\(\int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx\) [978]

Optimal result

Integrand size = 31, antiderivative size = 165 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {1}{16} a^2 (7 A+2 B) x-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d} \]

[Out]

1/16*a^2*(7*A+2*B)*x-1/30*a^2*(7*A+2*B)*cos(d*x+c)^5/d+1/16*a^2*(7*A+2*B)*cos(d*x+c)*sin(d*x+c)/d+1/24*a^2*(7*
A+2*B)*cos(d*x+c)^3*sin(d*x+c)/d-1/7*B*cos(d*x+c)^5*(a+a*sin(d*x+c))^2/d-1/42*(7*A+2*B)*cos(d*x+c)^5*(a^2+a^2*
sin(d*x+c))/d

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {2939, 2757, 2748, 2715, 8} \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2 \sin (c+d x)+a^2\right )}{42 d}+\frac {a^2 (7 A+2 B) \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac {a^2 (7 A+2 B) \sin (c+d x) \cos (c+d x)}{16 d}+\frac {1}{16} a^2 x (7 A+2 B)-\frac {B \cos ^5(c+d x) (a \sin (c+d x)+a)^2}{7 d} \]

[In]

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

[Out]

(a^2*(7*A + 2*B)*x)/16 - (a^2*(7*A + 2*B)*Cos[c + d*x]^5)/(30*d) + (a^2*(7*A + 2*B)*Cos[c + d*x]*Sin[c + d*x])
/(16*d) + (a^2*(7*A + 2*B)*Cos[c + d*x]^3*Sin[c + d*x])/(24*d) - (B*Cos[c + d*x]^5*(a + a*Sin[c + d*x])^2)/(7*
d) - ((7*A + 2*B)*Cos[c + d*x]^5*(a^2 + a^2*Sin[c + d*x]))/(42*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 ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}+\frac {1}{7} (7 A+2 B) \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 \, dx \\ & = -\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{6} (a (7 A+2 B)) \int \cos ^4(c+d x) (a+a \sin (c+d x)) \, dx \\ & = -\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{6} \left (a^2 (7 A+2 B)\right ) \int \cos ^4(c+d x) \, dx \\ & = -\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{8} \left (a^2 (7 A+2 B)\right ) \int \cos ^2(c+d x) \, dx \\ & = -\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d}+\frac {1}{16} \left (a^2 (7 A+2 B)\right ) \int 1 \, dx \\ & = \frac {1}{16} a^2 (7 A+2 B) x-\frac {a^2 (7 A+2 B) \cos ^5(c+d x)}{30 d}+\frac {a^2 (7 A+2 B) \cos (c+d x) \sin (c+d x)}{16 d}+\frac {a^2 (7 A+2 B) \cos ^3(c+d x) \sin (c+d x)}{24 d}-\frac {B \cos ^5(c+d x) (a+a \sin (c+d x))^2}{7 d}-\frac {(7 A+2 B) \cos ^5(c+d x) \left (a^2+a^2 \sin (c+d x)\right )}{42 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.22 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.04 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {a^2 \cos (c+d x) \left (504 A+354 B+\frac {420 (7 A+2 B) \arcsin \left (\frac {\sqrt {1-\sin (c+d x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(c+d x)}}+(672 A+447 B) \cos (2 (c+d x))+6 (28 A+13 B) \cos (4 (c+d x))-15 B \cos (6 (c+d x))-1645 A \sin (c+d x)-350 B \sin (c+d x)-140 A \sin (3 (c+d x))+140 B \sin (3 (c+d x))+35 A \sin (5 (c+d x))+70 B \sin (5 (c+d x))\right )}{3360 d} \]

[In]

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

[Out]

-1/3360*(a^2*Cos[c + d*x]*(504*A + 354*B + (420*(7*A + 2*B)*ArcSin[Sqrt[1 - Sin[c + d*x]]/Sqrt[2]])/Sqrt[Cos[c
+ d*x]^2] + (672*A + 447*B)*Cos[2*(c + d*x)] + 6*(28*A + 13*B)*Cos[4*(c + d*x)] - 15*B*Cos[6*(c + d*x)] - 164
5*A*Sin[c + d*x] - 350*B*Sin[c + d*x] - 140*A*Sin[3*(c + d*x)] + 140*B*Sin[3*(c + d*x)] + 35*A*Sin[5*(c + d*x)
] + 70*B*Sin[5*(c + d*x)]))/d

