\(\int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx\) [393]

Optimal result

Integrand size = 29, antiderivative size = 182 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {21 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \]

[Out]

21/256*a^3*x-4/5*a^3*cos(d*x+c)^5/d+a^3*cos(d*x+c)^7/d-1/3*a^3*cos(d*x+c)^9/d+21/256*a^3*cos(d*x+c)*sin(d*x+c)
/d+7/128*a^3*cos(d*x+c)^3*sin(d*x+c)/d-7/32*a^3*cos(d*x+c)^5*sin(d*x+c)/d-7/16*a^3*cos(d*x+c)^5*sin(d*x+c)^3/d
-1/10*a^3*cos(d*x+c)^5*sin(d*x+c)^5/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 182, 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, 2645, 14, 2648, 2715, 8, 276} \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}-\frac {a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac {7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac {7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac {7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac {21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac {21 a^3 x}{256} \]

[In]

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

[Out]

(21*a^3*x)/256 - (4*a^3*Cos[c + d*x]^5)/(5*d) + (a^3*Cos[c + d*x]^7)/d - (a^3*Cos[c + d*x]^9)/(3*d) + (21*a^3*
Cos[c + d*x]*Sin[c + d*x])/(256*d) + (7*a^3*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (7*a^3*Cos[c + d*x]^5*Sin[c
+ d*x])/(32*d) - (7*a^3*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^3*Cos[c + d*x]^5*Sin[c + d*x]^5)/(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 ^4(c+d x) \sin ^3(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^4(c+d x)+3 a^3 \cos ^4(c+d x) \sin ^5(c+d x)+a^3 \cos ^4(c+d x) \sin ^6(c+d x)\right ) \, dx \\ & = a^3 \int \cos ^4(c+d x) \sin ^3(c+d x) \, dx+a^3 \int \cos ^4(c+d x) \sin ^6(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+\left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^5(c+d x) \, dx \\ & = -\frac {3 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{8 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{2} a^3 \int \cos ^4(c+d x) \sin ^4(c+d x) \, dx+\frac {1}{8} \left (9 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int x^4 \left (1-x^2\right )^2 \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {3 a^3 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{16} \left (3 a^3\right ) \int \cos ^4(c+d x) \, dx+\frac {1}{16} \left (3 a^3\right ) \int \cos ^4(c+d x) \sin ^2(c+d x) \, dx-\frac {a^3 \text {Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac {\left (3 a^3\right ) \text {Subst}\left (\int \left (x^4-2 x^6+x^8\right ) \, dx,x,\cos (c+d x)\right )}{d} \\ & = -\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {3 a^3 \cos ^3(c+d x) \sin (c+d x)}{64 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{32} a^3 \int \cos ^4(c+d x) \, dx+\frac {1}{64} \left (9 a^3\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {9 a^3 \cos (c+d x) \sin (c+d x)}{128 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{128} \left (3 a^3\right ) \int \cos ^2(c+d x) \, dx+\frac {1}{128} \left (9 a^3\right ) \int 1 \, dx \\ & = \frac {9 a^3 x}{128}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {21 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d}+\frac {1}{256} \left (3 a^3\right ) \int 1 \, dx \\ & = \frac {21 a^3 x}{256}-\frac {4 a^3 \cos ^5(c+d x)}{5 d}+\frac {a^3 \cos ^7(c+d x)}{d}-\frac {a^3 \cos ^9(c+d x)}{3 d}+\frac {21 a^3 \cos (c+d x) \sin (c+d x)}{256 d}+\frac {7 a^3 \cos ^3(c+d x) \sin (c+d x)}{128 d}-\frac {7 a^3 \cos ^5(c+d x) \sin (c+d x)}{32 d}-\frac {7 a^3 \cos ^5(c+d x) \sin ^3(c+d x)}{16 d}-\frac {a^3 \cos ^5(c+d x) \sin ^5(c+d x)}{10 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.58 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.64 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {a^3 (2700 c+2520 d x-3600 \cos (c+d x)-960 \cos (3 (c+d x))+384 \cos (5 (c+d x))+120 \cos (7 (c+d x))-40 \cos (9 (c+d x))-60 \sin (2 (c+d x))-840 \sin (4 (c+d x))+30 \sin (6 (c+d x))+105 \sin (8 (c+d x))-6 \sin (10 (c+d x)))}{30720 d} \]

[In]

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

[Out]

