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

Optimal. Leaf size=146 \[ \frac{\left (8 a^2+b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x \left (8 a^2+b^2\right )-\frac{9 a b \cos ^7(c+d x)}{56 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \]

[Out]

(5*(8*a^2 + b^2)*x)/128 - (9*a*b*Cos[c + d*x]^7)/(56*d) + (5*(8*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d)
+ (5*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((8*a^2 + b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) -
(b*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/(8*d)

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Rubi [A]  time = 0.133561, antiderivative size = 146, 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 = {2692, 2669, 2635, 8} \[ \frac{\left (8 a^2+b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac{5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac{5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac{5}{128} x \left (8 a^2+b^2\right )-\frac{9 a b \cos ^7(c+d x)}{56 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(5*(8*a^2 + b^2)*x)/128 - (9*a*b*Cos[c + d*x]^7)/(56*d) + (5*(8*a^2 + b^2)*Cos[c + d*x]*Sin[c + d*x])/(128*d)
+ (5*(8*a^2 + b^2)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + ((8*a^2 + b^2)*Cos[c + d*x]^5*Sin[c + d*x])/(48*d) -
(b*Cos[c + d*x]^7*(a + b*Sin[c + d*x]))/(8*d)

Rule 2692

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[1/(m + p), Int[(g*Cos[e + f*x])^
p*(a + b*Sin[e + f*x])^(m - 2)*(b^2*(m - 1) + a^2*(m + p) + a*b*(2*m + p - 1)*Sin[e + f*x]), x], x] /; FreeQ[{
a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && GtQ[m, 1] && NeQ[m + p, 0] && (IntegersQ[2*m, 2*p] || IntegerQ[m
])

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+b \sin (c+d x))^2 \, dx &=-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac{1}{8} \int \cos ^6(c+d x) \left (8 a^2+b^2+9 a b \sin (c+d x)\right ) \, dx\\ &=-\frac{9 a b \cos ^7(c+d x)}{56 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac{1}{8} \left (8 a^2+b^2\right ) \int \cos ^6(c+d x) \, dx\\ &=-\frac{9 a b \cos ^7(c+d x)}{56 d}+\frac{\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac{1}{48} \left (5 \left (8 a^2+b^2\right )\right ) \int \cos ^4(c+d x) \, dx\\ &=-\frac{9 a b \cos ^7(c+d x)}{56 d}+\frac{5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac{1}{64} \left (5 \left (8 a^2+b^2\right )\right ) \int \cos ^2(c+d x) \, dx\\ &=-\frac{9 a b \cos ^7(c+d x)}{56 d}+\frac{5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}+\frac{1}{128} \left (5 \left (8 a^2+b^2\right )\right ) \int 1 \, dx\\ &=\frac{5}{128} \left (8 a^2+b^2\right ) x-\frac{9 a b \cos ^7(c+d x)}{56 d}+\frac{5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac{5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac{\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac{b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d}\\ \end{align*}

Mathematica [A]  time = 0.374819, size = 141, normalized size = 0.97 \[ \frac{840 \left (8 a^2+b^2\right ) (c+d x)+336 \left (15 a^2+b^2\right ) \sin (2 (c+d x))+168 \left (6 a^2-b^2\right ) \sin (4 (c+d x))+112 (a-b) (a+b) \sin (6 (c+d x))-3360 a b \cos (c+d x)-2016 a b \cos (3 (c+d x))-672 a b \cos (5 (c+d x))-96 a b \cos (7 (c+d x))-21 b^2 \sin (8 (c+d x))}{21504 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(840*(8*a^2 + b^2)*(c + d*x) - 3360*a*b*Cos[c + d*x] - 2016*a*b*Cos[3*(c + d*x)] - 672*a*b*Cos[5*(c + d*x)] -
96*a*b*Cos[7*(c + d*x)] + 336*(15*a^2 + b^2)*Sin[2*(c + d*x)] + 168*(6*a^2 - b^2)*Sin[4*(c + d*x)] + 112*(a -
b)*(a + b)*Sin[6*(c + d*x)] - 21*b^2*Sin[8*(c + d*x)])/(21504*d)

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Maple [A]  time = 0.043, size = 128, normalized size = 0.9 \begin{align*}{\frac{1}{d} \left ({b}^{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\,ab \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+b*sin(d*x+c))^2,x)

[Out]

1/d*(b^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*b*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))

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Maxima [A]  time = 0.975867, size = 154, normalized size = 1.05 \begin{align*} -\frac{6144 \, a b \cos \left (d x + c\right )^{7} + 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} - 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 )} b^{2}}{21504 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21504*(6144*a*b*cos(d*x + c)^7 + 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 - 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))*b^2)/
d

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Fricas [A]  time = 2.46034, size = 270, normalized size = 1.85 \begin{align*} -\frac{768 \, a b \cos \left (d x + c\right )^{7} - 105 \,{\left (8 \, a^{2} + b^{2}\right )} d x + 7 \,{\left (48 \, b^{2} \cos \left (d x + c\right )^{7} - 8 \,{\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \,{\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \,{\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/2688*(768*a*b*cos(d*x + c)^7 - 105*(8*a^2 + b^2)*d*x + 7*(48*b^2*cos(d*x + c)^7 - 8*(8*a^2 + b^2)*cos(d*x +
c)^5 - 10*(8*a^2 + b^2)*cos(d*x + c)^3 - 15*(8*a^2 + b^2)*cos(d*x + c))*sin(d*x + c))/d

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Sympy [A]  time = 13.7406, size = 398, normalized size = 2.73 \begin{align*} \begin{cases} \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 ^{2}{\left (c + d x \right )}}{16} + \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 ^{6}{\left (c + d x \right )}}{16} + \frac{5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{16 d} + \frac{5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac{11 a^{2} \sin{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac{2 a b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac{5 b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac{5 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac{15 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac{5 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac{5 b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac{5 b^{2} \sin ^{7}{\left (c + d x \right )} \cos{\left (c + d x \right )}}{128 d} + \frac{55 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac{73 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac{5 b^{2} \sin{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text{for}\: d \neq 0 \\x \left (a + b \sin{\left (c \right )}\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+b*sin(d*x+c))**2,x)

[Out]

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

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Giac [A]  time = 1.10233, size = 219, normalized size = 1.5 \begin{align*} \frac{5}{128} \,{\left (8 \, a^{2} + b^{2}\right )} x - \frac{a b \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac{a b \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac{3 \, a b \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac{5 \, a b \cos \left (d x + c\right )}{32 \, d} - \frac{b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac{{\left (a^{2} - b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac{{\left (6 \, a^{2} - b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac{{\left (15 \, a^{2} + b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5/128*(8*a^2 + b^2)*x - 1/224*a*b*cos(7*d*x + 7*c)/d - 1/32*a*b*cos(5*d*x + 5*c)/d - 3/32*a*b*cos(3*d*x + 3*c)
/d - 5/32*a*b*cos(d*x + c)/d - 1/1024*b^2*sin(8*d*x + 8*c)/d + 1/192*(a^2 - b^2)*sin(6*d*x + 6*c)/d + 1/128*(6
*a^2 - b^2)*sin(4*d*x + 4*c)/d + 1/64*(15*a^2 + b^2)*sin(2*d*x + 2*c)/d