3.378 \(\int \cos ^4(c+d x) \sin ^4(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=185 \[ -\frac{2 a^2 \cos ^9(c+d x)}{9 d}+\frac{4 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{9 a^2 x}{256} \]

[Out]

(9*a^2*x)/256 - (2*a^2*Cos[c + d*x]^5)/(5*d) + (4*a^2*Cos[c + d*x]^7)/(7*d) - (2*a^2*Cos[c + d*x]^9)/(9*d) + (
9*a^2*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (3*a^2*Cos[c + d*x]^3*Sin[c + d*x])/(128*d) - (3*a^2*Cos[c + d*x]^5
*Sin[c + d*x])/(32*d) - (3*a^2*Cos[c + d*x]^5*Sin[c + d*x]^3)/(16*d) - (a^2*Cos[c + d*x]^5*Sin[c + d*x]^5)/(10
*d)

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Rubi [A]  time = 0.339205, antiderivative size = 185, normalized size of antiderivative = 1., number of steps used = 16, 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} \[ -\frac{2 a^2 \cos ^9(c+d x)}{9 d}+\frac{4 a^2 \cos ^7(c+d x)}{7 d}-\frac{2 a^2 \cos ^5(c+d x)}{5 d}-\frac{a^2 \sin ^5(c+d x) \cos ^5(c+d x)}{10 d}-\frac{3 a^2 \sin ^3(c+d x) \cos ^5(c+d x)}{16 d}-\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{32 d}+\frac{3 a^2 \sin (c+d x) \cos ^3(c+d x)}{128 d}+\frac{9 a^2 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{9 a^2 x}{256} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

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

Mathematica [A]  time = 0.701515, size = 116, normalized size = 0.63 \[ \frac{a^2 (-1260 \sin (2 (c+d x))-7560 \sin (4 (c+d x))+630 \sin (6 (c+d x))+945 \sin (8 (c+d x))-126 \sin (10 (c+d x))-30240 \cos (c+d x)-6720 \cos (3 (c+d x))+4032 \cos (5 (c+d x))+720 \cos (7 (c+d x))-560 \cos (9 (c+d x))+22680 c+22680 d x)}{645120 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^4*(a + a*Sin[c + d*x])^2,x]

[Out]

(a^2*(22680*c + 22680*d*x - 30240*Cos[c + d*x] - 6720*Cos[3*(c + d*x)] + 4032*Cos[5*(c + d*x)] + 720*Cos[7*(c
+ d*x)] - 560*Cos[9*(c + d*x)] - 1260*Sin[2*(c + d*x)] - 7560*Sin[4*(c + d*x)] + 630*Sin[6*(c + d*x)] + 945*Si
n[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)

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Maple [A]  time = 0.041, size = 218, normalized size = 1.2 \begin{align*}{\frac{1}{d} \left ({a}^{2} \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 ) +2\,{a}^{2} \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 ) +{a}^{2} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{8}}-{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{16}}+{\frac{\sin \left ( dx+c \right ) }{64} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{3\,dx}{128}}+{\frac{3\,c}{128}} \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)^4*(a+a*sin(d*x+c))^2,x)

[Out]

1/d*(a^2*(-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)+2*a^2*(-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)+a^2*(-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*(co
s(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+3/128*d*x+3/128*c))

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Maxima [A]  time = 1.07072, size = 166, normalized size = 0.9 \begin{align*} -\frac{4096 \,{\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^{2} + 63 \,{\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^{2} - 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^{2}}{645120 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/645120*(4096*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^2 + 63*(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^2 - 630*(24*d*x + 24*c + sin(8*d*x + 8*c) - 8*si
n(4*d*x + 4*c))*a^2)/d

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Fricas [A]  time = 1.62081, size = 333, normalized size = 1.8 \begin{align*} -\frac{17920 \, a^{2} \cos \left (d x + c\right )^{9} - 46080 \, a^{2} \cos \left (d x + c\right )^{7} + 32256 \, a^{2} \cos \left (d x + c\right )^{5} - 2835 \, a^{2} d x + 63 \,{\left (128 \, a^{2} \cos \left (d x + c\right )^{9} - 496 \, a^{2} \cos \left (d x + c\right )^{7} + 488 \, a^{2} \cos \left (d x + c\right )^{5} - 30 \, a^{2} \cos \left (d x + c\right )^{3} - 45 \, a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{80640 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^4*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/80640*(17920*a^2*cos(d*x + c)^9 - 46080*a^2*cos(d*x + c)^7 + 32256*a^2*cos(d*x + c)^5 - 2835*a^2*d*x + 63*(
128*a^2*cos(d*x + c)^9 - 496*a^2*cos(d*x + c)^7 + 488*a^2*cos(d*x + c)^5 - 30*a^2*cos(d*x + c)^3 - 45*a^2*cos(
d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 35.374, size = 554, normalized size = 2.99 \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)**4*(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((3*a**2*x*sin(c + d*x)**10/256 + 15*a**2*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 3*a**2*x*sin(c + d*
x)**8/128 + 15*a**2*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 3*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 15*a
**2*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 9*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a**2*x*sin(c + d*
x)**2*cos(c + d*x)**8/256 + 3*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 3*a**2*x*cos(c + d*x)**10/256 + 3*a*
*2*x*cos(c + d*x)**8/128 + 3*a**2*sin(c + d*x)**9*cos(c + d*x)/(256*d) + 7*a**2*sin(c + d*x)**7*cos(c + d*x)**
3/(128*d) + 3*a**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) - a**2*sin(c + d*x)**5*cos(c + d*x)**5/(10*d) + 11*a**
2*sin(c + d*x)**5*cos(c + d*x)**3/(128*d) - 2*a**2*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 7*a**2*sin(c + d*x)
**3*cos(c + d*x)**7/(128*d) - 11*a**2*sin(c + d*x)**3*cos(c + d*x)**5/(128*d) - 8*a**2*sin(c + d*x)**2*cos(c +
d*x)**7/(35*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**9/(256*d) - 3*a**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) -
16*a**2*cos(c + d*x)**9/(315*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*sin(c)**4*cos(c)**4, True))

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Giac [A]  time = 1.41972, size = 235, normalized size = 1.27 \begin{align*} \frac{9}{256} \, a^{2} x - \frac{a^{2} \cos \left (9 \, d x + 9 \, c\right )}{1152 \, d} + \frac{a^{2} \cos \left (7 \, d x + 7 \, c\right )}{896 \, d} + \frac{a^{2} \cos \left (5 \, d x + 5 \, c\right )}{160 \, d} - \frac{a^{2} \cos \left (3 \, d x + 3 \, c\right )}{96 \, d} - \frac{3 \, a^{2} \cos \left (d x + c\right )}{64 \, d} - \frac{a^{2} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{3 \, a^{2} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} + \frac{a^{2} \sin \left (6 \, d x + 6 \, c\right )}{1024 \, d} - \frac{3 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} - \frac{a^{2} \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)^4*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

9/256*a^2*x - 1/1152*a^2*cos(9*d*x + 9*c)/d + 1/896*a^2*cos(7*d*x + 7*c)/d + 1/160*a^2*cos(5*d*x + 5*c)/d - 1/
96*a^2*cos(3*d*x + 3*c)/d - 3/64*a^2*cos(d*x + c)/d - 1/5120*a^2*sin(10*d*x + 10*c)/d + 3/2048*a^2*sin(8*d*x +
8*c)/d + 1/1024*a^2*sin(6*d*x + 6*c)/d - 3/256*a^2*sin(4*d*x + 4*c)/d - 1/512*a^2*sin(2*d*x + 2*c)/d