\(\int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx\) [605]

Optimal result

Integrand size = 29, antiderivative size = 224 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27 a^3 x}{1024}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^{13}(c+d x)}{13 d}+\frac {27 a^3 \cos (c+d x) \sin (c+d x)}{1024 d}+\frac {9 a^3 \cos ^3(c+d x) \sin (c+d x)}{512 d}+\frac {9 a^3 \cos ^5(c+d x) \sin (c+d x)}{640 d}-\frac {27 a^3 \cos ^7(c+d x) \sin (c+d x)}{320 d}-\frac {9 a^3 \cos ^7(c+d x) \sin ^3(c+d x)}{40 d}-\frac {a^3 \cos ^7(c+d x) \sin ^5(c+d x)}{4 d} \]

[Out]

27/1024*a^3*x-4/7*a^3*cos(d*x+c)^7/d+a^3*cos(d*x+c)^9/d-6/11*a^3*cos(d*x+c)^11/d+1/13*a^3*cos(d*x+c)^13/d+27/1
024*a^3*cos(d*x+c)*sin(d*x+c)/d+9/512*a^3*cos(d*x+c)^3*sin(d*x+c)/d+9/640*a^3*cos(d*x+c)^5*sin(d*x+c)/d-27/320
*a^3*cos(d*x+c)^7*sin(d*x+c)/d-9/40*a^3*cos(d*x+c)^7*sin(d*x+c)^3/d-1/4*a^3*cos(d*x+c)^7*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.00, number of steps used = 21, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {2952, 2648, 2715, 8, 2645, 276} \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 \cos ^{13}(c+d x)}{13 d}-\frac {6 a^3 \cos ^{11}(c+d x)}{11 d}+\frac {a^3 \cos ^9(c+d x)}{d}-\frac {4 a^3 \cos ^7(c+d x)}{7 d}-\frac {a^3 \sin ^5(c+d x) \cos ^7(c+d x)}{4 d}-\frac {9 a^3 \sin ^3(c+d x) \cos ^7(c+d x)}{40 d}-\frac {27 a^3 \sin (c+d x) \cos ^7(c+d x)}{320 d}+\frac {9 a^3 \sin (c+d x) \cos ^5(c+d x)}{640 d}+\frac {9 a^3 \sin (c+d x) \cos ^3(c+d x)}{512 d}+\frac {27 a^3 \sin (c+d x) \cos (c+d x)}{1024 d}+\frac {27 a^3 x}{1024} \]

[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 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^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 \text {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 ) \text {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 \text {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 ) \text {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] (verified)

Time = 1.24 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.65 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (720720 c+1081080 d x-1401400 \cos (c+d x)-450450 \cos (3 (c+d x))+150150 \cos (5 (c+d x))+94380 \cos (7 (c+d x))-20020 \cos (9 (c+d x))-11830 \cos (11 (c+d x))+770 \cos (13 (c+d x))+80080 \sin (2 (c+d x))-385385 \sin (4 (c+d x))-40040 \sin (6 (c+d x))+65065 \sin (8 (c+d x))+8008 \sin (10 (c+d x))-5005 \sin (12 (c+d x)))}{41000960 d} \]

[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)

Maple [A] (verified)

