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

Optimal. Leaf size=182 \[ -\frac{a^3 \cos ^9(c+d x)}{3 d}+\frac{a^3 \cos ^7(c+d x)}{d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{21 a^3 x}{256} \]

[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)

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Rubi [A]  time = 0.382367, antiderivative size = 182, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 7, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.241, Rules used = {2873, 2565, 14, 2568, 2635, 8, 270} \[ -\frac{a^3 \cos ^9(c+d x)}{3 d}+\frac{a^3 \cos ^7(c+d x)}{d}-\frac{4 a^3 \cos ^5(c+d x)}{5 d}-\frac{a^3 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{7 a^3 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{7 a^3 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{7 a^3 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{21 a^3 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{21 a^3 x}{256} \]

Antiderivative was successfully verified.

[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 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 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 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 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 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 ^4(c+d x) \sin ^3(c+d x) (a+a \sin (c+d x))^3 \, dx &=\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 \operatorname{Subst}\left (\int x^4 \left (1-x^2\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{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 \operatorname{Subst}\left (\int \left (x^4-x^6\right ) \, dx,x,\cos (c+d x)\right )}{d}-\frac{\left (3 a^3\right ) \operatorname{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]  time = 1.04624, size = 116, normalized size = 0.64 \[ \frac{a^3 (-60 \sin (2 (c+d x))-840 \sin (4 (c+d x))+30 \sin (6 (c+d x))+105 \sin (8 (c+d x))-6 \sin (10 (c+d x))-3600 \cos (c+d x)-960 \cos (3 (c+d x))+384 \cos (5 (c+d x))+120 \cos (7 (c+d x))-40 \cos (9 (c+d x))+2700 c+2520 d x)}{30720 d} \]

Antiderivative was successfully verified.

[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)

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Maple [A]  time = 0.046, size = 252, normalized size = 1.4 \begin{align*}{\frac{1}{d} \left ({a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{10}}-{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{32}}+{\frac{\sin \left ( dx+c \right ) }{128} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{256}}+{\frac{3\,c}{256}} \right ) +3\,{a}^{3} \left ( -1/9\, \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{4\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{63}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{315}} \right ) +3\,{a}^{3} \left ( -1/8\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-1/16\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{ \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+3/2\,\cos \left ( dx+c \right ) \right ) \sin \left ( dx+c \right ) }{64}}+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \right ) +{a}^{3} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/d*(a^3*(-1/10*sin(d*x+c)^5*cos(d*x+c)^5-1/16*sin(d*x+c)^3*cos(d*x+c)^5-1/32*sin(d*x+c)*cos(d*x+c)^5+1/128*(c
os(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/256*d*x+3/256*c)+3*a^3*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*sin(d*x+c
)^2*cos(d*x+c)^5-8/315*cos(d*x+c)^5)+3*a^3*(-1/8*sin(d*x+c)^3*cos(d*x+c)^5-1/16*sin(d*x+c)*cos(d*x+c)^5+1/64*(
cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c)+a^3*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)
^5))

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Maxima [A]  time = 1.1205, size = 201, normalized size = 1.1 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 1.6892, size = 327, normalized size = 1.8 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 35.7566, size = 595, normalized size = 3.27 \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)**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))

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Giac [A]  time = 1.39211, size = 235, normalized size = 1.29 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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