(a^3*(2700*c + 2520*d*x - 3600*Cos[c + d*x] - 960*Cos[3*(c + d*x)] + 384*Cos[5*(c + d*x)] + 120*Cos[7*(c + d*x
)] - 40*Cos[9*(c + d*x)] - 60*Sin[2*(c + d*x)] - 840*Sin[4*(c + d*x)] + 30*Sin[6*(c + d*x)] + 105*Sin[8*(c + d
*x)] - 6*Sin[10*(c + d*x)]))/(30720*d)

Maple [A] (verified)

Time = 0.67 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.66

method	result	size
parallelrisch	\(-\frac {\left (-42 d x +\sin \left (2 d x +2 c \right )+14 \sin \left (4 d x +4 c \right )-\frac {\sin \left (6 d x +6 c \right )}{2}-\frac {7 \sin \left (8 d x +8 c \right )}{4}+\frac {\sin \left (10 d x +10 c \right )}{10}+60 \cos \left (d x +c \right )+16 \cos \left (3 d x +3 c \right )-\frac {32 \cos \left (5 d x +5 c \right )}{5}-2 \cos \left (7 d x +7 c \right )+\frac {2 \cos \left (9 d x +9 c \right )}{3}+\frac {1024}{15}\right ) a^{3}}{512 d}\)	\(120\)
risch	\(\frac {21 a^{3} x}{256}-\frac {15 a^{3} \cos \left (d x +c \right )}{128 d}-\frac {a^{3} \sin \left (10 d x +10 c \right )}{5120 d}-\frac {a^{3} \cos \left (9 d x +9 c \right )}{768 d}+\frac {7 a^{3} \sin \left (8 d x +8 c \right )}{2048 d}+\frac {a^{3} \cos \left (7 d x +7 c \right )}{256 d}+\frac {a^{3} \sin \left (6 d x +6 c \right )}{1024 d}+\frac {a^{3} \cos \left (5 d x +5 c \right )}{80 d}-\frac {7 a^{3} \sin \left (4 d x +4 c \right )}{256 d}-\frac {a^{3} \cos \left (3 d x +3 c \right )}{32 d}-\frac {a^{3} \sin \left (2 d x +2 c \right )}{512 d}\)	\(175\)
derivativedivides	\(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\)	\(252\)
default	\(\frac {a^{3} \left (-\frac {\left (\sin ^{5}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{10}-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{32}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{128}+\frac {3 d x}{256}+\frac {3 c}{256}\right )+3 a^{3} \left (-\frac {\left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{9}-\frac {4 \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{63}-\frac {8 \left (\cos ^{5}\left (d x +c \right )\right )}{315}\right )+3 a^{3} \left (-\frac {\left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{8}-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{16}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{64}+\frac {3 d x}{128}+\frac {3 c}{128}\right )+a^{3} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )}{d}\)	\(252\)

[In]

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

[Out]

-1/512*(-42*d*x+sin(2*d*x+2*c)+14*sin(4*d*x+4*c)-1/2*sin(6*d*x+6*c)-7/4*sin(8*d*x+8*c)+1/10*sin(10*d*x+10*c)+6
0*cos(d*x+c)+16*cos(3*d*x+3*c)-32/5*cos(5*d*x+5*c)-2*cos(7*d*x+7*c)+2/3*cos(9*d*x+9*c)+1024/15)*a^3/d

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.68 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {1280 \, a^{3} \cos \left (d x + c\right )^{9} - 3840 \, a^{3} \cos \left (d x + c\right )^{7} + 3072 \, a^{3} \cos \left (d x + c\right )^{5} - 315 \, a^{3} d x + 3 \, {\left (128 \, a^{3} \cos \left (d x + c\right )^{9} - 816 \, a^{3} \cos \left (d x + c\right )^{7} + 968 \, a^{3} \cos \left (d x + c\right )^{5} - 70 \, a^{3} \cos \left (d x + c\right )^{3} - 105 \, a^{3} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{3840 \, d} \]

[In]

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

[Out]

-1/3840*(1280*a^3*cos(d*x + c)^9 - 3840*a^3*cos(d*x + c)^7 + 3072*a^3*cos(d*x + c)^5 - 315*a^3*d*x + 3*(128*a^
3*cos(d*x + c)^9 - 816*a^3*cos(d*x + c)^7 + 968*a^3*cos(d*x + c)^5 - 70*a^3*cos(d*x + c)^3 - 105*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. 595 vs. \(2 (172) = 344\).

