\(\int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx\) [393]

Optimal result

Integrand size = 21, antiderivative size = 146 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5}{128} \left (8 a^2+b^2\right ) x-\frac {9 a b \cos ^7(c+d x)}{56 d}+\frac {5 \left (8 a^2+b^2\right ) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 \left (8 a^2+b^2\right ) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {\left (8 a^2+b^2\right ) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \]

[Out]

5/128*(8*a^2+b^2)*x-9/56*a*b*cos(d*x+c)^7/d+5/128*(8*a^2+b^2)*cos(d*x+c)*sin(d*x+c)/d+5/192*(8*a^2+b^2)*cos(d*
x+c)^3*sin(d*x+c)/d+1/48*(8*a^2+b^2)*cos(d*x+c)^5*sin(d*x+c)/d-1/8*b*cos(d*x+c)^7*(a+b*sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.10 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2771, 2748, 2715, 8} \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {\left (8 a^2+b^2\right ) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 \left (8 a^2+b^2\right ) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} x \left (8 a^2+b^2\right )-\frac {9 a b \cos ^7(c+d x)}{56 d}-\frac {b \cos ^7(c+d x) (a+b \sin (c+d x))}{8 d} \]

[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 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2715

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] && Integ
erQ[2*n]

Rule 2748

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-b)*((g*Co
s[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 2771

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])

Rubi steps \begin{align*} \text {integral}& = -\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] (verified)

Time = 0.42 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {840 \left (8 a^2+b^2\right ) (c+d x)-3360 a b \cos (c+d x)-2016 a b \cos (3 (c+d x))-672 a b \cos (5 (c+d x))-96 a b \cos (7 (c+d x))+336 \left (15 a^2+b^2\right ) \sin (2 (c+d x))+168 \left (6 a^2-b^2\right ) \sin (4 (c+d x))+112 (a-b) (a+b) \sin (6 (c+d x))-21 b^2 \sin (8 (c+d x))}{21504 d} \]

[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)

Maple [A] (verified)

Time = 1.76 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7}+b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\)	\(128\)
default	\(\frac {a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {2 a b \left (\cos ^{7}\left (d x +c \right )\right )}{7}+b^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{7}\left (d x +c \right )\right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )}{d}\)	\(128\)
parallelrisch	\(\frac {336 \left (15 a^{2}+b^{2}\right ) \sin \left (2 d x +2 c \right )+168 \left (6 a^{2}-b^{2}\right ) \sin \left (4 d x +4 c \right )+112 \left (a^{2}-b^{2}\right ) \sin \left (6 d x +6 c \right )+6720 a^{2} d x +840 b^{2} d x -3360 \cos \left (d x +c \right ) a b -2016 a b \cos \left (3 d x +3 c \right )-672 a b \cos \left (5 d x +5 c \right )-96 a b \cos \left (7 d x +7 c \right )-21 b^{2} \sin \left (8 d x +8 c \right )-6144 a b}{21504 d}\)	\(150\)
risch	\(\frac {5 a^{2} x}{16}+\frac {5 b^{2} x}{128}-\frac {5 a b \cos \left (d x +c \right )}{32 d}-\frac {b^{2} \sin \left (8 d x +8 c \right )}{1024 d}-\frac {a b \cos \left (7 d x +7 c \right )}{224 d}+\frac {\sin \left (6 d x +6 c \right ) a^{2}}{192 d}-\frac {\sin \left (6 d x +6 c \right ) b^{2}}{192 d}-\frac {a b \cos \left (5 d x +5 c \right )}{32 d}+\frac {3 a^{2} \sin \left (4 d x +4 c \right )}{64 d}-\frac {\sin \left (4 d x +4 c \right ) b^{2}}{128 d}-\frac {3 a b \cos \left (3 d x +3 c \right )}{32 d}+\frac {15 a^{2} \sin \left (2 d x +2 c \right )}{64 d}+\frac {\sin \left (2 d x +2 c \right ) b^{2}}{64 d}\)	\(194\)
norman	\(\frac {-\frac {12 a b \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\left (\frac {5 a^{2}}{16}+\frac {5 b^{2}}{128}\right ) x +\frac {5 \left (136 a^{2}+353 b^{2}\right ) \left (\tan ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{16}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {4 a b}{7 d}+\frac {\left (88 a^{2}-5 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{64 d}-\frac {4 a b \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{7 d}-\frac {12 a b \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {20 a b \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {4 a b \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}+\left (\frac {175 a^{2}}{8}+\frac {175 b^{2}}{64}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{4}+\frac {35 b^{2}}{32}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{2}+\frac {35 b^{2}}{16}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{4}+\frac {35 b^{2}}{32}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {35 a^{2}}{2}+\frac {35 b^{2}}{16}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (488 a^{2}+397 b^{2}\right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {5 \left (136 a^{2}+353 b^{2}\right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {20 a b \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (488 a^{2}+397 b^{2}\right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}-\frac {\left (904 a^{2}-895 b^{2}\right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\left (\frac {5 a^{2}}{16}+\frac {5 b^{2}}{128}\right ) x \left (\tan ^{16}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\frac {\left (904 a^{2}-895 b^{2}\right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{192 d}+\left (\frac {5 a^{2}}{2}+\frac {5 b^{2}}{16}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {\left (88 a^{2}-5 b^{2}\right ) \left (\tan ^{15}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}\)	\(570\)