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 138, normalized size of antiderivative = 0.84

method	result	size
parallelrisch	\(\frac {17 \left (\frac {\left (-8 A -5 B \right ) \cos \left (3 d x +3 c \right )}{17}+\frac {\left (-8 A -3 B \right ) \cos \left (5 d x +5 c \right )}{85}+\left (A +\frac {2 B}{17}\right ) \sin \left (2 d x +2 c \right )+\frac {\left (A -2 B \right ) \sin \left (4 d x +4 c \right )}{17}+\frac {\left (-A -2 B \right ) \sin \left (6 d x +6 c \right )}{51}+\frac {B \cos \left (7 d x +7 c \right )}{119}+\frac {\left (-16 A -11 B \right ) \cos \left (d x +c \right )}{17}+\frac {28 d x A}{17}+\frac {8 d x B}{17}-\frac {128 A}{85}-\frac {576 B}{595}\right ) a^{2}}{64 d}\)	\(138\)
derivativedivides	\(\frac {A \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-\frac {2 A \,a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+A \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {B \,a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(215\)
default	\(\frac {A \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+B \,a^{2} \left (-\frac {\left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{7}-\frac {2 \left (\cos ^{5}\left (d x +c \right )\right )}{35}\right )-\frac {2 A \,a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}+2 B \,a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{5}\left (d x +c \right )\right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )+A \,a^{2} \left (\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{4}+\frac {3 d x}{8}+\frac {3 c}{8}\right )-\frac {B \,a^{2} \left (\cos ^{5}\left (d x +c \right )\right )}{5}}{d}\)	\(215\)
risch	\(\frac {7 a^{2} x A}{16}+\frac {a^{2} x B}{8}-\frac {A \,a^{2} \cos \left (d x +c \right )}{4 d}-\frac {11 a^{2} \cos \left (d x +c \right ) B}{64 d}+\frac {a^{2} \cos \left (7 d x +7 c \right ) B}{448 d}-\frac {\sin \left (6 d x +6 c \right ) A \,a^{2}}{192 d}-\frac {B \,a^{2} \sin \left (6 d x +6 c \right )}{96 d}-\frac {a^{2} \cos \left (5 d x +5 c \right ) A}{40 d}-\frac {3 a^{2} \cos \left (5 d x +5 c \right ) B}{320 d}+\frac {\sin \left (4 d x +4 c \right ) A \,a^{2}}{64 d}-\frac {\sin \left (4 d x +4 c \right ) B \,a^{2}}{32 d}-\frac {a^{2} \cos \left (3 d x +3 c \right ) A}{8 d}-\frac {5 a^{2} \cos \left (3 d x +3 c \right ) B}{64 d}+\frac {17 \sin \left (2 d x +2 c \right ) A \,a^{2}}{64 d}+\frac {\sin \left (2 d x +2 c \right ) B \,a^{2}}{32 d}\)	\(244\)
norman	\(\frac {\left (\frac {7}{16} A \,a^{2}+\frac {1}{8} B \,a^{2}\right ) x +\left (\frac {7}{16} A \,a^{2}+\frac {1}{8} B \,a^{2}\right ) x \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {49}{16} A \,a^{2}+\frac {7}{8} B \,a^{2}\right ) x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {49}{16} A \,a^{2}+\frac {7}{8} B \,a^{2}\right ) x \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {147}{16} A \,a^{2}+\frac {21}{8} B \,a^{2}\right ) x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {147}{16} A \,a^{2}+\frac {21}{8} B \,a^{2}\right ) x \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {245}{16} A \,a^{2}+\frac {35}{8} B \,a^{2}\right ) x \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (\frac {245}{16} A \,a^{2}+\frac {35}{8} B \,a^{2}\right ) x \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\frac {28 A \,a^{2}+18 B \,a^{2}}{35 d}-\frac {\left (4 A \,a^{2}+2 B \,a^{2}\right ) \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (8 A \,a^{2}+8 B \,a^{2}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}-\frac {\left (8 A \,a^{2}+8 B \,a^{2}\right ) \left (\tan ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (12 A \,a^{2}+2 B \,a^{2}\right ) \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (16 A \,a^{2}+16 B \,a^{2}\right ) \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d}-\frac {\left (44 A \,a^{2}+14 B \,a^{2}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{5 d}+\frac {a^{2} \left (9 A -2 B \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d}-\frac {a^{2} \left (9 A -2 B \right ) \left (\tan ^{13}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 d}+\frac {a^{2} \left (23 A -62 B \right ) \left (\tan ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}-\frac {a^{2} \left (23 A -62 B \right ) \left (\tan ^{9}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{24 d}+\frac {a^{2} \left (29 A +22 B \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}-\frac {a^{2} \left (29 A +22 B \right ) \left (\tan ^{11}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{6 d}}{\left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}\)	\(561\)

