\(\int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx\) [977]

Optimal result

Integrand size = 31, antiderivative size = 196 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {5}{128} a^2 (9 A+2 B) x-\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac {5 a^2 (9 A+2 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d} \]

[Out]

5/128*a^2*(9*A+2*B)*x-1/56*a^2*(9*A+2*B)*cos(d*x+c)^7/d+5/128*a^2*(9*A+2*B)*cos(d*x+c)*sin(d*x+c)/d+5/192*a^2*
(9*A+2*B)*cos(d*x+c)^3*sin(d*x+c)/d+1/48*a^2*(9*A+2*B)*cos(d*x+c)^5*sin(d*x+c)/d-1/9*B*cos(d*x+c)^7*(a+a*sin(d
*x+c))^2/d-1/72*(9*A+2*B)*cos(d*x+c)^7*(a^2+a^2*sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.14 (sec) , antiderivative size = 196, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{72 d}+\frac {a^2 (9 A+2 B) \sin (c+d x) \cos ^5(c+d x)}{48 d}+\frac {5 a^2 (9 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{192 d}+\frac {5 a^2 (9 A+2 B) \sin (c+d x) \cos (c+d x)}{128 d}+\frac {5}{128} a^2 x (9 A+2 B)-\frac {B \cos ^7(c+d x) (a \sin (c+d x)+a)^2}{9 d} \]

[In]

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

[Out]

(5*a^2*(9*A + 2*B)*x)/128 - (a^2*(9*A + 2*B)*Cos[c + d*x]^7)/(56*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]*Sin[c +
d*x])/(128*d) + (5*a^2*(9*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(192*d) + (a^2*(9*A + 2*B)*Cos[c + d*x]^5*Sin[
c + d*x])/(48*d) - (B*Cos[c + d*x]^7*(a + a*Sin[c + d*x])^2)/(9*d) - ((9*A + 2*B)*Cos[c + d*x]^7*(a^2 + a^2*Si
n[c + d*x]))/(72*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 2757

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 2939

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

Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}+\frac {1}{9} (9 A+2 B) \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac {1}{8} (a (9 A+2 B)) \int \cos ^6(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac {1}{8} \left (a^2 (9 A+2 B)\right ) \int \cos ^6(c+d x) \, dx \\ & = -\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac {a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac {1}{48} \left (5 a^2 (9 A+2 B)\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac {5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac {1}{64} \left (5 a^2 (9 A+2 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac {5 a^2 (9 A+2 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d}+\frac {1}{128} \left (5 a^2 (9 A+2 B)\right ) \int 1 \, dx \\ & = \frac {5}{128} a^2 (9 A+2 B) x-\frac {a^2 (9 A+2 B) \cos ^7(c+d x)}{56 d}+\frac {5 a^2 (9 A+2 B) \cos (c+d x) \sin (c+d x)}{128 d}+\frac {5 a^2 (9 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{192 d}+\frac {a^2 (9 A+2 B) \cos ^5(c+d x) \sin (c+d x)}{48 d}-\frac {B \cos ^7(c+d x) (a+a \sin (c+d x))^2}{9 d}-\frac {(9 A+2 B) \cos ^7(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{72 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.49 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.10 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2 \cos (c+d x) \left (2880 A+1900 B+\frac {2520 (9 A+2 B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+32 (135 A+86 B) \cos (2 (c+d x))+16 (108 A+59 B) \cos (4 (c+d x))+288 A \cos (6 (c+d x))+64 B \cos (6 (c+d x))-28 B \cos (8 (c+d x))-13671 A \sin (c+d x)-2478 B \sin (c+d x)-2457 A \sin (3 (c+d x))+462 B \sin (3 (c+d x))-63 A \sin (5 (c+d x))+546 B \sin (5 (c+d x))+63 A \sin (7 (c+d x))+126 B \sin (7 (c+d x))\right )}{32256 d} \]

[In]

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

[Out]

