3.605 \(\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\)

Optimal. Leaf size=224 \[ \frac{a^3 \cos ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac{9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac{27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac{9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac{27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{27 a^3 x}{1024} \]

[Out]

(27*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (a^3*Cos[c + d*x]^9)/d - (6*a^3*Cos[c + d*x]^11)/(11*d) + (a^
3*Cos[c + d*x]^13)/(13*d) + (27*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/
(512*d) + (9*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(640*d) - (27*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (9*a^3*
Cos[c + d*x]^7*Sin[c + d*x]^3)/(40*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(4*d)

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Rubi [A]  time = 0.422832, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 21, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207, Rules used = {2873, 2568, 2635, 8, 2565, 270} \[ \frac{a^3 \cos ^{13}(c+d x)}{13 d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}-\frac{a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac{9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac{27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac{9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac{9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac{27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac{27 a^3 x}{1024} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

(27*a^3*x)/1024 - (4*a^3*Cos[c + d*x]^7)/(7*d) + (a^3*Cos[c + d*x]^9)/d - (6*a^3*Cos[c + d*x]^11)/(11*d) + (a^
3*Cos[c + d*x]^13)/(13*d) + (27*a^3*Cos[c + d*x]*Sin[c + d*x])/(1024*d) + (9*a^3*Cos[c + d*x]^3*Sin[c + d*x])/
(512*d) + (9*a^3*Cos[c + d*x]^5*Sin[c + d*x])/(640*d) - (27*a^3*Cos[c + d*x]^7*Sin[c + d*x])/(320*d) - (9*a^3*
Cos[c + d*x]^7*Sin[c + d*x]^3)/(40*d) - (a^3*Cos[c + d*x]^7*Sin[c + d*x]^5)/(4*d)

Rule 2873

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]

Rule 2568

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 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2565

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 270

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]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx &=\int \left (a^3 \cos ^6(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^5(c+d x)+3 a^3 \cos ^6(c+d x) \sin ^6(c+d x)+a^3 \cos ^6(c+d x) \sin ^7(c+d x)\right ) \, dx\\ &=a^3 \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx+a^3 \int \cos ^6(c+d x) \sin ^7(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^5(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^6(c+d x) \, dx\\ &=-\frac{a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{10 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac{1}{10} \left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx+\frac{1}{4} \left (5 a^3\right ) \int \cos ^6(c+d x) \sin ^4(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^3 \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int x^6 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{3 a^3 \cos ^7(c+d x) \sin (c+d x)}{80 d}-\frac{9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac{1}{80} \left (3 a^3\right ) \int \cos ^6(c+d x) \, dx+\frac{1}{8} \left (3 a^3\right ) \int \cos ^6(c+d x) \sin ^2(c+d x) \, dx-\frac{a^3 \operatorname{Subst}\left (\int \left (x^6-3 x^8+3 x^{10}-x^{12}\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{Subst}\left (\int \left (x^6-2 x^8+x^{10}\right ) \, dx,x,\cos (c+d x)\right )}{d}\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^{13}(c+d x)}{13 d}+\frac{a^3 \cos ^5(c+d x) \sin (c+d x)}{160 d}-\frac{27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac{1}{32} a^3 \int \cos ^4(c+d x) \, dx+\frac{1}{64} \left (3 a^3\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^{13}(c+d x)}{13 d}+\frac{a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}+\frac{9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac{27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac{1}{128} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac{1}{128} \left (5 a^3\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^{13}(c+d x)}{13 d}+\frac{3 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac{9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac{27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac{1}{256} \left (3 a^3\right ) \int 1 \, dx+\frac{1}{512} \left (15 a^3\right ) \int \cos ^2(c+d x) \, dx\\ &=\frac{3 a^3 x}{256}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^{13}(c+d x)}{13 d}+\frac{27 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac{9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac{27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}+\frac{\left (15 a^3\right ) \int 1 \, dx}{1024}\\ &=\frac{27 a^3 x}{1024}-\frac{4 a^3 \cos ^7(c+d x)}{7 d}+\frac{a^3 \cos ^9(c+d x)}{d}-\frac{6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac{a^3 \cos ^{13}(c+d x)}{13 d}+\frac{27 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac{9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac{9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac{27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac{9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac{a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d}\\ \end{align*}

