\(\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\) [588]

Optimal result

Integrand size = 29, antiderivative size = 209 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {17 a^2 x}{1024}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d} \]

[Out]

17/1024*a^2*x-2/7*a^2*cos(d*x+c)^7/d+4/9*a^2*cos(d*x+c)^9/d-2/11*a^2*cos(d*x+c)^11/d+17/1024*a^2*cos(d*x+c)*si
n(d*x+c)/d+17/1536*a^2*cos(d*x+c)^3*sin(d*x+c)/d+17/1920*a^2*cos(d*x+c)^5*sin(d*x+c)/d-17/320*a^2*cos(d*x+c)^7
*sin(d*x+c)/d-17/120*a^2*cos(d*x+c)^7*sin(d*x+c)^3/d-1/12*a^2*cos(d*x+c)^7*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 18, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2648, 2715, 8, 2645, 276} \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}-\frac {a^2 \sin ^5(c+d x) \cos ^7(c+d x)}{12 d}-\frac {17 a^2 \sin ^3(c+d x) \cos ^7(c+d x)}{120 d}-\frac {17 a^2 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {17 a^2 \sin (c+d x) \cos ^5(c+d x)}{1920 d}+\frac {17 a^2 \sin (c+d x) \cos ^3(c+d x)}{1536 d}+\frac {17 a^2 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {17 a^2 x}{1024} \]

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(17*a^2*x)/1024 - (2*a^2*Cos[c + d*x]^7)/(7*d) + (4*a^2*Cos[c + d*x]^9)/(9*d) - (2*a^2*Cos[c + d*x]^11)/(11*d)
+ (17*a^2*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (17*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(1536*d) + (17*a^2*Cos[c
+ d*x]^5*Sin[c + d*x])/(1920*d) - (17*a^2*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (17*a^2*Cos[c + d*x]^7*Sin[c
+ d*x]^3)/(120*d) - (a^2*Cos[c + d*x]^7*Sin[c + d*x]^5)/(12*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 ^6(c+d x) \sin ^4(c+d x)+2 a^2 \cos ^6(c+d x) \sin ^5(c+d x)+a^2 \cos ^6(c+d x) \sin ^6(c+d x)\right ) \, dx \\ & = a^2 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+a^2 \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx+\left (2 a^2\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{10} \left (3 a^2\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac {1}{12} \left (5 a^2\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {3 a^2 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{80} \left (3 a^2\right ) \int \cos ^6(c+d x) \, dx+\frac {1}{8} a^2 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {\left (2 a^2\right ) \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{64} a^2 \int \cos ^6(c+d x) \, dx+\frac {1}{32} a^2 \int \cos ^4(c+d x) \, dx \\ & = -\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {a^2 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{384} \left (5 a^2\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{128} \left (3 a^2\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {3 a^2 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {1}{512} \left (5 a^2\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{256} \left (3 a^2\right ) \int 1 \, dx \\ & = \frac {3 a^2 x}{256}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d}+\frac {\left (5 a^2\right ) \int 1 \, dx}{1024} \\ & = \frac {17 a^2 x}{1024}-\frac {2 a^2 \cos ^7(c+d x)}{7 d}+\frac {4 a^2 \cos ^9(c+d x)}{9 d}-\frac {2 a^2 \cos ^{11}(c+d x)}{11 d}+\frac {17 a^2 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {17 a^2 \cos ^3(c+d x) \sin (c+d x)}{1536 d}+\frac {17 a^2 \cos ^5(c+d x) \sin (c+d x)}{1920 d}-\frac {17 a^2 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {17 a^2 \cos ^7(c+d x) \sin ^3(c+d x)}{120 d}-\frac {a^2 \cos ^7(c+d x) \sin ^5(c+d x)}{12 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.83 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2 (166320 c+471240 d x-554400 \cos (c+d x)-184800 \cos (3 (c+d x))+55440 \cos (5 (c+d x))+39600 \cos (7 (c+d x))-6160 \cos (9 (c+d x))-5040 \cos (11 (c+d x))+55440 \sin (2 (c+d x))-162855 \sin (4 (c+d x))-27720 \sin (6 (c+d x))+24255 \sin (8 (c+d x))+5544 \sin (10 (c+d x))-1155 \sin (12 (c+d x)))}{28385280 d} \]