[In]

int(cos(d*x+c)^6*(a+b*sin(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.74 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {768 \, a b \cos \left (d x + c\right )^{7} - 105 \, {\left (8 \, a^{2} + b^{2}\right )} d x + 7 \, {\left (48 \, b^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{5} - 10 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )^{3} - 15 \, {\left (8 \, a^{2} + b^{2}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2688 \, d} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (141) = 282\).

Time = 0.70 (sec) , antiderivative size = 398, normalized size of antiderivative = 2.73 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\begin {cases} \frac {5 a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {5 a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {11 a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 a b \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 b^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {15 b^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {5 b^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {5 b^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 b^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 b^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {73 b^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} - \frac {5 b^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} & \text {for}\: d \neq 0 \\x \left (a + b \sin {\left (c \right )}\right )^{2} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.78 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=-\frac {6144 \, a b \cos \left (d x + c\right )^{7} + 112 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} a^{2} - 7 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} b^{2}}{21504 \, d} \]

[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

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.11 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5}{128} \, {\left (8 \, a^{2} + b^{2}\right )} x - \frac {a b \cos \left (7 \, d x + 7 \, c\right )}{224 \, d} - \frac {a b \cos \left (5 \, d x + 5 \, c\right )}{32 \, d} - \frac {3 \, a b \cos \left (3 \, d x + 3 \, c\right )}{32 \, d} - \frac {5 \, a b \cos \left (d x + c\right )}{32 \, d} - \frac {b^{2} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (a^{2} - b^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (6 \, a^{2} - b^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (15 \, a^{2} + b^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

[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

Mupad [B] (verification not implemented)

Time = 5.16 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.22 \[ \int \cos ^6(c+d x) (a+b \sin (c+d x))^2 \, dx=\frac {5\,a^2\,x}{16}+\frac {5\,b^2\,x}{128}+\frac {5\,a^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{24\,d}+\frac {a^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{6\,d}+\frac {5\,b^2\,{\cos \left (c+d\,x\right )}^3\,\sin \left (c+d\,x\right )}{192\,d}+\frac {b^2\,{\cos \left (c+d\,x\right )}^5\,\sin \left (c+d\,x\right )}{48\,d}-\frac {b^2\,{\cos \left (c+d\,x\right )}^7\,\sin \left (c+d\,x\right )}{8\,d}-\frac {2\,a\,b\,{\cos \left (c+d\,x\right )}^7}{7\,d}+\frac {5\,a^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{16\,d}+\frac {5\,b^2\,\cos \left (c+d\,x\right )\,\sin \left (c+d\,x\right )}{128\,d} \]

[In]

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

[Out]

(5*a^2*x)/16 + (5*b^2*x)/128 + (5*a^2*cos(c + d*x)^3*sin(c + d*x))/(24*d) + (a^2*cos(c + d*x)^5*sin(c + d*x))/
(6*d) + (5*b^2*cos(c + d*x)^3*sin(c + d*x))/(192*d) + (b^2*cos(c + d*x)^5*sin(c + d*x))/(48*d) - (b^2*cos(c +
d*x)^7*sin(c + d*x))/(8*d) - (2*a*b*cos(c + d*x)^7)/(7*d) + (5*a^2*cos(c + d*x)*sin(c + d*x))/(16*d) + (5*b^2*
cos(c + d*x)*sin(c + d*x))/(128*d)