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

Optimal. Leaf size=187 \[ -\frac{11 a^4 \cos ^7(c+d x)}{112 d}-\frac{\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4 \sin (c+d x)+a^4\right )}{144 d}+\frac{11 a^4 \sin (c+d x) \cos ^5(c+d x)}{96 d}+\frac{55 a^4 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{55 a^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{55 a^4 x}{256}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d} \]

[Out]

(55*a^4*x)/256 - (11*a^4*Cos[c + d*x]^7)/(112*d) + (55*a^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (55*a^4*Cos[c
+ d*x]^3*Sin[c + d*x])/(384*d) + (11*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(96*d) - (Cos[c + d*x]^5*(a + a*Sin[c +
d*x])^5)/(10*a*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x])^2)/(18*d) - (11*Cos[c + d*x]^7*(a^4 + a^4*Sin[c +
d*x]))/(144*d)

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Rubi [A]  time = 0.234233, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172, Rules used = {2870, 2678, 2669, 2635, 8} \[ -\frac{11 a^4 \cos ^7(c+d x)}{112 d}-\frac{\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4 \sin (c+d x)+a^4\right )}{144 d}+\frac{11 a^4 \sin (c+d x) \cos ^5(c+d x)}{96 d}+\frac{55 a^4 \sin (c+d x) \cos ^3(c+d x)}{384 d}+\frac{55 a^4 \sin (c+d x) \cos (c+d x)}{256 d}+\frac{55 a^4 x}{256}-\frac{\cos ^5(c+d x) (a \sin (c+d x)+a)^5}{10 a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(55*a^4*x)/256 - (11*a^4*Cos[c + d*x]^7)/(112*d) + (55*a^4*Cos[c + d*x]*Sin[c + d*x])/(256*d) + (55*a^4*Cos[c
+ d*x]^3*Sin[c + d*x])/(384*d) + (11*a^4*Cos[c + d*x]^5*Sin[c + d*x])/(96*d) - (Cos[c + d*x]^5*(a + a*Sin[c +
d*x])^5)/(10*a*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c + d*x])^2)/(18*d) - (11*Cos[c + d*x]^7*(a^4 + a^4*Sin[c +
d*x]))/(144*d)

Rule 2870

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*sin[(e_.) + (f_.)*(x_)]^2*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> -Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(2*b*f*g*(m + 1)), x] + Dist[a/(2
*g^2), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] &&
EqQ[a^2 - b^2, 0] && EqQ[m - p, 0]

Rule 2678

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m - 1))/(f*g*(m + p)), x] + Dist[(a*(2*m + p - 1))/(m + p), Int[(
g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0]
&& GtQ[m, 0] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
e + f*x])^(p + 1))/(f*g*(p + 1)), x] + Dist[a, Int[(g*Cos[e + f*x])^p, x], x] /; FreeQ[{a, b, e, f, g, p}, x]
&& (IntegerQ[2*p] || NeQ[a^2 - b^2, 0])

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]

Rubi steps

\begin{align*} \int \cos ^4(c+d x) \sin ^2(c+d x) (a+a \sin (c+d x))^4 \, dx &=-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}+\frac{1}{2} a \int \cos ^6(c+d x) (a+a \sin (c+d x))^3 \, dx\\ &=-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}+\frac{1}{18} \left (11 a^2\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\\ &=-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac{1}{16} \left (11 a^3\right ) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{11 a^4 \cos ^7(c+d x)}{112 d}-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac{1}{16} \left (11 a^4\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac{11 a^4 \cos ^7(c+d x)}{112 d}+\frac{11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac{1}{96} \left (55 a^4\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{11 a^4 \cos ^7(c+d x)}{112 d}+\frac{55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac{11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac{1}{128} \left (55 a^4\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{11 a^4 \cos ^7(c+d x)}{112 d}+\frac{55 a^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac{11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}+\frac{1}{256} \left (55 a^4\right ) \int 1 \, dx\\ &=\frac{55 a^4 x}{256}-\frac{11 a^4 \cos ^7(c+d x)}{112 d}+\frac{55 a^4 \cos (c+d x) \sin (c+d x)}{256 d}+\frac{55 a^4 \cos ^3(c+d x) \sin (c+d x)}{384 d}+\frac{11 a^4 \cos ^5(c+d x) \sin (c+d x)}{96 d}-\frac{\cos ^5(c+d x) (a+a \sin (c+d x))^5}{10 a d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )^2}{18 d}-\frac{11 \cos ^7(c+d x) \left (a^4+a^4 \sin (c+d x)\right )}{144 d}\\ \end{align*}