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(166320*c + 471240*d*x - 554400*Cos[c + d*x] - 184800*Cos[3*(c + d*x)] + 55440*Cos[5*(c + d*x)] + 39600*C
os[7*(c + d*x)] - 6160*Cos[9*(c + d*x)] - 5040*Cos[11*(c + d*x)] + 55440*Sin[2*(c + d*x)] - 162855*Sin[4*(c +
d*x)] - 27720*Sin[6*(c + d*x)] + 24255*Sin[8*(c + d*x)] + 5544*Sin[10*(c + d*x)] - 1155*Sin[12*(c + d*x)]))/(2
8385280*d)

Maple [A] (verified)

Time = 1.04 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.67

method	result	size
parallelrisch	\(\frac {a^{2} \left (\frac {17 d x}{2}+\sin \left (2 d x +2 c \right )-\frac {47 \sin \left (4 d x +4 c \right )}{16}-\frac {\sin \left (6 d x +6 c \right )}{2}+\frac {7 \sin \left (8 d x +8 c \right )}{16}-\frac {\cos \left (11 d x +11 c \right )}{11}+\frac {\sin \left (10 d x +10 c \right )}{10}-\frac {\sin \left (12 d x +12 c \right )}{48}-10 \cos \left (d x +c \right )-\frac {10 \cos \left (3 d x +3 c \right )}{3}+\cos \left (5 d x +5 c \right )+\frac {5 \cos \left (7 d x +7 c \right )}{7}-\frac {\cos \left (9 d x +9 c \right )}{9}-\frac {8192}{693}\right )}{512 d}\)	\(140\)
risch	\(\frac {17 a^{2} x}{1024}-\frac {5 a^{2} \cos \left (d x +c \right )}{256 d}-\frac {a^{2} \sin \left (12 d x +12 c \right )}{24576 d}+\frac {a^{2} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a^{2} \cos \left (11 d x +11 c \right )}{5632 d}-\frac {a^{2} \cos \left (9 d x +9 c \right )}{4608 d}+\frac {7 a^{2} \sin \left (8 d x +8 c \right )}{8192 d}+\frac {5 a^{2} \cos \left (7 d x +7 c \right )}{3584 d}-\frac {a^{2} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {a^{2} \cos \left (5 d x +5 c \right )}{512 d}-\frac {47 a^{2} \sin \left (4 d x +4 c \right )}{8192 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right )}{768 d}+\frac {a^{2} \sin \left (2 d x +2 c \right )}{512 d}\)	\(209\)
derivativedivides	\(\frac {a^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\)	\(238\)
default	\(\frac {a^{2} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+2 a^{2} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a^{2} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\)	\(238\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

1/512*a^2*(17/2*d*x+sin(2*d*x+2*c)-47/16*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)+7/16*sin(8*d*x+8*c)-1/11*cos(11*d*x
+11*c)+1/10*sin(10*d*x+10*c)-1/48*sin(12*d*x+12*c)-10*cos(d*x+c)-10/3*cos(3*d*x+3*c)+cos(5*d*x+5*c)+5/7*cos(7*
d*x+7*c)-1/9*cos(9*d*x+9*c)-8192/693)/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.66 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {645120 \, a^{2} \cos \left (d x + c\right )^{11} - 1576960 \, a^{2} \cos \left (d x + c\right )^{9} + 1013760 \, a^{2} \cos \left (d x + c\right )^{7} - 58905 \, a^{2} d x + 231 \, {\left (1280 \, a^{2} \cos \left (d x + c\right )^{11} - 4736 \, a^{2} \cos \left (d x + c\right )^{9} + 4272 \, a^{2} \cos \left (d x + c\right )^{7} - 136 \, a^{2} \cos \left (d x + c\right )^{5} - 170 \, a^{2} \cos \left (d x + c\right )^{3} - 255 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3548160 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/3548160*(645120*a^2*cos(d*x + c)^11 - 1576960*a^2*cos(d*x + c)^9 + 1013760*a^2*cos(d*x + c)^7 - 58905*a^2*d
*x + 231*(1280*a^2*cos(d*x + c)^11 - 4736*a^2*cos(d*x + c)^9 + 4272*a^2*cos(d*x + c)^7 - 136*a^2*cos(d*x + c)^
5 - 170*a^2*cos(d*x + c)^3 - 255*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. 656 vs. \(2 (201) = 402\).