[In]

int(cos(d*x+c)^4*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

17/64*(1/17*(-8*A-5*B)*cos(3*d*x+3*c)+1/85*(-8*A-3*B)*cos(5*d*x+5*c)+(A+2/17*B)*sin(2*d*x+2*c)+1/17*(A-2*B)*si
n(4*d*x+4*c)+1/51*(-A-2*B)*sin(6*d*x+6*c)+1/119*B*cos(7*d*x+7*c)+1/17*(-16*A-11*B)*cos(d*x+c)+28/17*d*x*A+8/17
*d*x*B-128/85*A-576/595*B)*a^2/d

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 115, normalized size of antiderivative = 0.70 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {240 \, B a^{2} \cos \left (d x + c\right )^{7} - 672 \, {\left (A + B\right )} a^{2} \cos \left (d x + c\right )^{5} + 105 \, {\left (7 \, A + 2 \, B\right )} a^{2} d x - 35 \, {\left (8 \, {\left (A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{5} - 2 \, {\left (7 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )^{3} - 3 \, {\left (7 \, A + 2 \, B\right )} a^{2} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{1680 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/1680*(240*B*a^2*cos(d*x + c)^7 - 672*(A + B)*a^2*cos(d*x + c)^5 + 105*(7*A + 2*B)*a^2*d*x - 35*(8*(A + 2*B)*
a^2*cos(d*x + c)^5 - 2*(7*A + 2*B)*a^2*cos(d*x + c)^3 - 3*(7*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. 539 vs. \(2 (146) = 292\).

Time = 0.50 (sec) , antiderivative size = 539, normalized size of antiderivative = 3.27 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\begin {cases} \frac {A a^{2} x \sin ^{6}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \sin ^{4}{\left (c + d x \right )}}{8} + \frac {3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{4} + \frac {A a^{2} x \cos ^{6}{\left (c + d x \right )}}{16} + \frac {3 A a^{2} x \cos ^{4}{\left (c + d x \right )}}{8} + \frac {A a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{16 d} + \frac {A a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{6 d} + \frac {3 A a^{2} \sin ^{3}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} - \frac {A a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{16 d} + \frac {5 A a^{2} \sin {\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{8 d} - \frac {2 A a^{2} \cos ^{5}{\left (c + d x \right )}}{5 d} + \frac {B a^{2} x \sin ^{6}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{4}{\left (c + d x \right )} \cos ^{2}{\left (c + d x \right )}}{8} + \frac {3 B a^{2} x \sin ^{2}{\left (c + d x \right )} \cos ^{4}{\left (c + d x \right )}}{8} + \frac {B a^{2} x \cos ^{6}{\left (c + d x \right )}}{8} + \frac {B a^{2} \sin ^{5}{\left (c + d x \right )} \cos {\left (c + d x \right )}}{8 d} + \frac {B a^{2} \sin ^{3}{\left (c + d x \right )} \cos ^{3}{\left (c + d x \right )}}{3 d} - \frac {B a^{2} \sin ^{2}{\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{5 d} - \frac {B a^{2} \sin {\left (c + d x \right )} \cos ^{5}{\left (c + d x \right )}}{8 d} - \frac {2 B a^{2} \cos ^{7}{\left (c + d x \right )}}{35 d} - \frac {B a^{2} \cos ^{5}{\left (c + d x \right )}}{5 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 ^{4}{\left (c \right )} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**4*(a+a*sin(d*x+c))**2*(A+B*sin(d*x+c)),x)

[Out]

