3.13 \(\int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx\)

Optimal. Leaf size=126 \[ -\frac{9 a^2 \cos ^7(c+d x)}{56 d}-\frac{\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}+\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{15 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{45 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{45 a^2 x}{128} \]

[Out]

(45*a^2*x)/128 - (9*a^2*Cos[c + d*x]^7)/(56*d) + (45*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*Cos[c +
d*x]^3*Sin[c + d*x])/(64*d) + (3*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c +
d*x]))/(8*d)

________________________________________________________________________________________

Rubi [A]  time = 0.106751, antiderivative size = 126, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2678, 2669, 2635, 8} \[ -\frac{9 a^2 \cos ^7(c+d x)}{56 d}-\frac{\cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{8 d}+\frac{3 a^2 \sin (c+d x) \cos ^5(c+d x)}{16 d}+\frac{15 a^2 \sin (c+d x) \cos ^3(c+d x)}{64 d}+\frac{45 a^2 \sin (c+d x) \cos (c+d x)}{128 d}+\frac{45 a^2 x}{128} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]

[Out]

(45*a^2*x)/128 - (9*a^2*Cos[c + d*x]^7)/(56*d) + (45*a^2*Cos[c + d*x]*Sin[c + d*x])/(128*d) + (15*a^2*Cos[c +
d*x]^3*Sin[c + d*x])/(64*d) + (3*a^2*Cos[c + d*x]^5*Sin[c + d*x])/(16*d) - (Cos[c + d*x]^7*(a^2 + a^2*Sin[c +
d*x]))/(8*d)

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 ^6(c+d x) (a+a \sin (c+d x))^2 \, dx &=-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac{1}{8} (9 a) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx\\ &=-\frac{9 a^2 \cos ^7(c+d x)}{56 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac{1}{8} \left (9 a^2\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac{9 a^2 \cos ^7(c+d x)}{56 d}+\frac{3 a^2 \cos ^5(c+d x) \sin (c+d x)}{16 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac{1}{16} \left (15 a^2\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{9 a^2 \cos ^7(c+d x)}{56 d}+\frac{15 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)}{16 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac{1}{64} \left (45 a^2\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{9 a^2 \cos ^7(c+d x)}{56 d}+\frac{45 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{15 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)}{16 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}+\frac{1}{128} \left (45 a^2\right ) \int 1 \, dx\\ &=\frac{45 a^2 x}{128}-\frac{9 a^2 \cos ^7(c+d x)}{56 d}+\frac{45 a^2 \cos (c+d x) \sin (c+d x)}{128 d}+\frac{15 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)}{16 d}-\frac{\cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{8 d}\\ \end{align*}

Mathematica [A]  time = 1.55813, size = 171, normalized size = 1.36 \[ -\frac{a^2 \left (630 \sqrt{1-\sin (c+d x)} \sin ^{-1}\left (\frac{\sqrt{1-\sin (c+d x)}}{\sqrt{2}}\right )+\sqrt{\sin (c+d x)+1} \left (112 \sin ^8(c+d x)+144 \sin ^7(c+d x)-424 \sin ^6(c+d x)-600 \sin ^5(c+d x)+558 \sin ^4(c+d x)+978 \sin ^3(c+d x)-187 \sin ^2(c+d x)-837 \sin (c+d x)+256\right )\right ) \cos ^7(c+d x)}{896 d (\sin (c+d x)-1)^4 (\sin (c+d x)+1)^{7/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + a*Sin[c + d*x])^2,x]

[Out]

-(a^2*Cos[c + d*x]^7*(630*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]]*Sqrt[1 - Sin[c + d*x]] + Sqrt[1 + Sin[c + d*x
]]*(256 - 837*Sin[c + d*x] - 187*Sin[c + d*x]^2 + 978*Sin[c + d*x]^3 + 558*Sin[c + d*x]^4 - 600*Sin[c + d*x]^5
- 424*Sin[c + d*x]^6 + 144*Sin[c + d*x]^7 + 112*Sin[c + d*x]^8)))/(896*d*(-1 + Sin[c + d*x])^4*(1 + Sin[c + d
*x])^(7/2))