Time = 2.57 (sec) , antiderivative size = 656, normalized size of antiderivative = 3.14 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\begin {cases} \frac {5 a^{2} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {15 a^{2} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {3 a^{2} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {75 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {15 a^{2} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {25 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {15 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {75 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {5 a^{2} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {3 a^{2} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a^{2} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 a^{2} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3072 d} + \frac {3 a^{2} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {33 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} + \frac {7 a^{2} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {33 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} + \frac {a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {2 a^{2} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {85 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{3072 d} - \frac {7 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {5 a^{2} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {3 a^{2} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {16 a^{2} \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{2} \sin ^{4}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**4*(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((5*a**2*x*sin(c + d*x)**12/1024 + 15*a**2*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 3*a**2*x*sin(c +
d*x)**10/256 + 75*a**2*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 15*a**2*x*sin(c + d*x)**8*cos(c + d*x)**2/256
+ 25*a**2*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 15*a**2*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 75*a**2*x*si
n(c + d*x)**4*cos(c + d*x)**8/1024 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**2*x*sin(c + d*x)**2
*cos(c + d*x)**10/512 + 15*a**2*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 5*a**2*x*cos(c + d*x)**12/1024 + 3*a**
2*x*cos(c + d*x)**10/256 + 5*a**2*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85*a**2*sin(c + d*x)**9*cos(c + d*x
)**3/(3072*d) + 3*a**2*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 33*a**2*sin(c + d*x)**7*cos(c + d*x)**5/(512*d)
+ 7*a**2*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) - 33*a**2*sin(c + d*x)**5*cos(c + d*x)**7/(512*d) + a**2*sin(
c + d*x)**5*cos(c + d*x)**5/(10*d) - 2*a**2*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 85*a**2*sin(c + d*x)**3*co
s(c + d*x)**9/(3072*d) - 7*a**2*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 8*a**2*sin(c + d*x)**2*cos(c + d*x)*
*9/(63*d) - 5*a**2*sin(c + d*x)*cos(c + d*x)**11/(1024*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 16*a
**2*cos(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)**4*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.66 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=-\frac {81920 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{2} - 2772 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} - 1155 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2}}{28385280 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/28385280*(81920*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^2 - 2772*(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 - 1155*(4*sin(4*d*x + 4*c)^3 + 120*d*x
+ 120*c + 9*sin(8*d*x + 8*c) - 48*sin(4*d*x + 4*c))*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.49 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {17}{1024} \, a^{2} x - \frac {a^{2} \cos \left (11 \, d x + 11 \, c\right )}{5632 \, d} - \frac {a^{2} \cos \left (9 \, d x + 9 \, c\right )}{4608 \, d} + \frac {5 \, a^{2} \cos \left (7 \, d x + 7 \, c\right )}{3584 \, d} + \frac {a^{2} \cos \left (5 \, d x + 5 \, c\right )}{512 \, d} - \frac {5 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{768 \, d} - \frac {5 \, a^{2} \cos \left (d x + c\right )}{256 \, d} - \frac {a^{2} \sin \left (12 \, d x + 12 \, c\right )}{24576 \, d} + \frac {a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {7 \, a^{2} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {47 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {a^{2} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

