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

Optimal. Leaf size=165 \[ -\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]

[Out]

3/256*a*x-1/7*a*cos(d*x+c)^7/d+2/9*a*cos(d*x+c)^9/d-1/11*a*cos(d*x+c)^11/d+3/256*a*cos(d*x+c)*sin(d*x+c)/d+1/1
28*a*cos(d*x+c)^3*sin(d*x+c)/d+1/160*a*cos(d*x+c)^5*sin(d*x+c)/d-3/80*a*cos(d*x+c)^7*sin(d*x+c)/d-1/10*a*cos(d
*x+c)^7*sin(d*x+c)^3/d

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Rubi [A]  time = 0.20, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2838, 2568, 2635, 8, 2565, 270} \[ -\frac {a \cos ^{11}(c+d x)}{11 d}+\frac {2 a \cos ^9(c+d x)}{9 d}-\frac {a \cos ^7(c+d x)}{7 d}-\frac {a \sin ^3(c+d x) \cos ^7(c+d x)}{10 d}-\frac {3 a \sin (c+d x) \cos ^7(c+d x)}{80 d}+\frac {a \sin (c+d x) \cos ^5(c+d x)}{160 d}+\frac {a \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {3 a \sin (c+d x) \cos (c+d x)}{256 d}+\frac {3 a x}{256} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(3*a*x)/256 - (a*Cos[c + d*x]^7)/(7*d) + (2*a*Cos[c + d*x]^9)/(9*d) - (a*Cos[c + d*x]^11)/(11*d) + (3*a*Cos[c
+ d*x]*Sin[c + d*x])/(256*d) + (a*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(160*
d) - (3*a*Cos[c + d*x]^7*Sin[c + d*x])/(80*d) - (a*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 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]

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 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 2838

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_.)*((a_) + (b_.)*sin[(e_.) + (f_.)
*(x_)]), x_Symbol] :> Dist[a, Int[(g*Cos[e + f*x])^p*(d*Sin[e + f*x])^n, x], x] + Dist[b/d, Int[(g*Cos[e + f*x
])^p*(d*Sin[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, g, n, p}, x]

Rubi steps

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

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Mathematica [A]  time = 0.57, size = 121, normalized size = 0.73 \[ \frac {a (13860 \sin (2 (c+d x))-27720 \sin (4 (c+d x))-6930 \sin (6 (c+d x))+3465 \sin (8 (c+d x))+1386 \sin (10 (c+d x))-69300 \cos (c+d x)-23100 \cos (3 (c+d x))+6930 \cos (5 (c+d x))+4950 \cos (7 (c+d x))-770 \cos (9 (c+d x))-630 \cos (11 (c+d x))+83160 d x)}{7096320 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(83160*d*x - 69300*Cos[c + d*x] - 23100*Cos[3*(c + d*x)] + 6930*Cos[5*(c + d*x)] + 4950*Cos[7*(c + d*x)] -
770*Cos[9*(c + d*x)] - 630*Cos[11*(c + d*x)] + 13860*Sin[2*(c + d*x)] - 27720*Sin[4*(c + d*x)] - 6930*Sin[6*(c
+ d*x)] + 3465*Sin[8*(c + d*x)] + 1386*Sin[10*(c + d*x)]))/(7096320*d)

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fricas [A]  time = 0.70, size = 106, normalized size = 0.64 \[ -\frac {80640 \, a \cos \left (d x + c\right )^{11} - 197120 \, a \cos \left (d x + c\right )^{9} + 126720 \, a \cos \left (d x + c\right )^{7} - 10395 \, a d x - 693 \, {\left (128 \, a \cos \left (d x + c\right )^{9} - 176 \, a \cos \left (d x + c\right )^{7} + 8 \, a \cos \left (d x + c\right )^{5} + 10 \, a \cos \left (d x + c\right )^{3} + 15 \, a \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{887040 \, d} \]

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)),x, algorithm="fricas")

[Out]

-1/887040*(80640*a*cos(d*x + c)^11 - 197120*a*cos(d*x + c)^9 + 126720*a*cos(d*x + c)^7 - 10395*a*d*x - 693*(12
8*a*cos(d*x + c)^9 - 176*a*cos(d*x + c)^7 + 8*a*cos(d*x + c)^5 + 10*a*cos(d*x + c)^3 + 15*a*cos(d*x + c))*sin(
d*x + c))/d

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giac [A]  time = 0.30, size = 167, normalized size = 1.01 \[ \frac {3}{256} \, a x - \frac {a \cos \left (11 \, d x + 11 \, c\right )}{11264 \, d} - \frac {a \cos \left (9 \, d x + 9 \, c\right )}{9216 \, d} + \frac {5 \, a \cos \left (7 \, d x + 7 \, c\right )}{7168 \, d} + \frac {a \cos \left (5 \, d x + 5 \, c\right )}{1024 \, d} - \frac {5 \, a \cos \left (3 \, d x + 3 \, c\right )}{1536 \, d} - \frac {5 \, a \cos \left (d x + c\right )}{512 \, d} + \frac {a \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {a \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac {a \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {a \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac {a \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

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)),x, algorithm="giac")

[Out]

3/256*a*x - 1/11264*a*cos(11*d*x + 11*c)/d - 1/9216*a*cos(9*d*x + 9*c)/d + 5/7168*a*cos(7*d*x + 7*c)/d + 1/102
4*a*cos(5*d*x + 5*c)/d - 5/1536*a*cos(3*d*x + 3*c)/d - 5/512*a*cos(d*x + c)/d + 1/5120*a*sin(10*d*x + 10*c)/d
+ 1/2048*a*sin(8*d*x + 8*c)/d - 1/1024*a*sin(6*d*x + 6*c)/d - 1/256*a*sin(4*d*x + 4*c)/d + 1/512*a*sin(2*d*x +
2*c)/d