Mathematica [A]  time = 1.17601, size = 116, normalized size = 0.62 \[ \frac{a^4 (8820 \sin (2 (c+d x))-42840 \sin (4 (c+d x))-2730 \sin (6 (c+d x))+4095 \sin (8 (c+d x))-126 \sin (10 (c+d x))-181440 \cos (c+d x)-53760 \cos (3 (c+d x))+16128 \cos (5 (c+d x))+7200 \cos (7 (c+d x))-1120 \cos (9 (c+d x))+136080 c+138600 d x)}{645120 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a^4*(136080*c + 138600*d*x - 181440*Cos[c + d*x] - 53760*Cos[3*(c + d*x)] + 16128*Cos[5*(c + d*x)] + 7200*Cos
[7*(c + d*x)] - 1120*Cos[9*(c + d*x)] + 8820*Sin[2*(c + d*x)] - 42840*Sin[4*(c + d*x)] - 2730*Sin[6*(c + d*x)]
+ 4095*Sin[8*(c + d*x)] - 126*Sin[10*(c + d*x)]))/(645120*d)

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Maple [A]  time = 0.044, size = 306, normalized size = 1.6 \begin{align*}{\frac{1}{d} \left ({a}^{4} \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 ) +4\,{a}^{4} \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 ) +6\,{a}^{4} \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 ) +4\,{a}^{4} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) +{a}^{4} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{6}}+{\frac{\sin \left ( dx+c \right ) }{24} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{3}+{\frac{3\,\cos \left ( dx+c \right ) }{2}} \right ) }+{\frac{dx}{16}}+{\frac{c}{16}} \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)^2*(a+a*sin(d*x+c))^4,x)

[Out]

1/d*(a^4*(-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)+4*a^4*(-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)+6*a^4*(-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)+4*a^4*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+
c)^5)+a^4*(-1/6*sin(d*x+c)*cos(d*x+c)^5+1/24*(cos(d*x+c)^3+3/2*cos(d*x+c))*sin(d*x+c)+1/16*d*x+1/16*c))

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Maxima [A]  time = 1.11004, size = 251, normalized size = 1.34 \begin{align*} -\frac{8192 \,{\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^{4} - 73728 \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{4} + 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^{4} - 3360 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{4} - 3780 \,{\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^{4}}{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)^2*(a+a*sin(d*x+c))^4,x, algorithm="maxima")

[Out]

-1/645120*(8192*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*cos(d*x + c)^5)*a^4 - 73728*(5*cos(d*x + c)^7 - 7*
cos(d*x + c)^5)*a^4 + 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^4 - 3360*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*a^4 - 3780*(24*d*x + 24*c + sin(8*d*x +
8*c) - 8*sin(4*d*x + 4*c))*a^4)/d

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Fricas [A]  time = 1.23926, size = 343, normalized size = 1.83 \begin{align*} -\frac{35840 \, a^{4} \cos \left (d x + c\right )^{9} - 138240 \, a^{4} \cos \left (d x + c\right )^{7} + 129024 \, a^{4} \cos \left (d x + c\right )^{5} - 17325 \, a^{4} d x + 21 \,{\left (384 \, a^{4} \cos \left (d x + c\right )^{9} - 3888 \, a^{4} \cos \left (d x + c\right )^{7} + 5704 \, a^{4} \cos \left (d x + c\right )^{5} - 550 \, a^{4} \cos \left (d x + c\right )^{3} - 825 \, a^{4} \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)^2*(a+a*sin(d*x+c))^4,x, algorithm="fricas")

[Out]

-1/80640*(35840*a^4*cos(d*x + c)^9 - 138240*a^4*cos(d*x + c)^7 + 129024*a^4*cos(d*x + c)^5 - 17325*a^4*d*x + 2
1*(384*a^4*cos(d*x + c)^9 - 3888*a^4*cos(d*x + c)^7 + 5704*a^4*cos(d*x + c)^5 - 550*a^4*cos(d*x + c)^3 - 825*a
^4*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 36.57, size = 746, normalized size = 3.99 \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)**2*(a+a*sin(d*x+c))**4,x)