17/1024*a^2*x - 1/5632*a^2*cos(11*d*x + 11*c)/d - 1/4608*a^2*cos(9*d*x + 9*c)/d + 5/3584*a^2*cos(7*d*x + 7*c)/
d + 1/512*a^2*cos(5*d*x + 5*c)/d - 5/768*a^2*cos(3*d*x + 3*c)/d - 5/256*a^2*cos(d*x + c)/d - 1/24576*a^2*sin(1
2*d*x + 12*c)/d + 1/5120*a^2*sin(10*d*x + 10*c)/d + 7/8192*a^2*sin(8*d*x + 8*c)/d - 1/1024*a^2*sin(6*d*x + 6*c
)/d - 47/8192*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 = 12.33 (sec) , antiderivative size = 518, normalized size of antiderivative = 2.48 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx=\frac {a^2\,\left (\frac {17\,c}{1024}-\frac {17\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{512}-\frac {128\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{231}-\frac {595\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{1536}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{21}+\frac {11097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{2560}+\frac {704\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{63}+\frac {27449\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{2560}-\frac {384\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8}{7}-\frac {202307\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{3840}+\frac {192\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{7}+\frac {28659\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{256}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}-\frac {28659\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{256}-64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+\frac {202307\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{3840}+32\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-\frac {27449\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{2560}-\frac {64\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}}{3}-\frac {11097\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{2560}+\frac {595\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{1536}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{23}}{512}+\frac {17\,d\,x}{1024}+\frac {51\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (c+d\,x\right )}{256}+\frac {561\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (c+d\,x\right )}{512}+\frac {935\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (c+d\,x\right )}{256}+\frac {8415\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (c+d\,x\right )}{1024}+\frac {1683\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (c+d\,x\right )}{128}+\frac {3927\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (c+d\,x\right )}{256}+\frac {1683\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (c+d\,x\right )}{128}+\frac {8415\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (c+d\,x\right )}{1024}+\frac {935\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (c+d\,x\right )}{256}+\frac {561\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}\,\left (c+d\,x\right )}{512}+\frac {51\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{22}\,\left (c+d\,x\right )}{256}+\frac {17\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{24}\,\left (c+d\,x\right )}{1024}-\frac {32}{693}\right )}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{12}} \]

[In]

int(cos(c + d*x)^6*sin(c + d*x)^4*(a + a*sin(c + d*x))^2,x)

[Out]

(a^2*((17*c)/1024 - (17*tan(c/2 + (d*x)/2))/512 - (128*tan(c/2 + (d*x)/2)^2)/231 - (595*tan(c/2 + (d*x)/2)^3)/
1536 - (64*tan(c/2 + (d*x)/2)^4)/21 + (11097*tan(c/2 + (d*x)/2)^5)/2560 + (704*tan(c/2 + (d*x)/2)^6)/63 + (274
49*tan(c/2 + (d*x)/2)^7)/2560 - (384*tan(c/2 + (d*x)/2)^8)/7 - (202307*tan(c/2 + (d*x)/2)^9)/3840 + (192*tan(c
/2 + (d*x)/2)^10)/7 + (28659*tan(c/2 + (d*x)/2)^11)/256 - (64*tan(c/2 + (d*x)/2)^12)/3 - (28659*tan(c/2 + (d*x
)/2)^13)/256 - 64*tan(c/2 + (d*x)/2)^14 + (202307*tan(c/2 + (d*x)/2)^15)/3840 + 32*tan(c/2 + (d*x)/2)^16 - (27
449*tan(c/2 + (d*x)/2)^17)/2560 - (64*tan(c/2 + (d*x)/2)^18)/3 - (11097*tan(c/2 + (d*x)/2)^19)/2560 + (595*tan
(c/2 + (d*x)/2)^21)/1536 + (17*tan(c/2 + (d*x)/2)^23)/512 + (17*d*x)/1024 + (51*tan(c/2 + (d*x)/2)^2*(c + d*x)
)/256 + (561*tan(c/2 + (d*x)/2)^4*(c + d*x))/512 + (935*tan(c/2 + (d*x)/2)^6*(c + d*x))/256 + (8415*tan(c/2 +
(d*x)/2)^8*(c + d*x))/1024 + (1683*tan(c/2 + (d*x)/2)^10*(c + d*x))/128 + (3927*tan(c/2 + (d*x)/2)^12*(c + d*x
))/256 + (1683*tan(c/2 + (d*x)/2)^14*(c + d*x))/128 + (8415*tan(c/2 + (d*x)/2)^16*(c + d*x))/1024 + (935*tan(c
/2 + (d*x)/2)^18*(c + d*x))/256 + (561*tan(c/2 + (d*x)/2)^20*(c + d*x))/512 + (51*tan(c/2 + (d*x)/2)^22*(c + d
*x))/256 + (17*tan(c/2 + (d*x)/2)^24*(c + d*x))/1024 - 32/693))/(d*(tan(c/2 + (d*x)/2)^2 + 1)^12)