Piecewise((A*a**2*x*sin(c + d*x)**6/16 + 3*A*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/16 + 3*A*a**2*x*sin(c + d*
x)**4/8 + 3*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/16 + 3*A*a**2*x*sin(c + d*x)**2*cos(c + d*x)**2/4 + A*a**
2*x*cos(c + d*x)**6/16 + 3*A*a**2*x*cos(c + d*x)**4/8 + A*a**2*sin(c + d*x)**5*cos(c + d*x)/(16*d) + A*a**2*si
n(c + d*x)**3*cos(c + d*x)**3/(6*d) + 3*A*a**2*sin(c + d*x)**3*cos(c + d*x)/(8*d) - A*a**2*sin(c + d*x)*cos(c
+ d*x)**5/(16*d) + 5*A*a**2*sin(c + d*x)*cos(c + d*x)**3/(8*d) - 2*A*a**2*cos(c + d*x)**5/(5*d) + B*a**2*x*sin
(c + d*x)**6/8 + 3*B*a**2*x*sin(c + d*x)**4*cos(c + d*x)**2/8 + 3*B*a**2*x*sin(c + d*x)**2*cos(c + d*x)**4/8 +
B*a**2*x*cos(c + d*x)**6/8 + B*a**2*sin(c + d*x)**5*cos(c + d*x)/(8*d) + B*a**2*sin(c + d*x)**3*cos(c + d*x)*
*3/(3*d) - B*a**2*sin(c + d*x)**2*cos(c + d*x)**5/(5*d) - B*a**2*sin(c + d*x)*cos(c + d*x)**5/(8*d) - 2*B*a**2
*cos(c + d*x)**7/(35*d) - B*a**2*cos(c + d*x)**5/(5*d), Ne(d, 0)), (x*(A + B*sin(c))*(a*sin(c) + a)**2*cos(c)*
*4, True))

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.04 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=-\frac {2688 \, A a^{2} \cos \left (d x + c\right )^{5} + 1344 \, B a^{2} \cos \left (d x + c\right )^{5} - 35 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} A a^{2} - 210 \, {\left (12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (2 \, d x + 2 \, c\right )\right )} A a^{2} - 192 \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} B a^{2} - 70 \, {\left (4 \, \sin \left (2 \, d x + 2 \, c\right )^{3} + 12 \, d x + 12 \, c - 3 \, \sin \left (4 \, d x + 4 \, c\right )\right )} B a^{2}}{6720 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/6720*(2688*A*a^2*cos(d*x + c)^5 + 1344*B*a^2*cos(d*x + c)^5 - 35*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*
sin(4*d*x + 4*c))*A*a^2 - 210*(12*d*x + 12*c + sin(4*d*x + 4*c) + 8*sin(2*d*x + 2*c))*A*a^2 - 192*(5*cos(d*x +
c)^7 - 7*cos(d*x + c)^5)*B*a^2 - 70*(4*sin(2*d*x + 2*c)^3 + 12*d*x + 12*c - 3*sin(4*d*x + 4*c))*B*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.38 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.16 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {B a^{2} \cos \left (7 \, d x + 7 \, c\right )}{448 \, d} + \frac {1}{16} \, {\left (7 \, A a^{2} + 2 \, B a^{2}\right )} x - \frac {{\left (8 \, A a^{2} + 3 \, B a^{2}\right )} \cos \left (5 \, d x + 5 \, c\right )}{320 \, d} - \frac {{\left (8 \, A a^{2} + 5 \, B a^{2}\right )} \cos \left (3 \, d x + 3 \, c\right )}{64 \, d} - \frac {{\left (16 \, A a^{2} + 11 \, B a^{2}\right )} \cos \left (d x + c\right )}{64 \, d} - \frac {{\left (A a^{2} + 2 \, B a^{2}\right )} \sin \left (6 \, d x + 6 \, c\right )}{192 \, d} + \frac {{\left (A a^{2} - 2 \, B a^{2}\right )} \sin \left (4 \, d x + 4 \, c\right )}{64 \, d} + \frac {{\left (17 \, A a^{2} + 2 \, B a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )}{64 \, d} \]

[In]

integrate(cos(d*x+c)^4*(a+a*sin(d*x+c))^2*(A+B*sin(d*x+c)),x, algorithm="giac")

[Out]

