\(\int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx\) [607]

Optimal result

Integrand size = 29, antiderivative size = 183 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19 a^3 x}{256}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d} \]

[Out]

19/256*a^3*x-4/7*a^3*cos(d*x+c)^7/d+5/9*a^3*cos(d*x+c)^9/d-1/11*a^3*cos(d*x+c)^11/d+19/256*a^3*cos(d*x+c)*sin(
d*x+c)/d+19/384*a^3*cos(d*x+c)^3*sin(d*x+c)/d+19/480*a^3*cos(d*x+c)^5*sin(d*x+c)/d-19/80*a^3*cos(d*x+c)^7*sin(
d*x+c)/d-3/10*a^3*cos(d*x+c)^7*sin(d*x+c)^3/d

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {2952, 2648, 2715, 8, 2645, 14, 276} \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {3 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {19 a^3 \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {19 a^3 \sin (c+d x) \cos ^5(c+d x)}{480 d}+\frac {19 a^3 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac {19 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {19 a^3 x}{256} \]

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(19*a^3*x)/256 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (5*a^3*Cos[c + d*x]^9)/(9*d) - (a^3*Cos[c + d*x]^11)/(11*d) +
(19*a^3*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (19*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(384*d) + (19*a^3*Cos[c + d*
x]^5*Sin[c + d*x])/(480*d) - (19*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (3*a^3*Cos[c + d*x]^7*Sin[c + d*x]^
3)/(10*d)

Rule 8

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

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^3 \cos ^6(c+d x) \sin ^2(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^4(c+d x)+a^3 \cos ^6(c+d x) \sin ^5(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^3(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx \\ & = -\frac {a^3 \cos ^7(c+d x) \sin (c+d x)}{8 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{8} a^3 \int \cos ^6(c+d x) \, dx+\frac {1}{10} \left (9 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^6 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = \frac {a^3 \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{48} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{80} \left (9 a^3\right ) \int \cos ^6(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^6-x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{64} \left (5 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{32} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {5 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{128} \left (5 a^3\right ) \int 1 \, dx+\frac {1}{128} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = \frac {5 a^3 x}{128}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}+\frac {1}{256} \left (9 a^3\right ) \int 1 \, dx \\ & = \frac {19 a^3 x}{256}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {5 a^3 \cos ^9(c+d x)}{9 d}-\frac {a^3 \cos ^{11}(c+d x)}{11 d}+\frac {19 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {19 a^3 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac {19 a^3 \cos ^5(c+d x) \sin (c+d x)}{480 d}-\frac {19 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac {3 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (415800 c+526680 d x-568260 \cos (c+d x)-244860 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+40590 \cos (7 (c+d x))+8470 \cos (9 (c+d x))-630 \cos (11 (c+d x))+152460 \sin (2 (c+d x))-138600 \sin (4 (c+d x))-57750 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+4158 \sin (10 (c+d x)))}{7096320 d} \]

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^2*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(415800*c + 526680*d*x - 568260*Cos[c + d*x] - 244860*Cos[3*(c + d*x)] + 6930*Cos[5*(c + d*x)] + 40590*Co
s[7*(c + d*x)] + 8470*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 152460*Sin[2*(c + d*x)] - 138600*Sin[4*(c + d
*x)] - 57750*Sin[6*(c + d*x)] + 3465*Sin[8*(c + d*x)] + 4158*Sin[10*(c + d*x)]))/(7096320*d)

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.73

method	result	size
parallelrisch	\(-\frac {a^{3} \left (-526680 d x +568260 \cos \left (d x +c \right )-6930 \cos \left (5 d x +5 c \right )+244860 \cos \left (3 d x +3 c \right )+630 \cos \left (11 d x +11 c \right )-4158 \sin \left (10 d x +10 c \right )-8470 \cos \left (9 d x +9 c \right )-3465 \sin \left (8 d x +8 c \right )-40590 \cos \left (7 d x +7 c \right )+57750 \sin \left (6 d x +6 c \right )+138600 \sin \left (4 d x +4 c \right )-152460 \sin \left (2 d x +2 c \right )+757760\right )}{7096320 d}\)	\(133\)
risch	\(-\frac {a^{3} \cos \left (11 d x +11 c \right )}{11264 d}+\frac {19 a^{3} x}{256}-\frac {41 a^{3} \cos \left (d x +c \right )}{512 d}+\frac {3 a^{3} \sin \left (10 d x +10 c \right )}{5120 d}+\frac {11 a^{3} \cos \left (9 d x +9 c \right )}{9216 d}+\frac {a^{3} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {41 a^{3} \cos \left (7 d x +7 c \right )}{7168 d}-\frac {25 a^{3} \sin \left (6 d x +6 c \right )}{3072 d}+\frac {a^{3} \cos \left (5 d x +5 c \right )}{1024 d}-\frac {5 a^{3} \sin \left (4 d x +4 c \right )}{256 d}-\frac {53 a^{3} \cos \left (3 d x +3 c \right )}{1536 d}+\frac {11 a^{3} \sin \left (2 d x +2 c \right )}{512 d}\)	\(192\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\)	\(236\)
default	\(\frac {a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )+a^{3} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\)	\(236\)