-1/32256*(a^2*Cos[c + d*x]*(2880*A + 1900*B + (2520*(9*A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[C
os[c + d*x]^2] + 32*(135*A + 86*B)*Cos[2*(c + d*x)] + 16*(108*A + 59*B)*Cos[4*(c + d*x)] + 288*A*Cos[6*(c + d*
x)] + 64*B*Cos[6*(c + d*x)] - 28*B*Cos[8*(c + d*x)] - 13671*A*Sin[c + d*x] - 2478*B*Sin[c + d*x] - 2457*A*Sin[
3*(c + d*x)] + 462*B*Sin[3*(c + d*x)] - 63*A*Sin[5*(c + d*x)] + 546*B*Sin[5*(c + d*x)] + 63*A*Sin[7*(c + d*x)]
+ 126*B*Sin[7*(c + d*x)]))/d

Maple [A] (verified)

Time = 1.28 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(-\frac {\left (7 \left (3 A +\frac {11 B}{6}\right ) \cos \left (3 d x +3 c \right )+7 \left (A +\frac {B}{2}\right ) \cos \left (5 d x +5 c \right )+\left (A +\frac {B}{8}\right ) \cos \left (7 d x +7 c \right )+7 \left (-8 A -B \right ) \sin \left (2 d x +2 c \right )+\frac {7 \left (-\frac {5 A}{2}+B \right ) \sin \left (4 d x +4 c \right )}{2}+\frac {7 \left (\frac {A}{2}+B \right ) \sin \left (8 d x +8 c \right )}{16}-\frac {7 B \cos \left (9 d x +9 c \right )}{72}+\frac {7 B \sin \left (6 d x +6 c \right )}{3}+7 \left (5 A +\frac {13 B}{4}\right ) \cos \left (d x +c \right )-\frac {315 d x A}{4}-\frac {35 d x B}{2}+64 A +\frac {352 B}{9}\right ) a^{2}}{224 d}\)	\(164\)
derivativedivides	\(\frac {A \,a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+B \,a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-\frac {2 A \left (\cos ^{7}\left (d x +c \right )\right ) a^{2}}{7}+2 B \,a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+A \,a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {B \,a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(245\)
default	\(\frac {A \,a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+B \,a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{9}-\frac {2 \left (\cos ^{7}\left (d x +c \right )\right )}{63}\right )-\frac {2 A \left (\cos ^{7}\left (d x +c \right )\right ) a^{2}}{7}+2 B \,a^{2} \left (-\frac {\left (\cos ^{7}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{8}+\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{48}+\frac {5 d x}{128}+\frac {5 c}{128}\right )+A \,a^{2} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )-\frac {B \,a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}}{d}\)	\(245\)
risch	\(\frac {45 a^{2} x A}{128}+\frac {5 a^{2} x B}{64}-\frac {5 A \,a^{2} \cos \left (d x +c \right )}{32 d}-\frac {13 a^{2} \cos \left (d x +c \right ) B}{128 d}+\frac {B \,a^{2} \cos \left (9 d x +9 c \right )}{2304 d}-\frac {\sin \left (8 d x +8 c \right ) A \,a^{2}}{1024 d}-\frac {\sin \left (8 d x +8 c \right ) B \,a^{2}}{512 d}-\frac {a^{2} \cos \left (7 d x +7 c \right ) A}{224 d}-\frac {a^{2} \cos \left (7 d x +7 c \right ) B}{1792 d}-\frac {B \,a^{2} \sin \left (6 d x +6 c \right )}{96 d}-\frac {a^{2} \cos \left (5 d x +5 c \right ) A}{32 d}-\frac {a^{2} \cos \left (5 d x +5 c \right ) B}{64 d}+\frac {5 \sin \left (4 d x +4 c \right ) A \,a^{2}}{128 d}-\frac {\sin \left (4 d x +4 c \right ) B \,a^{2}}{64 d}-\frac {3 a^{2} \cos \left (3 d x +3 c \right ) A}{32 d}-\frac {11 a^{2} \cos \left (3 d x +3 c \right ) B}{192 d}+\frac {\sin \left (2 d x +2 c \right ) A \,a^{2}}{4 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{32 d}\)	\(298\)
norman	\(\text {Expression too large to display}\)	\(723\)

[In]

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

[Out]