1/448*B*a^2*cos(7*d*x + 7*c)/d + 1/16*(7*A*a^2 + 2*B*a^2)*x - 1/320*(8*A*a^2 + 3*B*a^2)*cos(5*d*x + 5*c)/d - 1
/64*(8*A*a^2 + 5*B*a^2)*cos(3*d*x + 3*c)/d - 1/64*(16*A*a^2 + 11*B*a^2)*cos(d*x + c)/d - 1/192*(A*a^2 + 2*B*a^
2)*sin(6*d*x + 6*c)/d + 1/64*(A*a^2 - 2*B*a^2)*sin(4*d*x + 4*c)/d + 1/64*(17*A*a^2 + 2*B*a^2)*sin(2*d*x + 2*c)
/d

Mupad [B] (verification not implemented)

Time = 11.94 (sec) , antiderivative size = 494, normalized size of antiderivative = 2.99 \[ \int \cos ^4(c+d x) (a+a \sin (c+d x))^2 (A+B \sin (c+d x)) \, dx=\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (7\,A+2\,B\right )}{8\,\left (\frac {7\,A\,a^2}{8}+\frac {B\,a^2}{4}\right )}\right )\,\left (7\,A+2\,B\right )}{8\,d}-\frac {a^2\,\left (7\,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 )}{8\,d}-\frac {\frac {4\,A\,a^2}{5}-\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (\frac {9\,A\,a^2}{8}-\frac {B\,a^2}{4}\right )+\frac {18\,B\,a^2}{35}+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}\,\left (4\,A\,a^2+2\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (12\,A\,a^2+2\,B\,a^2\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\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}{5}+\frac {8\,B\,a^2}{5}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}\,\left (\frac {9\,A\,a^2}{8}-\frac {B\,a^2}{4}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (16\,A\,a^2+16\,B\,a^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (\frac {29\,A\,a^2}{6}+\frac {11\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (\frac {29\,A\,a^2}{6}+\frac {11\,B\,a^2}{3}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (\frac {44\,A\,a^2}{5}+\frac {14\,B\,a^2}{5}\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (\frac {23\,A\,a^2}{24}-\frac {31\,B\,a^2}{12}\right )+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (\frac {23\,A\,a^2}{24}-\frac {31\,B\,a^2}{12}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \]

[In]

int(cos(c + d*x)^4*(A + B*sin(c + d*x))*(a + a*sin(c + d*x))^2,x)

[Out]

(a^2*atan((a^2*tan(c/2 + (d*x)/2)*(7*A + 2*B))/(8*((7*A*a^2)/8 + (B*a^2)/4)))*(7*A + 2*B))/(8*d) - (a^2*(7*A +
2*B)*(atan(tan(c/2 + (d*x)/2)) - (d*x)/2))/(8*d) - ((4*A*a^2)/5 - tan(c/2 + (d*x)/2)*((9*A*a^2)/8 - (B*a^2)/4
) + (18*B*a^2)/35 + tan(c/2 + (d*x)/2)^12*(4*A*a^2 + 2*B*a^2) + tan(c/2 + (d*x)/2)^8*(12*A*a^2 + 2*B*a^2) + ta
n(c/2 + (d*x)/2)^10*(8*A*a^2 + 8*B*a^2) + tan(c/2 + (d*x)/2)^2*((8*A*a^2)/5 + (8*B*a^2)/5) + tan(c/2 + (d*x)/2
)^13*((9*A*a^2)/8 - (B*a^2)/4) + tan(c/2 + (d*x)/2)^6*(16*A*a^2 + 16*B*a^2) - tan(c/2 + (d*x)/2)^3*((29*A*a^2)
/6 + (11*B*a^2)/3) + tan(c/2 + (d*x)/2)^11*((29*A*a^2)/6 + (11*B*a^2)/3) + tan(c/2 + (d*x)/2)^4*((44*A*a^2)/5
+ (14*B*a^2)/5) - tan(c/2 + (d*x)/2)^5*((23*A*a^2)/24 - (31*B*a^2)/12) + tan(c/2 + (d*x)/2)^9*((23*A*a^2)/24 -
(31*B*a^2)/12))/(d*(7*tan(c/2 + (d*x)/2)^2 + 21*tan(c/2 + (d*x)/2)^4 + 35*tan(c/2 + (d*x)/2)^6 + 35*tan(c/2 +
(d*x)/2)^8 + 21*tan(c/2 + (d*x)/2)^10 + 7*tan(c/2 + (d*x)/2)^12 + tan(c/2 + (d*x)/2)^14 + 1))