[In]

int(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-1/7096320*a^3*(-526680*d*x+568260*cos(d*x+c)-6930*cos(5*d*x+5*c)+244860*cos(3*d*x+3*c)+630*cos(11*d*x+11*c)-4
158*sin(10*d*x+10*c)-8470*cos(9*d*x+9*c)-3465*sin(8*d*x+8*c)-40590*cos(7*d*x+7*c)+57750*sin(6*d*x+6*c)+138600*
sin(4*d*x+4*c)-152460*sin(2*d*x+2*c)+757760)/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.68 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {80640 \, a^{3} \cos \left (d x + c\right )^{11} - 492800 \, a^{3} \cos \left (d x + c\right )^{9} + 506880 \, a^{3} \cos \left (d x + c\right )^{7} - 65835 \, a^{3} d x - 231 \, {\left (1152 \, a^{3} \cos \left (d x + c\right )^{9} - 2064 \, a^{3} \cos \left (d x + c\right )^{7} + 152 \, a^{3} \cos \left (d x + c\right )^{5} + 190 \, a^{3} \cos \left (d x + c\right )^{3} + 285 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

-1/887040*(80640*a^3*cos(d*x + c)^11 - 492800*a^3*cos(d*x + c)^9 + 506880*a^3*cos(d*x + c)^7 - 65835*a^3*d*x -
231*(1152*a^3*cos(d*x + c)^9 - 2064*a^3*cos(d*x + c)^7 + 152*a^3*cos(d*x + c)^5 + 190*a^3*cos(d*x + c)^3 + 28
5*a^3*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. 597 vs. \(2 (175) = 350\).

Time = 1.85 (sec) , antiderivative size = 597, normalized size of antiderivative = 3.26 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {9 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {45 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {5 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {45 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {9 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {5 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {9 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {21 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {5 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {3 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {55 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} - \frac {a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {21 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {73 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {3 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {5 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \cos ^{11}{\left (c + d x \right )}}{693 d} - \frac {2 a^{3} \cos ^{9}{\left (c + d x \right )}}{21 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{2}{\left (c \right )} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**2*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((9*a**3*x*sin(c + d*x)**10/256 + 45*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 5*a**3*x*sin(c + d*
x)**8/128 + 45*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 5*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 45*a
**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 45*a**3*x*sin(c + d
*x)**2*cos(c + d*x)**8/256 + 5*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 9*a**3*x*cos(c + d*x)**10/256 + 5*a
**3*x*cos(c + d*x)**8/128 + 9*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 21*a**3*sin(c + d*x)**7*cos(c + d*x)
**3/(128*d) + 5*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 3*a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 55
*a**3*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) - a**3*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 21*a**3*sin(c + d
*x)**3*cos(c + d*x)**7/(128*d) + 73*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(384*d) - 4*a**3*sin(c + d*x)**2*cos(
c + d*x)**9/(63*d) - 3*a**3*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 9*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d
) - 5*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 8*a**3*cos(c + d*x)**11/(693*d) - 2*a**3*cos(c + d*x)**9/(21
*d), Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**2*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.90 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {10240 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 337920 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 2079 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 2310 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{7096320 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

-1/7096320*(10240*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 - 337920*(7*cos(d*x + c)^9
- 9*cos(d*x + c)^7)*a^3 - 2079*(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^3 - 2310*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 24*sin(4*d*x + 4*c))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.04 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19}{256} \, a^{3} x - \frac {a^{3} \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} + \frac {11 \, a^{3} \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {41 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {53 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {41 \, a^{3} \cos \left (d x + c\right )}{512 \, d} + \frac {3 \, a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {25 \, a^{3} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac {5 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {11 \, a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^2*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