-1/224*(7*(3*A+11/6*B)*cos(3*d*x+3*c)+7*(A+1/2*B)*cos(5*d*x+5*c)+(A+1/8*B)*cos(7*d*x+7*c)+7*(-8*A-B)*sin(2*d*x
+2*c)+7/2*(-5/2*A+B)*sin(4*d*x+4*c)+7/16*(1/2*A+B)*sin(8*d*x+8*c)-7/72*B*cos(9*d*x+9*c)+7/3*B*sin(6*d*x+6*c)+7
*(5*A+13/4*B)*cos(d*x+c)-315/4*d*x*A-35/2*d*x*B+64*A+352/9*B)*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.69 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {896 \, B a^{2} \cos \left (d x + c\right )^{9} - 2304 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{7} + 315 \, {\left (9 \, A + 2 \, B\right )} a^{2} d x - 21 \, {\left (48 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{7} - 8 \, {\left (9 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} - 10 \, {\left (9 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} - 15 \, {\left (9 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{8064 \, d} \]

[In]

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

[Out]

1/8064*(896*B*a^2*cos(d*x + c)^9 - 2304*(A + B)*a^2*cos(d*x + c)^7 + 315*(9*A + 2*B)*a^2*d*x - 21*(48*(A + 2*B
)*a^2*cos(d*x + c)^7 - 8*(9*A + 2*B)*a^2*cos(d*x + c)^5 - 10*(9*A + 2*B)*a^2*cos(d*x + c)^3 - 15*(9*A + 2*B)*a
^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. 719 vs. \(2 (180) = 360\).

Time = 0.97 (sec) , antiderivative size = 719, normalized size of antiderivative = 3.67 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {5 A a^{2} x \sin ^{8}{\left (c + d x \right )}}{128} + \frac {5 A a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{32} + \frac {5 A a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {15 A a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{64} + \frac {15 A a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {5 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{32} + \frac {15 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {5 A a^{2} x \cos ^{8}{\left (c + d x \right )}}{128} + \frac {5 A a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 A a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{128 d} + \frac {55 A a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{384 d} + \frac {5 A a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {73 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{384 d} + \frac {5 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} - \frac {5 A a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{128 d} + \frac {11 A a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} - \frac {2 A a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} + \frac {5 B a^{2} x \sin ^{8}{\left (c + d x \right )}}{64} + \frac {5 B a^{2} x \sin ^{6}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {15 B a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{32} + \frac {5 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{6}{\left (c + d x \right )}}{16} + \frac {5 B a^{2} x \cos ^{8}{\left (c + d x \right )}}{64} + \frac {5 B a^{2} \sin ^{7}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{64 d} + \frac {55 B a^{2} \sin ^{5}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{192 d} + \frac {73 B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{192 d} - \frac {B a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{7 d} - \frac {5 B a^{2} \sin {\left (c + d x \right )} \cos ^{7}{\left (c + d x \right )}}{64 d} - \frac {2 B a^{2} \cos ^{9}{\left (c + d x \right )}}{63 d} - \frac {B a^{2} \cos ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (A + B \sin {\left (c \right )}\right ) \left (a \sin {\left (c \right )} + a\right )^{2} \cos ^{6}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((5*A*a**2*x*sin(c + d*x)**8/128 + 5*A*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/32 + 5*A*a**2*x*sin(c +
d*x)**6/16 + 15*A*a**2*x*sin(c + d*x)**4*cos(c + d*x)**4/64 + 15*A*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16
+ 5*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/32 + 15*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 5*A*a**2*x*
cos(c + d*x)**8/128 + 5*A*a**2*x*cos(c + d*x)**6/16 + 5*A*a**2*sin(c + d*x)**7*cos(c + d*x)/(128*d) + 55*A*a**
2*sin(c + d*x)**5*cos(c + d*x)**3/(384*d) + 5*A*a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + 73*A*a**2*sin(c + d
*x)**3*cos(c + d*x)**5/(384*d) + 5*A*a**2*sin(c + d*x)**3*cos(c + d*x)**3/(6*d) - 5*A*a**2*sin(c + d*x)*cos(c
+ d*x)**7/(128*d) + 11*A*a**2*sin(c + d*x)*cos(c + d*x)**5/(16*d) - 2*A*a**2*cos(c + d*x)**7/(7*d) + 5*B*a**2*
x*sin(c + d*x)**8/64 + 5*B*a**2*x*sin(c + d*x)**6*cos(c + d*x)**2/16 + 15*B*a**2*x*sin(c + d*x)**4*cos(c + d*x
)**4/32 + 5*B*a**2*x*sin(c + d*x)**2*cos(c + d*x)**6/16 + 5*B*a**2*x*cos(c + d*x)**8/64 + 5*B*a**2*sin(c + d*x
)**7*cos(c + d*x)/(64*d) + 55*B*a**2*sin(c + d*x)**5*cos(c + d*x)**3/(192*d) + 73*B*a**2*sin(c + d*x)**3*cos(c
+ d*x)**5/(192*d) - B*a**2*sin(c + d*x)**2*cos(c + d*x)**7/(7*d) - 5*B*a**2*sin(c + d*x)*cos(c + d*x)**7/(64*
d) - 2*B*a**2*cos(c + d*x)**9/(63*d) - B*a**2*cos(c + d*x)**7/(7*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) +
a)**2*cos(c)**6, True))