[Out]

Piecewise((3*a**4*x*sin(c + d*x)**10/256 + 15*a**4*x*sin(c + d*x)**8*cos(c + d*x)**2/256 + 9*a**4*x*sin(c + d*
x)**8/64 + 15*a**4*x*sin(c + d*x)**6*cos(c + d*x)**4/128 + 9*a**4*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + a**4*
x*sin(c + d*x)**6/16 + 15*a**4*x*sin(c + d*x)**4*cos(c + d*x)**6/128 + 27*a**4*x*sin(c + d*x)**4*cos(c + d*x)*
*4/32 + 3*a**4*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 15*a**4*x*sin(c + d*x)**2*cos(c + d*x)**8/256 + 9*a**4*x
*sin(c + d*x)**2*cos(c + d*x)**6/16 + 3*a**4*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*a**4*x*cos(c + d*x)**10/
256 + 9*a**4*x*cos(c + d*x)**8/64 + a**4*x*cos(c + d*x)**6/16 + 3*a**4*sin(c + d*x)**9*cos(c + d*x)/(256*d) +
7*a**4*sin(c + d*x)**7*cos(c + d*x)**3/(128*d) + 9*a**4*sin(c + d*x)**7*cos(c + d*x)/(64*d) - a**4*sin(c + d*x
)**5*cos(c + d*x)**5/(10*d) + 33*a**4*sin(c + d*x)**5*cos(c + d*x)**3/(64*d) + a**4*sin(c + d*x)**5*cos(c + d*
x)/(16*d) - 4*a**4*sin(c + d*x)**4*cos(c + d*x)**5/(5*d) - 7*a**4*sin(c + d*x)**3*cos(c + d*x)**7/(128*d) - 33
*a**4*sin(c + d*x)**3*cos(c + d*x)**5/(64*d) + a**4*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 16*a**4*sin(c + d*
x)**2*cos(c + d*x)**7/(35*d) - 4*a**4*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - 3*a**4*sin(c + d*x)*cos(c + d*x)
**9/(256*d) - 9*a**4*sin(c + d*x)*cos(c + d*x)**7/(64*d) - a**4*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 32*a**4*
cos(c + d*x)**9/(315*d) - 8*a**4*cos(c + d*x)**7/(35*d), Ne(d, 0)), (x*(a*sin(c) + a)**4*sin(c)**2*cos(c)**4,
True))

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Giac [A]  time = 1.33955, size = 235, normalized size = 1.26 \begin{align*} \frac{55}{256} \, a^{4} x - \frac{a^{4} \cos \left (9 \, d x + 9 \, c\right )}{576 \, d} + \frac{5 \, a^{4} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac{a^{4} \cos \left (5 \, d x + 5 \, c\right )}{40 \, d} - \frac{a^{4} \cos \left (3 \, d x + 3 \, c\right )}{12 \, d} - \frac{9 \, a^{4} \cos \left (d x + c\right )}{32 \, d} - \frac{a^{4} \sin \left (10 \, d x + 10 \, c\right )}{5120 \, d} + \frac{13 \, a^{4} \sin \left (8 \, d x + 8 \, c\right )}{2048 \, d} - \frac{13 \, a^{4} \sin \left (6 \, d x + 6 \, c\right )}{3072 \, d} - \frac{17 \, a^{4} \sin \left (4 \, d x + 4 \, c\right )}{256 \, d} + \frac{7 \, a^{4} \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)^2*(a+a*sin(d*x+c))^4,x, algorithm="giac")

[Out]

55/256*a^4*x - 1/576*a^4*cos(9*d*x + 9*c)/d + 5/448*a^4*cos(7*d*x + 7*c)/d + 1/40*a^4*cos(5*d*x + 5*c)/d - 1/1
2*a^4*cos(3*d*x + 3*c)/d - 9/32*a^4*cos(d*x + c)/d - 1/5120*a^4*sin(10*d*x + 10*c)/d + 13/2048*a^4*sin(8*d*x +
8*c)/d - 13/3072*a^4*sin(6*d*x + 6*c)/d - 17/256*a^4*sin(4*d*x + 4*c)/d + 7/512*a^4*sin(2*d*x + 2*c)/d