+7*c)+4/9*cos(9*d*x+9*c)+8192/315)*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.67 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {17920 \, a^{2} \cos \left (d x + c\right )^{9} - 46080 \, a^{2} \cos \left (d x + c\right )^{7} + 32256 \, a^{2} \cos \left (d x + c\right )^{5} - 2835 \, a^{2} d x + 63 \, {\left (128 \, a^{2} \cos \left (d x + c\right )^{9} - 496 \, a^{2} \cos \left (d x + c\right )^{7} + 488 \, a^{2} \cos \left (d x + c\right )^{5} - 30 \, a^{2} \cos \left (d x + c\right )^{3} - 45 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/80640*(17920*a^2*cos(d*x + c)^9 - 46080*a^2*cos(d*x + c)^7 + 32256*a^2*cos(d*x + c)^5 - 2835*a^2*d*x + 63*(
128*a^2*cos(d*x + c)^9 - 496*a^2*cos(d*x + c)^7 + 488*a^2*cos(d*x + c)^5 - 30*a^2*cos(d*x + c)^3 - 45*a^2*cos(
d*x + c))*sin(d*x + c))/d

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (177) = 354\).

Time = 1.30 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.99 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {3 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {9 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {3 a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {11 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {2 a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {11 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {8 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {16 a^{2} \cos ^{9}{\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{4}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**4*(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((3*a**2*x*sin(c + d*x)**10/256 + 15*a**2*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 3*a**2*x*sin(c + d*
x)**8/128 + 15*a**2*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 3*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a
**2*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 9*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a**2*x*sin(c + d*
x)**2*cos(c + d*x)**8/256 + 3*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*a**2*x*cos(c + d*x)**10/256 + 3*a*
*2*x*cos(c + d*x)**8/128 + 3*a**2*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**2*sin(c + d*x)**7*cos(c + d*x)**
3/(128*d) + 3*a**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) - a**2*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 11*a**
2*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 2*a**2*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 7*a**2*sin(c + d*x)
**3*cos(c + d*x)**7/(128*d) - 11*a**2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 8*a**2*sin(c + d*x)**2*cos(c +
d*x)**7/(35*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) -
16*a**2*cos(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)**4*cos(c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.66 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {4096 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{2} + 63 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} - 630 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{645120 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/645120*(4096*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^2 + 63*(32*sin(2*d*x + 2*c)^5 -
120*d*x - 120*c - 5*sin(8*d*x + 8*c) + 40*sin(4*d*x + 4*c))*a^2 - 630*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*si
n(4*d*x + 4*c))*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.60 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.94 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {9}{256} \, a^{2} x - \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac {a^{2} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac {a^{2} \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac {3 \, a^{2} \cos \left (d x + c\right )}{64 \, d} - \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {3 \, a^{2} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {3 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

9/256*a^2*x - 1/1152*a^2*cos(9*d*x + 9*c)/d + 1/896*a^2*cos(7*d*x + 7*c)/d + 1/160*a^2*cos(5*d*x + 5*c)/d - 1/
96*a^2*cos(3*d*x + 3*c)/d - 3/64*a^2*cos(d*x + c)/d - 1/5120*a^2*sin(10*d*x + 10*c)/d + 3/2048*a^2*sin(8*d*x +
8*c)/d + 1/1024*a^2*sin(6*d*x + 6*c)/d - 3/256*a^2*sin(4*d*x + 4*c)/d - 1/512*a^2*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 13.25 (sec) , antiderivative size = 469, normalized size of antiderivative = 2.54 \[ \int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {9\,a^2\,x}{256}-\frac {\frac {9\,a^2\,\left (c+d\,x\right )}{256}+\frac {87\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {553\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {491\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {2555\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {2555\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {491\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {553\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {87\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}-\frac {a^2\,\left (2835\,c+2835\,d\,x-8192\right )}{80640}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {45\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (28350\,c+28350\,d\,x-81920\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {405\,a^2\,\left (c+d\,x\right )}{256}-\frac {a^2\,\left (127575\,c+127575\,d\,x-368640\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {135\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (340200\,c+340200\,d\,x+737280\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {945\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (595350\,c+595350\,d\,x+860160\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {135\,a^2\,\left (c+d\,x\right )}{32}-\frac {a^2\,\left (340200\,c+340200\,d\,x-1720320\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {567\,a^2\,\left (c+d\,x\right )}{64}-\frac {a^2\,\left (714420\,c+714420\,d\,x-1032192\right )}{80640}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {945\,a^2\,\left (c+d\,x\right )}{128}-\frac {a^2\,\left (595350\,c+595350\,d\,x-2580480\right )}{80640}\right )+\frac {9\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{10}} \]

[In]

int(cos(c + d*x)^4*sin(c + d*x)^4*(a + a*sin(c + d*x))^2,x)

[Out]

(9*a^2*x)/256 - ((9*a^2*(c + d*x))/256 + (87*a^2*tan(c/2 + (d*x)/2)^3)/128 - (553*a^2*tan(c/2 + (d*x)/2)^5)/16
0 - (491*a^2*tan(c/2 + (d*x)/2)^7)/32 + (2555*a^2*tan(c/2 + (d*x)/2)^9)/64 - (2555*a^2*tan(c/2 + (d*x)/2)^11)/
64 + (491*a^2*tan(c/2 + (d*x)/2)^13)/32 + (553*a^2*tan(c/2 + (d*x)/2)^15)/160 - (87*a^2*tan(c/2 + (d*x)/2)^17)
/128 - (9*a^2*tan(c/2 + (d*x)/2)^19)/128 - (a^2*(2835*c + 2835*d*x - 8192))/80640 + tan(c/2 + (d*x)/2)^2*((45*
a^2*(c + d*x))/128 - (a^2*(28350*c + 28350*d*x - 81920))/80640) + tan(c/2 + (d*x)/2)^4*((405*a^2*(c + d*x))/25
6 - (a^2*(127575*c + 127575*d*x - 368640))/80640) + tan(c/2 + (d*x)/2)^6*((135*a^2*(c + d*x))/32 - (a^2*(34020
0*c + 340200*d*x + 737280))/80640) + tan(c/2 + (d*x)/2)^12*((945*a^2*(c + d*x))/128 - (a^2*(595350*c + 595350*
d*x + 860160))/80640) + tan(c/2 + (d*x)/2)^14*((135*a^2*(c + d*x))/32 - (a^2*(340200*c + 340200*d*x - 1720320)
)/80640) + tan(c/2 + (d*x)/2)^10*((567*a^2*(c + d*x))/64 - (a^2*(714420*c + 714420*d*x - 1032192))/80640) + ta
n(c/2 + (d*x)/2)^8*((945*a^2*(c + d*x))/128 - (a^2*(595350*c + 595350*d*x - 2580480))/80640) + (9*a^2*tan(c/2
+ (d*x)/2))/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^10)