19/256*a^3*x - 1/11264*a^3*cos(11*d*x + 11*c)/d + 11/9216*a^3*cos(9*d*x + 9*c)/d + 41/7168*a^3*cos(7*d*x + 7*c
)/d + 1/1024*a^3*cos(5*d*x + 5*c)/d - 53/1536*a^3*cos(3*d*x + 3*c)/d - 41/512*a^3*cos(d*x + c)/d + 3/5120*a^3*
sin(10*d*x + 10*c)/d + 1/2048*a^3*sin(8*d*x + 8*c)/d - 25/3072*a^3*sin(6*d*x + 6*c)/d - 5/256*a^3*sin(4*d*x +
4*c)/d + 11/512*a^3*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 15.04 (sec) , antiderivative size = 543, normalized size of antiderivative = 2.97 \[ \int \cos ^6(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {19\,a^3\,x}{256}-\frac {\frac {19\,a^3\,\left (c+d\,x\right )}{256}-\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{12}-\frac {32417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{1920}+\frac {466\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{15}-\frac {2937\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\frac {2937\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}-\frac {466\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{15}+\frac {32417\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{1920}+\frac {13\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{12}-\frac {19\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}-a^3\,\left (\frac {19\,c}{256}+\frac {19\,d\,x}{256}-\frac {148}{693}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {209\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {209\,c}{256}+\frac {209\,d\,x}{256}-\frac {148}{63}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {1045\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {1045\,c}{256}+\frac {1045\,d\,x}{256}-12\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {1045\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {1045\,c}{256}+\frac {1045\,d\,x}{256}+\frac {16}{63}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {3135\,c}{128}+\frac {3135\,d\,x}{128}-\frac {16}{3}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {3135\,c}{256}+\frac {3135\,d\,x}{256}-\frac {44}{3}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {3135\,c}{128}+\frac {3135\,d\,x}{128}-\frac {456}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {3135\,a^3\,\left (c+d\,x\right )}{256}-a^3\,\left (\frac {3135\,c}{256}+\frac {3135\,d\,x}{256}-\frac {144}{7}\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {4389\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {4389\,c}{128}+\frac {4389\,d\,x}{128}+24\right )\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {4389\,a^3\,\left (c+d\,x\right )}{128}-a^3\,\left (\frac {4389\,c}{128}+\frac {4389\,d\,x}{128}-\frac {368}{3}\right )\right )+\frac {19\,a^3\,\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 )}^{11}} \]

[In]

int(cos(c + d*x)^6*sin(c + d*x)^2*(a + a*sin(c + d*x))^3,x)

[Out]

(19*a^3*x)/256 - ((19*a^3*(c + d*x))/256 - (13*a^3*tan(c/2 + (d*x)/2)^3)/12 - (32417*a^3*tan(c/2 + (d*x)/2)^5)
/1920 + (466*a^3*tan(c/2 + (d*x)/2)^7)/15 - (2937*a^3*tan(c/2 + (d*x)/2)^9)/64 + (2937*a^3*tan(c/2 + (d*x)/2)^
13)/64 - (466*a^3*tan(c/2 + (d*x)/2)^15)/15 + (32417*a^3*tan(c/2 + (d*x)/2)^17)/1920 + (13*a^3*tan(c/2 + (d*x)
/2)^19)/12 - (19*a^3*tan(c/2 + (d*x)/2)^21)/128 - a^3*((19*c)/256 + (19*d*x)/256 - 148/693) + tan(c/2 + (d*x)/
2)^2*((209*a^3*(c + d*x))/256 - a^3*((209*c)/256 + (209*d*x)/256 - 148/63)) + tan(c/2 + (d*x)/2)^18*((1045*a^3
*(c + d*x))/256 - a^3*((1045*c)/256 + (1045*d*x)/256 - 12)) + tan(c/2 + (d*x)/2)^4*((1045*a^3*(c + d*x))/256 -
a^3*((1045*c)/256 + (1045*d*x)/256 + 16/63)) + tan(c/2 + (d*x)/2)^14*((3135*a^3*(c + d*x))/128 - a^3*((3135*c
)/128 + (3135*d*x)/128 - 16/3)) + tan(c/2 + (d*x)/2)^16*((3135*a^3*(c + d*x))/256 - a^3*((3135*c)/256 + (3135*
d*x)/256 - 44/3)) + tan(c/2 + (d*x)/2)^8*((3135*a^3*(c + d*x))/128 - a^3*((3135*c)/128 + (3135*d*x)/128 - 456/
7)) + tan(c/2 + (d*x)/2)^6*((3135*a^3*(c + d*x))/256 - a^3*((3135*c)/256 + (3135*d*x)/256 - 144/7)) + tan(c/2
+ (d*x)/2)^10*((4389*a^3*(c + d*x))/128 - a^3*((4389*c)/128 + (4389*d*x)/128 + 24)) + tan(c/2 + (d*x)/2)^12*((
4389*a^3*(c + d*x))/128 - a^3*((4389*c)/128 + (4389*d*x)/128 - 368/3)) + (19*a^3*tan(c/2 + (d*x)/2))/128)/(d*(
tan(c/2 + (d*x)/2)^2 + 1)^11)