________________________________________________________________________________________

Maple [A]  time = 0.037, size = 129, normalized size = 1. \begin{align*}{\frac{1}{d} \left ({a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{8}}+{\frac{\sin \left ( dx+c \right ) }{48} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{128}}+{\frac{5\,c}{128}} \right ) -{\frac{2\,{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{7}}+{a}^{2} \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+a*sin(d*x+c))^2,x)

[Out]

1/d*(a^2*(-1/8*sin(d*x+c)*cos(d*x+c)^7+1/48*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/128*d
*x+5/128*c)-2/7*a^2*cos(d*x+c)^7+a^2*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+
5/16*c))

________________________________________________________________________________________

Maxima [A]  time = 0.964566, size = 155, normalized size = 1.23 \begin{align*} -\frac{6144 \, a^{2} \cos \left (d x + c\right )^{7} - 7 \,{\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} a^{2} + 112 \,{\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2}}{21504 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/21504*(6144*a^2*cos(d*x + c)^7 - 7*(64*sin(2*d*x + 2*c)^3 + 120*d*x + 120*c - 3*sin(8*d*x + 8*c) - 24*sin(4
*d*x + 4*c))*a^2 + 112*(4*sin(2*d*x + 2*c)^3 - 60*d*x - 60*c - 9*sin(4*d*x + 4*c) - 48*sin(2*d*x + 2*c))*a^2)/
d

________________________________________________________________________________________

Fricas [A]  time = 1.80596, size = 216, normalized size = 1.71 \begin{align*} -\frac{256 \, a^{2} \cos \left (d x + c\right )^{7} - 315 \, a^{2} d x + 7 \,{\left (16 \, a^{2} \cos \left (d x + c\right )^{7} - 24 \, 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 )}{896 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="fricas")

[Out]

-1/896*(256*a^2*cos(d*x + c)^7 - 315*a^2*d*x + 7*(16*a^2*cos(d*x + c)^7 - 24*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

________________________________________________________________________________________

Sympy [A]  time = 14.6679, size = 398, normalized size = 3.16 \begin{align*} \begin{cases} \frac{5 a^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{5 a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac{15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac{5 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac{5 a^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac{5 a^{2} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{55 a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{73 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac{5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac{5 a^{2} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac{11 a^{2} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac{2 a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text{for}\: d \neq 0 \\x \left (a \sin{\left (c \right )} + a\right )^{2} \cos ^{6}{\left (c \right )} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+a*sin(d*x+c))**2,x)

[Out]

Piecewise((5*a**2*x*sin(c + d*x)**8/128 + 5*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 5*a**2*x*sin(c + d*x)*
*6/16 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 5*a**2*x
*sin(c + d*x)**2*cos(c + d*x)**6/32 + 15*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*a**2*x*cos(c + d*x)**8/
128 + 5*a**2*x*cos(c + d*x)**6/16 + 5*a**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*a**2*sin(c + d*x)**5*cos(
c + d*x)**3/(384*d) + 5*a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 73*a**2*sin(c + d*x)**3*cos(c + d*x)**5/(38
4*d) + 5*a**2*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 5*a**2*sin(c + d*x)*cos(c + d*x)**7/(128*d) + 11*a**2*si
n(c + d*x)*cos(c + d*x)**5/(16*d) - 2*a**2*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(a*sin(c) + a)**2*cos(c)**6, T
rue))

________________________________________________________________________________________

Giac [A]  time = 1.15243, size = 166, normalized size = 1.32 \begin{align*} \frac{45}{128} \, a^{2} x - \frac{a^{2} \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac{a^{2} \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac{3 \, a^{2} \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac{5 \, a^{2} \cos \left (d x + c\right )}{32 \, d} - \frac{a^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{5 \, a^{2} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{a^{2} \sin \left (2 \, d x + 2 \, c\right )}{4 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+a*sin(d*x+c))^2,x, algorithm="giac")

[Out]

45/128*a^2*x - 1/224*a^2*cos(7*d*x + 7*c)/d - 1/32*a^2*cos(5*d*x + 5*c)/d - 3/32*a^2*cos(3*d*x + 3*c)/d - 5/32
*a^2*cos(d*x + c)/d - 1/1024*a^2*sin(8*d*x + 8*c)/d + 5/128*a^2*sin(4*d*x + 4*c)/d + 1/4*a^2*sin(2*d*x + 2*c)/
d