Time = 1.45 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.69

method	result	size
parallelrisch	\(-\frac {a^{3} \left (-1081080 d x +1401400 \cos \left (d x +c \right )-150150 \cos \left (5 d x +5 c \right )+450450 \cos \left (3 d x +3 c \right )+20020 \cos \left (9 d x +9 c \right )+11830 \cos \left (11 d x +11 c \right )-8008 \sin \left (10 d x +10 c \right )+5005 \sin \left (12 d x +12 c \right )-770 \cos \left (13 d x +13 c \right )-65065 \sin \left (8 d x +8 c \right )-94380 \cos \left (7 d x +7 c \right )+40040 \sin \left (6 d x +6 c \right )+385385 \sin \left (4 d x +4 c \right )-80080 \sin \left (2 d x +2 c \right )+1638400\right )}{41000960 d}\)	\(155\)
risch	\(-\frac {a^{3} \cos \left (9 d x +9 c \right )}{2048 d}-\frac {13 a^{3} \cos \left (11 d x +11 c \right )}{45056 d}+\frac {27 a^{3} x}{1024}-\frac {35 a^{3} \cos \left (d x +c \right )}{1024 d}+\frac {a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a^{3} \sin \left (12 d x +12 c \right )}{8192 d}+\frac {a^{3} \cos \left (13 d x +13 c \right )}{53248 d}+\frac {13 a^{3} \sin \left (8 d x +8 c \right )}{8192 d}+\frac {33 a^{3} \cos \left (7 d x +7 c \right )}{14336 d}-\frac {a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {15 a^{3} \cos \left (5 d x +5 c \right )}{4096 d}-\frac {77 a^{3} \sin \left (4 d x +4 c \right )}{8192 d}-\frac {45 a^{3} \cos \left (3 d x +3 c \right )}{4096 d}+\frac {a^{3} \sin \left (2 d x +2 c \right )}{512 d}\)	\(226\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{13}-\frac {6 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{143}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{429}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\)	\(308\)
default	\(\frac {a^{3} \left (-\frac {\left (\sin ^{6}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{13}-\frac {6 \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{143}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{429}-\frac {16 \left (\cos ^{7}\left (d x +c \right )\right )}{3003}\right )+3 a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{12}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{24}-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{64}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{384}+\frac {5 d x}{1024}+\frac {5 c}{1024}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{11}-\frac {4 \left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{99}-\frac {8 \left (\cos ^{7}\left (d x +c \right )\right )}{693}\right )+a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{7}\left (d x +c \right )\right )}{10}-\frac {3 \left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{80}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{160}+\frac {3 d x}{256}+\frac {3 c}{256}\right )}{d}\)	\(308\)

[In]

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

[Out]

-1/41000960*a^3*(-1081080*d*x+1401400*cos(d*x+c)-150150*cos(5*d*x+5*c)+450450*cos(3*d*x+3*c)+20020*cos(9*d*x+9
*c)+11830*cos(11*d*x+11*c)-8008*sin(10*d*x+10*c)+5005*sin(12*d*x+12*c)-770*cos(13*d*x+13*c)-65065*sin(8*d*x+8*
c)-94380*cos(7*d*x+7*c)+40040*sin(6*d*x+6*c)+385385*sin(4*d*x+4*c)-80080*sin(2*d*x+2*c)+1638400)/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.67 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {394240 \, a^{3} \cos \left (d x + c\right )^{13} - 2795520 \, a^{3} \cos \left (d x + c\right )^{11} + 5125120 \, a^{3} \cos \left (d x + c\right )^{9} - 2928640 \, a^{3} \cos \left (d x + c\right )^{7} + 135135 \, a^{3} d x - 1001 \, {\left (1280 \, a^{3} \cos \left (d x + c\right )^{11} - 3712 \, a^{3} \cos \left (d x + c\right )^{9} + 2864 \, a^{3} \cos \left (d x + c\right )^{7} - 72 \, a^{3} \cos \left (d x + c\right )^{5} - 90 \, a^{3} \cos \left (d x + c\right )^{3} - 135 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{5125120 \, d} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 748 vs. \(2 (212) = 424\).

Time = 3.48 (sec) , antiderivative size = 748, normalized size of antiderivative = 3.34 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {15 a^{3} x \sin ^{12}{\left (c + d x \right )}}{1024} + \frac {45 a^{3} x \sin ^{10}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{512} + \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {225 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{1024} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {75 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {225 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{1024} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {45 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{10}{\left (c + d x \right )}}{512} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \cos ^{12}{\left (c + d x \right )}}{1024} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} \sin ^{11}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{1024 d} + \frac {85 a^{3} \sin ^{9}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{1024 d} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {99 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{512 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{6}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {99 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{512 d} + \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} - \frac {2 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {85 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{1024 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{231 d} - \frac {4 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{21 d} - \frac {15 a^{3} \sin {\left (c + d x \right )} \cos ^{11}{\left (c + d x \right )}}{1024 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {16 a^{3} \cos ^{13}{\left (c + d x \right )}}{3003 d} - \frac {8 a^{3} \cos ^{11}{\left (c + d x \right )}}{231 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \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))**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))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 184, normalized size of antiderivative = 0.82 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {40960 \, {\left (231 \, \cos \left (d x + c\right )^{13} - 819 \, \cos \left (d x + c\right )^{11} + 1001 \, \cos \left (d x + c\right )^{9} - 429 \, \cos \left (d x + c\right )^{7}\right )} a^{3} - 532480 \, {\left (63 \, \cos \left (d x + c\right )^{11} - 154 \, \cos \left (d x + c\right )^{9} + 99 \, \cos \left (d x + c\right )^{7}\right )} a^{3} + 12012 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} + 120 \, d x + 120 \, c + 5 \, \sin \left (8 \, d x + 8 \, c\right ) - 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} + 15015 \, {\left (4 \, \sin \left (4 \, d x + 4 \, c\right )^{3} + 120 \, d x + 120 \, c + 9 \, \sin \left (8 \, d x + 8 \, c\right ) - 48 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{123002880 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.37 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.00 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {27}{1024} \, a^{3} x + \frac {a^{3} \cos \left (13 \, d x + 13 \, c\right )}{53248 \, d} - \frac {13 \, a^{3} \cos \left (11 \, d x + 11 \, c\right )}{45056 \, d} - \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{2048 \, d} + \frac {33 \, a^{3} \cos \left (7 \, d x + 7 \, c\right )}{14336 \, d} + \frac {15 \, a^{3} \cos \left (5 \, d x + 5 \, c\right )}{4096 \, d} - \frac {45 \, a^{3} \cos \left (3 \, d x + 3 \, c\right )}{4096 \, d} - \frac {35 \, a^{3} \cos \left (d x + c\right )}{1024 \, d} - \frac {a^{3} \sin \left (12 \, d x + 12 \, c\right )}{8192 \, d} + \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {13 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{8192 \, d} - \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {77 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{8192 \, d} + \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 14.78 (sec) , antiderivative size = 612, normalized size of antiderivative = 2.73 \[ \int \cos ^6(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^3 \, dx=\text {Too large to display} \]