Time = 1.37 (sec) , antiderivative size = 595, normalized size of antiderivative = 3.27 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\begin {cases} \frac {3 a^{3} x \sin ^{10}{\left (c + d x \right )}}{256} + \frac {15 a^{3} x \sin ^{8}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {15 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{128} + \frac {9 a^{3} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{128} + \frac {27 a^{3} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{8}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {3 a^{3} x \cos ^{10}{\left (c + d x \right )}}{256} + \frac {9 a^{3} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {3 a^{3} \sin ^{9}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{256 d} + \frac {7 a^{3} \sin ^{7}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} + \frac {9 a^{3} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} - \frac {a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{10 d} + \frac {33 a^{3} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{128 d} - \frac {3 a^{3} \sin ^{4}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {7 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {33 a^{3} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{128 d} - \frac {12 a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {a^{3} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {3 a^{3} \sin {\left (c + d x \right )} \cos ^{9}{\left (c + d x \right )}}{256 d} - \frac {9 a^{3} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} - \frac {8 a^{3} \cos ^{9}{\left (c + d x \right )}}{105 d} - \frac {2 a^{3} \cos ^{7}{\left (c + d x \right )}}{35 d} & \text {for}\: d \neq 0 \\x \left (a \sin {\left (c \right )} + a\right )^{3} \sin ^{3}{\left (c \right )} \cos ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((3*a**3*x*sin(c + d*x)**10/256 + 15*a**3*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 9*a**3*x*sin(c + d*
x)**8/128 + 15*a**3*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 9*a**3*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a
**3*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 27*a**3*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a**3*x*sin(c + d
*x)**2*cos(c + d*x)**8/256 + 9*a**3*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*a**3*x*cos(c + d*x)**10/256 + 9*a
**3*x*cos(c + d*x)**8/128 + 3*a**3*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**3*sin(c + d*x)**7*cos(c + d*x)*
*3/(128*d) + 9*a**3*sin(c + d*x)**7*cos(c + d*x)/(128*d) - a**3*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 33*a*
*3*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 3*a**3*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 7*a**3*sin(c + d*x
)**3*cos(c + d*x)**7/(128*d) - 33*a**3*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 12*a**3*sin(c + d*x)**2*cos(c
+ d*x)**7/(35*d) - a**3*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**3*sin(c + d*x)*cos(c + d*x)**9/(256*d) -
9*a**3*sin(c + d*x)*cos(c + d*x)**7/(128*d) - 8*a**3*cos(c + d*x)**9/(105*d) - 2*a**3*cos(c + d*x)**7/(35*d),
Ne(d, 0)), (x*(a*sin(c) + a)**3*sin(c)**3*cos(c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 149, normalized size of antiderivative = 0.82 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=-\frac {2048 \, {\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{3} - 6144 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{3} + 21 \, {\left (32 \, \sin \left (2 \, d x + 2 \, c\right )^{5} - 120 \, d x - 120 \, c - 5 \, \sin \left (8 \, d x + 8 \, c\right ) + 40 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3} - 630 \, {\left (24 \, d x + 24 \, c + \sin \left (8 \, d x + 8 \, c\right ) - 8 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{3}}{215040 \, d} \]

[In]

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

[Out]

-1/215040*(2048*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^3 - 6144*(5*cos(d*x + c)^7 - 7*c
os(d*x + c)^5)*a^3 + 21*(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 - 630*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*sin(4*d*x + 4*c))*a^3)/d

Giac [A] (verification not implemented)

none

Time = 0.62 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21}{256} \, a^{3} x - \frac {a^{3} \cos \left (9 \, d x + 9 \, c\right )}{768 \, d} + \frac {a^{3} \cos \left (7 \, d x + 7 \, c\right )}{256 \, d} + \frac {a^{3} \cos \left (5 \, d x + 5 \, c\right )}{80 \, d} - \frac {a^{3} \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {15 \, a^{3} \cos \left (d x + c\right )}{128 \, d} - \frac {a^{3} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac {7 \, a^{3} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac {a^{3} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac {7 \, a^{3} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac {a^{3} \sin \left (2 \, d x + 2 \, c\right )}{512 \, d} \]

[In]

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

[Out]