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.06 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {18432 \, A a^{2} \cos \left (d x + c\right )^{7} + 9216 \, B a^{2} \cos \left (d x + c\right )^{7} - 21 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{2} + 336 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} - 60 \, d x - 60 \, c - 9 \, \sin \left (4 \, d x + 4 \, c\right ) - 48 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 1024 \, {\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} B a^{2} - 42 \, {\left (64 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 120 \, d x + 120 \, c - 3 \, \sin \left (8 \, d x + 8 \, c\right ) - 24 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{2}}{64512 \, d} \]

[In]

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

[Out]

-1/64512*(18432*A*a^2*cos(d*x + c)^7 + 9216*B*a^2*cos(d*x + c)^7 - 21*(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*a^2 + 336*(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*a^2 - 1024*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*B*a^2 - 42*(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*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.45 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.20 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {B a^{2} \cos \left (9 \, d x + 9 \, c\right )}{2304 \, d} - \frac {B a^{2} \sin \left (6 \, d x + 6 \, c\right )}{96 \, d} + \frac {5}{128} \, {\left (9 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac {{\left (8 \, A a^{2} + B a^{2}\right )} \cos \left (7 \, d x + 7 \, c\right )}{1792 \, d} - \frac {{\left (2 \, A a^{2} + B a^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{64 \, d} - \frac {{\left (18 \, A a^{2} + 11 \, B a^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{192 \, d} - \frac {{\left (20 \, A a^{2} + 13 \, B a^{2}\right )} \cos \left (d x + c\right )}{128 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (8 \, d x + 8 \, c\right )}{1024 \, d} + \frac {{\left (5 \, A a^{2} - 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{128 \, d} + \frac {{\left (8 \, A a^{2} + B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{32 \, d} \]

[In]

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

[Out]

1/2304*B*a^2*cos(9*d*x + 9*c)/d - 1/96*B*a^2*sin(6*d*x + 6*c)/d + 5/128*(9*A*a^2 + 2*B*a^2)*x - 1/1792*(8*A*a^
2 + B*a^2)*cos(7*d*x + 7*c)/d - 1/64*(2*A*a^2 + B*a^2)*cos(5*d*x + 5*c)/d - 1/192*(18*A*a^2 + 11*B*a^2)*cos(3*
d*x + 3*c)/d - 1/128*(20*A*a^2 + 13*B*a^2)*cos(d*x + c)/d - 1/1024*(A*a^2 + 2*B*a^2)*sin(8*d*x + 8*c)/d + 1/12
8*(5*A*a^2 - 2*B*a^2)*sin(4*d*x + 4*c)/d + 1/32*(8*A*a^2 + B*a^2)*sin(2*d*x + 2*c)/d

Mupad [B] (verification not implemented)