Mathematica [A]  time = 2.16765, size = 146, normalized size = 0.65 \[ \frac{a^3 (80080 \sin (2 (c+d x))-385385 \sin (4 (c+d x))-40040 \sin (6 (c+d x))+65065 \sin (8 (c+d x))+8008 \sin (10 (c+d x))-5005 \sin (12 (c+d x))-1401400 \cos (c+d x)-450450 \cos (3 (c+d x))+150150 \cos (5 (c+d x))+94380 \cos (7 (c+d x))-20020 \cos (9 (c+d x))-11830 \cos (11 (c+d x))+770 \cos (13 (c+d x))+720720 c+1081080 d x)}{41000960 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^3,x]

[Out]

(a^3*(720720*c + 1081080*d*x - 1401400*Cos[c + d*x] - 450450*Cos[3*(c + d*x)] + 150150*Cos[5*(c + d*x)] + 9438
0*Cos[7*(c + d*x)] - 20020*Cos[9*(c + d*x)] - 11830*Cos[11*(c + d*x)] + 770*Cos[13*(c + d*x)] + 80080*Sin[2*(c
+ d*x)] - 385385*Sin[4*(c + d*x)] - 40040*Sin[6*(c + d*x)] + 65065*Sin[8*(c + d*x)] + 8008*Sin[10*(c + d*x)]
- 5005*Sin[12*(c + d*x)]))/(41000960*d)

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Maple [A]  time = 0.046, size = 308, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{6} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{13}}-{\frac{6\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{143}}-{\frac{8\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{429}}-{\frac{16\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{3003}} \right ) +3\,{a}^{3} \left ( -1/12\, \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-1/24\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{64}}+{\frac{\sin \left ( dx+c \right ) }{384} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+5/4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{1024}}+{\frac{5\,c}{1024}} \right ) +3\,{a}^{3} \left ( -1/11\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{99}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{693}} \right ) +{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{10}}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{80}}+{\frac{\sin \left ( dx+c \right ) }{160} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x)

[Out]

1/d*(a^3*(-1/13*sin(d*x+c)^6*cos(d*x+c)^7-6/143*sin(d*x+c)^4*cos(d*x+c)^7-8/429*sin(d*x+c)^2*cos(d*x+c)^7-16/3
003*cos(d*x+c)^7)+3*a^3*(-1/12*sin(d*x+c)^5*cos(d*x+c)^7-1/24*sin(d*x+c)^3*cos(d*x+c)^7-1/64*sin(d*x+c)*cos(d*
x+c)^7+1/384*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/1024*d*x+5/1024*c)+3*a^3*(-1/11*sin(
d*x+c)^4*cos(d*x+c)^7-4/99*sin(d*x+c)^2*cos(d*x+c)^7-8/693*cos(d*x+c)^7)+a^3*(-1/10*sin(d*x+c)^3*cos(d*x+c)^7-
3/80*sin(d*x+c)*cos(d*x+c)^7+1/160*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*
c))

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Maxima [A]  time = 1.02185, size = 248, normalized size = 1.11 \begin{align*} \frac{40960 \,{\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 532480 \,{\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} + 12012 \,{\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} + 15015 \,{\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^{3}}{123002880 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/123002880*(40960*(231*cos(d*x + c)^13 - 819*cos(d*x + c)^11 + 1001*cos(d*x + c)^9 - 429*cos(d*x + c)^7)*a^3
- 532480*(63*cos(d*x + c)^11 - 154*cos(d*x + c)^9 + 99*cos(d*x + c)^7)*a^3 + 12012*(32*sin(2*d*x + 2*c)^5 + 12
0*d*x + 120*c + 5*sin(8*d*x + 8*c) - 40*sin(4*d*x + 4*c))*a^3 + 15015*(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^3)/d

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Fricas [A]  time = 1.38722, size = 428, normalized size = 1.91 \begin{align*} \frac{394240 \, a^{3} \cos \left (d x + c\right )^{13} - 2795520 \, a^{3} \cos \left (d x + c\right )^{11} + 5125120 \, a^{3} \cos \left (d x + c\right )^{9} - 2928640 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, a^{3} d x - 1001 \,{\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 3712 \, a^{3} \cos \left (d x + c\right )^{9} + 2864 \, a^{3} \cos \left (d x + c\right )^{7} - 72 \, a^{3} \cos \left (d x + c\right )^{5} - 90 \, a^{3} \cos \left (d x + c\right )^{3} - 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{5125120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="fricas")

[Out]