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maple [A]  time = 0.24, size = 134, normalized size = 0.81 \[ \frac {a \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d} \]

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)),x)

[Out]

1/d*(a*(-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*(-1/10*sin(d*x+c)
^3*cos(d*x+c)^7-3/80*cos(d*x+c)^7*sin(d*x+c)+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 = 0.38, size = 86, normalized size = 0.52 \[ -\frac {10240 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a - 693 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a}{7096320 \, d} \]

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)),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 - 693*(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)/d

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mupad [B]  time = 11.87, size = 447, normalized size = 2.71 \[ \frac {3\,a\,x}{256}+\frac {\frac {3\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{21}}{128}+\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{4}-\frac {3323\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{640}+\left (\frac {a\,\left (1715175\,c+1715175\,d\,x-9461760\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{5}+\left (\frac {a\,\left (3430350\,c+3430350\,d\,x+23654400\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-\frac {841\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{64}+\left (\frac {a\,\left (4802490\,c+4802490\,d\,x-52039680\right )}{887040}-\frac {693\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+\left (\frac {a\,\left (4802490\,c+4802490\,d\,x+42577920\right )}{887040}-\frac {693\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+\frac {841\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}+\left (\frac {a\,\left (3430350\,c+3430350\,d\,x-30412800\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{128}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-\frac {54\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{5}+\left (\frac {a\,\left (1715175\,c+1715175\,d\,x+6082560\right )}{887040}-\frac {495\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+\frac {3323\,a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{640}+\left (\frac {a\,\left (571725\,c+571725\,d\,x-1126400\right )}{887040}-\frac {165\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-\frac {a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{4}+\left (\frac {a\,\left (114345\,c+114345\,d\,x-225280\right )}{887040}-\frac {33\,a\,\left (c+d\,x\right )}{256}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-\frac {3\,a\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{128}+\frac {a\,\left (10395\,c+10395\,d\,x-20480\right )}{887040}-\frac {3\,a\,\left (c+d\,x\right )}{256}}{d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^{11}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(3*a*x)/256 + ((a*(10395*c + 10395*d*x - 20480))/887040 - (3*a*tan(c/2 + (d*x)/2))/128 - (3*a*(c + d*x))/256 +
tan(c/2 + (d*x)/2)^2*((a*(114345*c + 114345*d*x - 225280))/887040 - (33*a*(c + d*x))/256) + tan(c/2 + (d*x)/2
)^4*((a*(571725*c + 571725*d*x - 1126400))/887040 - (165*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^6*((a*(1715175
*c + 1715175*d*x + 6082560))/887040 - (495*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^16*((a*(1715175*c + 1715175*
d*x - 9461760))/887040 - (495*a*(c + d*x))/256) + tan(c/2 + (d*x)/2)^14*((a*(3430350*c + 3430350*d*x + 2365440
0))/887040 - (495*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^8*((a*(3430350*c + 3430350*d*x - 30412800))/887040 -
(495*a*(c + d*x))/128) + tan(c/2 + (d*x)/2)^10*((a*(4802490*c + 4802490*d*x + 42577920))/887040 - (693*a*(c +
d*x))/128) + tan(c/2 + (d*x)/2)^12*((a*(4802490*c + 4802490*d*x - 52039680))/887040 - (693*a*(c + d*x))/128) -
(a*tan(c/2 + (d*x)/2)^3)/4 + (3323*a*tan(c/2 + (d*x)/2)^5)/640 - (54*a*tan(c/2 + (d*x)/2)^7)/5 + (841*a*tan(c
/2 + (d*x)/2)^9)/64 - (841*a*tan(c/2 + (d*x)/2)^13)/64 + (54*a*tan(c/2 + (d*x)/2)^15)/5 - (3323*a*tan(c/2 + (d
*x)/2)^17)/640 + (a*tan(c/2 + (d*x)/2)^19)/4 + (3*a*tan(c/2 + (d*x)/2)^21)/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^
11)

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sympy [A]  time = 44.94, size = 318, normalized size = 1.93 \[ \begin {cases} \frac {3 a x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {15 a x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {15 a x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {3 a x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {3 a \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {a \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {a \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {7 a \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {4 a \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {3 a \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {8 a \cos ^{11}{\left (c + d x \right )}}{693 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\relax (c )} + a\right ) \sin ^{4}{\relax (c )} \cos ^{6}{\relax (c )} & \text {otherwise} \end {cases} \]

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)),x)

[Out]

Piecewise((3*a*x*sin(c + d*x)**10/256 + 15*a*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 15*a*x*sin(c + d*x)**6*co
s(c + d*x)**4/128 + 15*a*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 15*a*x*sin(c + d*x)**2*cos(c + d*x)**8/256 +
3*a*x*cos(c + d*x)**10/256 + 3*a*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a*sin(c + d*x)**7*cos(c + d*x)**3/(1
28*d) + a*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) - a*sin(c + d*x)**4*cos(c + d*x)**7/(7*d) - 7*a*sin(c + d*x)*
*3*cos(c + d*x)**7/(128*d) - 4*a*sin(c + d*x)**2*cos(c + d*x)**9/(63*d) - 3*a*sin(c + d*x)*cos(c + d*x)**9/(25
6*d) - 8*a*cos(c + d*x)**11/(693*d), Ne(d, 0)), (x*(a*sin(c) + a)*sin(c)**4*cos(c)**6, True))

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