[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 - ((27*a^3*(c + d*x))/1024 + (171*a^3*tan(c/2 + (d*x)/2)^3)/256 - (1603*a^3*tan(c/2 + (d*x)/2)
^5)/640 - (59523*a^3*tan(c/2 + (d*x)/2)^7)/1280 + (305539*a^3*tan(c/2 + (d*x)/2)^9)/2560 - (93973*a^3*tan(c/2
+ (d*x)/2)^11)/640 + (93973*a^3*tan(c/2 + (d*x)/2)^15)/640 - (305539*a^3*tan(c/2 + (d*x)/2)^17)/2560 + (59523*
a^3*tan(c/2 + (d*x)/2)^19)/1280 + (1603*a^3*tan(c/2 + (d*x)/2)^21)/640 - (171*a^3*tan(c/2 + (d*x)/2)^23)/256 -
(27*a^3*tan(c/2 + (d*x)/2)^25)/512 - a^3*((27*c)/1024 + (27*d*x)/1024 - 80/1001) + tan(c/2 + (d*x)/2)^2*((351
*a^3*(c + d*x))/1024 - a^3*((351*c)/1024 + (351*d*x)/1024 - 80/77)) + tan(c/2 + (d*x)/2)^4*((1053*a^3*(c + d*x
))/512 - a^3*((1053*c)/512 + (1053*d*x)/512 - 480/77)) + tan(c/2 + (d*x)/2)^20*((3861*a^3*(c + d*x))/512 - a^3
*((3861*c)/512 + (3861*d*x)/512 - 32)) + tan(c/2 + (d*x)/2)^6*((3861*a^3*(c + d*x))/512 - a^3*((3861*c)/512 +
(3861*d*x)/512 + 64/7)) + tan(c/2 + (d*x)/2)^14*((11583*a^3*(c + d*x))/256 - a^3*((11583*c)/256 + (11583*d*x)/
256 - 320)) + tan(c/2 + (d*x)/2)^12*((11583*a^3*(c + d*x))/256 - a^3*((11583*c)/256 + (11583*d*x)/256 + 1280/7
)) + tan(c/2 + (d*x)/2)^18*((19305*a^3*(c + d*x))/1024 - a^3*((19305*c)/1024 + (19305*d*x)/1024 - 16)) + tan(c
/2 + (d*x)/2)^8*((19305*a^3*(c + d*x))/1024 - a^3*((19305*c)/1024 + (19305*d*x)/1024 - 288/7)) + tan(c/2 + (d*
x)/2)^16*((34749*a^3*(c + d*x))/1024 - a^3*((34749*c)/1024 + (34749*d*x)/1024 + 48)) + tan(c/2 + (d*x)/2)^10*(
(34749*a^3*(c + d*x))/1024 - a^3*((34749*c)/1024 + (34749*d*x)/1024 - 1056/7)) + (27*a^3*tan(c/2 + (d*x)/2))/5
12)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^13)