1/5125120*(394240*a^3*cos(d*x + c)^13 - 2795520*a^3*cos(d*x + c)^11 + 5125120*a^3*cos(d*x + c)^9 - 2928640*a^3
*cos(d*x + c)^7 + 135135*a^3*d*x - 1001*(1280*a^3*cos(d*x + c)^11 - 3712*a^3*cos(d*x + c)^9 + 2864*a^3*cos(d*x
+ c)^7 - 72*a^3*cos(d*x + c)^5 - 90*a^3*cos(d*x + c)^3 - 135*a^3*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 127.425, size = 748, normalized size = 3.34 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*sin(d*x+c)**4*(a+a*sin(d*x+c))**3,x)

[Out]

Piecewise((15*a**3*x*sin(c + d*x)**12/1024 + 45*a**3*x*sin(c + d*x)**10*cos(c + d*x)**2/512 + 3*a**3*x*sin(c +
d*x)**10/256 + 225*a**3*x*sin(c + d*x)**8*cos(c + d*x)**4/1024 + 15*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/25
6 + 75*a**3*x*sin(c + d*x)**6*cos(c + d*x)**6/256 + 15*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 225*a**3*x
*sin(c + d*x)**4*cos(c + d*x)**8/1024 + 15*a**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 45*a**3*x*sin(c + d*x)
**2*cos(c + d*x)**10/512 + 15*a**3*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 15*a**3*x*cos(c + d*x)**12/1024 + 3
*a**3*x*cos(c + d*x)**10/256 + 15*a**3*sin(c + d*x)**11*cos(c + d*x)/(1024*d) + 85*a**3*sin(c + d*x)**9*cos(c
+ d*x)**3/(1024*d) + 3*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 99*a**3*sin(c + d*x)**7*cos(c + d*x)**5/(51
2*d) + 7*a**3*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) - a**3*sin(c + d*x)**6*cos(c + d*x)**7/(7*d) - 99*a**3*s
in(c + d*x)**5*cos(c + d*x)**7/(512*d) + a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) - 2*a**3*sin(c + d*x)**4*
cos(c + d*x)**9/(21*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 85*a**3*sin(c + d*x)**3*cos(c + d*x)**
9/(1024*d) - 7*a**3*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 8*a**3*sin(c + d*x)**2*cos(c + d*x)**11/(231*d)
- 4*a**3*sin(c + d*x)**2*cos(c + d*x)**9/(21*d) - 15*a**3*sin(c + d*x)*cos(c + d*x)**11/(1024*d) - 3*a**3*sin(
c + d*x)*cos(c + d*x)**9/(256*d) - 16*a**3*cos(c + d*x)**13/(3003*d) - 8*a**3*cos(c + d*x)**11/(231*d), Ne(d,
0)), (x*(a*sin(c) + a)**3*sin(c)**4*cos(c)**6, True))

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Giac [A]  time = 1.39393, size = 304, normalized size = 1.36 \begin{align*} \frac{27}{1024} \, a^{3} x + \frac{a^{3} \cos \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac{13 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{45056 \, d} - \frac{a^{3} \cos \left (9 \, d x + 9 \, c\right )}{2048 \, d} + \frac{33 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{14336 \, d} + \frac{15 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac{45 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{4096 \, d} - \frac{35 \, a^{3} \cos \left (d x + c\right )}{1024 \, d} - \frac{a^{3} \sin \left (12 \, d x + 12 \, c\right )}{8192 \, d} + \frac{a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{13 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac{a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{77 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac{a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*sin(d*x+c)^4*(a+a*sin(d*x+c))^3,x, algorithm="giac")

[Out]

27/1024*a^3*x + 1/53248*a^3*cos(13*d*x + 13*c)/d - 13/45056*a^3*cos(11*d*x + 11*c)/d - 1/2048*a^3*cos(9*d*x +
9*c)/d + 33/14336*a^3*cos(7*d*x + 7*c)/d + 15/4096*a^3*cos(5*d*x + 5*c)/d - 45/4096*a^3*cos(3*d*x + 3*c)/d - 3
5/1024*a^3*cos(d*x + c)/d - 1/8192*a^3*sin(12*d*x + 12*c)/d + 1/5120*a^3*sin(10*d*x + 10*c)/d + 13/8192*a^3*si
n(8*d*x + 8*c)/d - 1/1024*a^3*sin(6*d*x + 6*c)/d - 77/8192*a^3*sin(4*d*x + 4*c)/d + 1/512*a^3*sin(2*d*x + 2*c)
/d