21/256*a^3*x - 1/768*a^3*cos(9*d*x + 9*c)/d + 1/256*a^3*cos(7*d*x + 7*c)/d + 1/80*a^3*cos(5*d*x + 5*c)/d - 1/3
2*a^3*cos(3*d*x + 3*c)/d - 15/128*a^3*cos(d*x + c)/d - 1/5120*a^3*sin(10*d*x + 10*c)/d + 7/2048*a^3*sin(8*d*x
+ 8*c)/d + 1/1024*a^3*sin(6*d*x + 6*c)/d - 7/256*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 = 11.89 (sec) , antiderivative size = 572, normalized size of antiderivative = 3.14 \[ \int \cos ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx=\frac {21\,a^3\,x}{256}-\frac {\frac {203\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{128}-\frac {1973\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{160}-\frac {463\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{32}+\frac {3231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9}{64}-\frac {3231\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}}{64}+\frac {463\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}}{32}+\frac {1973\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}}{160}-\frac {203\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}}{128}-\frac {21\,a^3\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{19}}{128}+\frac {a^3\,\left (315\,c+315\,d\,x\right )}{3840}-\frac {a^3\,\left (315\,c+315\,d\,x-1024\right )}{3840}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{384}-\frac {a^3\,\left (3150\,c+3150\,d\,x\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{384}-\frac {a^3\,\left (3150\,c+3150\,d\,x-10240\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{256}-\frac {a^3\,\left (14175\,c+14175\,d\,x-15360\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {3\,a^3\,\left (315\,c+315\,d\,x\right )}{256}-\frac {a^3\,\left (14175\,c+14175\,d\,x-30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^3\,\left (37800\,c+37800\,d\,x+30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (\frac {7\,a^3\,\left (315\,c+315\,d\,x\right )}{128}-\frac {a^3\,\left (66150\,c+66150\,d\,x+30720\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (\frac {a^3\,\left (315\,c+315\,d\,x\right )}{32}-\frac {a^3\,\left (37800\,c+37800\,d\,x-153600\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (\frac {21\,a^3\,\left (315\,c+315\,d\,x\right )}{320}-\frac {a^3\,\left (79380\,c+79380\,d\,x-129024\right )}{3840}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (\frac {7\,a^3\,\left (315\,c+315\,d\,x\right )}{128}-\frac {a^3\,\left (66150\,c+66150\,d\,x-245760\right )}{3840}\right )+\frac {21\,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 )}^{10}} \]

[In]

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

[Out]

(21*a^3*x)/256 - ((203*a^3*tan(c/2 + (d*x)/2)^3)/128 - (1973*a^3*tan(c/2 + (d*x)/2)^5)/160 - (463*a^3*tan(c/2
+ (d*x)/2)^7)/32 + (3231*a^3*tan(c/2 + (d*x)/2)^9)/64 - (3231*a^3*tan(c/2 + (d*x)/2)^11)/64 + (463*a^3*tan(c/2
+ (d*x)/2)^13)/32 + (1973*a^3*tan(c/2 + (d*x)/2)^15)/160 - (203*a^3*tan(c/2 + (d*x)/2)^17)/128 - (21*a^3*tan(
c/2 + (d*x)/2)^19)/128 + (a^3*(315*c + 315*d*x))/3840 - (a^3*(315*c + 315*d*x - 1024))/3840 + tan(c/2 + (d*x)/
2)^18*((a^3*(315*c + 315*d*x))/384 - (a^3*(3150*c + 3150*d*x))/3840) + tan(c/2 + (d*x)/2)^2*((a^3*(315*c + 315
*d*x))/384 - (a^3*(3150*c + 3150*d*x - 10240))/3840) + tan(c/2 + (d*x)/2)^16*((3*a^3*(315*c + 315*d*x))/256 -
(a^3*(14175*c + 14175*d*x - 15360))/3840) + tan(c/2 + (d*x)/2)^4*((3*a^3*(315*c + 315*d*x))/256 - (a^3*(14175*
c + 14175*d*x - 30720))/3840) + tan(c/2 + (d*x)/2)^6*((a^3*(315*c + 315*d*x))/32 - (a^3*(37800*c + 37800*d*x +
30720))/3840) + tan(c/2 + (d*x)/2)^12*((7*a^3*(315*c + 315*d*x))/128 - (a^3*(66150*c + 66150*d*x + 30720))/38
40) + tan(c/2 + (d*x)/2)^14*((a^3*(315*c + 315*d*x))/32 - (a^3*(37800*c + 37800*d*x - 153600))/3840) + tan(c/2
+ (d*x)/2)^10*((21*a^3*(315*c + 315*d*x))/320 - (a^3*(79380*c + 79380*d*x - 129024))/3840) + tan(c/2 + (d*x)/
2)^8*((7*a^3*(315*c + 315*d*x))/128 - (a^3*(66150*c + 66150*d*x - 245760))/3840) + (21*a^3*tan(c/2 + (d*x)/2))
/128)/(d*(tan(c/2 + (d*x)/2)^2 + 1)^10)