Time = 11.42 (sec) , antiderivative size = 622, normalized size of antiderivative = 3.17 \[ \int \cos ^6(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {5\,a^2\,\mathrm {atan}\left (\frac {5\,a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (9\,A+2\,B\right )}{64\,\left (\frac {45\,A\,a^2}{64}+\frac {5\,B\,a^2}{32}\right )}\right )\,\left (9\,A+2\,B\right )}{64\,d}-\frac {5\,a^2\,\left (9\,A+2\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )-\frac {d\,x}{2}\right )}{64\,d}-\frac {\frac {4\,A\,a^2}{7}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {83\,A\,a^2}{64}-\frac {5\,B\,a^2}{32}\right )+\frac {22\,B\,a^2}{63}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}\,\left (4\,A\,a^2+2\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}\,\left (8\,A\,a^2+8\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^2}{7}+\frac {8\,B\,a^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (32\,A\,a^2+4\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (24\,A\,a^2+24\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (24\,A\,a^2+\frac {16\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (40\,A\,a^2+40\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {88\,A\,a^2}{7}+\frac {32\,B\,a^2}{7}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{17}\,\left (\frac {83\,A\,a^2}{64}-\frac {5\,B\,a^2}{32}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {149\,A\,a^2}{32}-\frac {83\,B\,a^2}{16}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {149\,A\,a^2}{32}-\frac {83\,B\,a^2}{16}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {189\,A\,a^2}{32}+\frac {191\,B\,a^2}{48}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{15}\,\left (\frac {189\,A\,a^2}{32}+\frac {191\,B\,a^2}{48}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (\frac {409\,A\,a^2}{32}+\frac {145\,B\,a^2}{16}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {409\,A\,a^2}{32}+\frac {145\,B\,a^2}{16}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+126\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+36\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(5*a^2*atan((5*a^2*tan(c/2 + (d*x)/2)*(9*A + 2*B))/(64*((45*A*a^2)/64 + (5*B*a^2)/32)))*(9*A + 2*B))/(64*d) -
(5*a^2*(9*A + 2*B)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(64*d) - ((4*A*a^2)/7 - tan(c/2 + (d*x)/2)*((83*A*a^2
)/64 - (5*B*a^2)/32) + (22*B*a^2)/63 + tan(c/2 + (d*x)/2)^16*(4*A*a^2 + 2*B*a^2) + tan(c/2 + (d*x)/2)^14*(8*A*
a^2 + 8*B*a^2) + tan(c/2 + (d*x)/2)^2*((8*A*a^2)/7 + (8*B*a^2)/7) + tan(c/2 + (d*x)/2)^8*(32*A*a^2 + 4*B*a^2)
+ tan(c/2 + (d*x)/2)^6*(24*A*a^2 + 24*B*a^2) + tan(c/2 + (d*x)/2)^12*(24*A*a^2 + (16*B*a^2)/3) + tan(c/2 + (d*
x)/2)^10*(40*A*a^2 + 40*B*a^2) + tan(c/2 + (d*x)/2)^4*((88*A*a^2)/7 + (32*B*a^2)/7) + tan(c/2 + (d*x)/2)^17*((
83*A*a^2)/64 - (5*B*a^2)/32) - tan(c/2 + (d*x)/2)^5*((149*A*a^2)/32 - (83*B*a^2)/16) + tan(c/2 + (d*x)/2)^13*(
(149*A*a^2)/32 - (83*B*a^2)/16) - tan(c/2 + (d*x)/2)^3*((189*A*a^2)/32 + (191*B*a^2)/48) + tan(c/2 + (d*x)/2)^
15*((189*A*a^2)/32 + (191*B*a^2)/48) - tan(c/2 + (d*x)/2)^7*((409*A*a^2)/32 + (145*B*a^2)/16) + tan(c/2 + (d*x
)/2)^11*((409*A*a^2)/32 + (145*B*a^2)/16))/(d*(9*tan(c/2 + (d*x)/2)^2 + 36*tan(c/2 + (d*x)/2)^4 + 84*tan(c/2 +
(d*x)/2)^6 + 126*tan(c/2 + (d*x)/2)^8 + 126*tan(c/2 + (d*x)/2)^10 + 84*tan(c/2 + (d*x)/2)^12 + 36*tan(c/2 + (
d*x)/2)^14 + 9*tan(c/2 + (d*x)/2)^16 + tan(c/2 + (d*x)/2)^18 + 1))