\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx\) [260]

Optimal result

Integrand size = 33, antiderivative size = 201 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {1}{8} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{5 f}+\frac {a^3 (20 A c+15 B c+15 A d+13 B d) \cos ^3(e+f x)}{60 f}-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f} \]

[Out]

1/8*a^3*(20*A*c+15*A*d+15*B*c+13*B*d)*x-1/5*a^3*(20*A*c+15*A*d+15*B*c+13*B*d)*cos(f*x+e)/f+1/60*a^3*(20*A*c+15
*A*d+15*B*c+13*B*d)*cos(f*x+e)^3/f-3/40*a^3*(20*A*c+15*A*d+15*B*c+13*B*d)*cos(f*x+e)*sin(f*x+e)/f-1/20*(5*A*d+
5*B*c-B*d)*cos(f*x+e)*(a+a*sin(f*x+e))^3/f-1/5*B*d*cos(f*x+e)*(a+a*sin(f*x+e))^4/a/f

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {3047, 3102, 2830, 2724, 2718, 2715, 8, 2713} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {a^3 (20 A c+15 A d+15 B c+13 B d) \cos ^3(e+f x)}{60 f}-\frac {a^3 (20 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{5 f}-\frac {3 a^3 (20 A c+15 A d+15 B c+13 B d) \sin (e+f x) \cos (e+f x)}{40 f}+\frac {1}{8} a^3 x (20 A c+15 A d+15 B c+13 B d)-\frac {(5 A d+5 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac {B d \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f} \]

[In]

Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]

[Out]

(a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*x)/8 - (a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x])/(5*f) +
(a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*Cos[e + f*x]^3)/(60*f) - (3*a^3*(20*A*c + 15*B*c + 15*A*d + 13*B*d)*
Cos[e + f*x]*Sin[e + f*x])/(40*f) - ((5*B*c + 5*A*d - B*d)*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(20*f) - (B*d*
Cos[e + f*x]*(a + a*Sin[e + f*x])^4)/(5*a*f)

Rule 8

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

Rule 2713

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

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 2718

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2724

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(a + b*sin[c + d*x])^n, x], x] /;
FreeQ[{a, b, c, d, n}, x] && EqQ[a^2 - b^2, 0] && IGtQ[n, 0]

Rule 2830

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-d
)*Cos[e + f*x]*((a + b*Sin[e + f*x])^m/(f*(m + 1))), x] + Dist[(a*d*m + b*c*(m + 1))/(b*(m + 1)), Int[(a + b*S
in[e + f*x])^m, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &&  !LtQ[m
, -2^(-1)]

Rule 3047

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(
e_.) + (f_.)*(x_)]), x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*(A*c + (B*c + A*d)*Sin[e + f*x] + B*d*Sin[e + f*x
]^2), x] /; FreeQ[{a, b, c, d, e, f, A, B, m}, x] && NeQ[b*c - a*d, 0]

Rule 3102

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (
f_.)*(x_)]^2), x_Symbol] :> Simp[(-C)*Cos[e + f*x]*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + 2))), x] + Dist[1/(
b*(m + 2)), Int[(a + b*Sin[e + f*x])^m*Simp[A*b*(m + 2) + b*C*(m + 1) + (b*B*(m + 2) - a*C)*Sin[e + f*x], x],
x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] &&  !LtQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \int (a+a \sin (e+f x))^3 \left (A c+(B c+A d) \sin (e+f x)+B d \sin ^2(e+f x)\right ) \, dx \\ & = -\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {\int (a+a \sin (e+f x))^3 (a (5 A c+4 B d)+a (5 B c+5 A d-B d) \sin (e+f x)) \, dx}{5 a} \\ & = -\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} (20 A c+15 B c+15 A d+13 B d) \int (a+a \sin (e+f x))^3 \, dx \\ & = -\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} (20 A c+15 B c+15 A d+13 B d) \int \left (a^3+3 a^3 \sin (e+f x)+3 a^3 \sin ^2(e+f x)+a^3 \sin ^3(e+f x)\right ) \, dx \\ & = \frac {1}{20} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{20} \left (a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int \sin ^3(e+f x) \, dx+\frac {1}{20} \left (3 a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int \sin (e+f x) \, dx+\frac {1}{20} \left (3 a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int \sin ^2(e+f x) \, dx \\ & = \frac {1}{20} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{20 f}-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f}+\frac {1}{40} \left (3 a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \int 1 \, dx-\frac {\left (a^3 (20 A c+15 B c+15 A d+13 B d)\right ) \text {Subst}\left (\int \left (1-x^2\right ) \, dx,x,\cos (e+f x)\right )}{20 f} \\ & = \frac {1}{8} a^3 (20 A c+15 B c+15 A d+13 B d) x-\frac {a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x)}{5 f}+\frac {a^3 (20 A c+15 B c+15 A d+13 B d) \cos ^3(e+f x)}{60 f}-\frac {3 a^3 (20 A c+15 B c+15 A d+13 B d) \cos (e+f x) \sin (e+f x)}{40 f}-\frac {(5 B c+5 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^3}{20 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^4}{5 a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.78 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {\cos (e+f x) \left (-\frac {1}{4} a^4 (5 B c+5 A d-B d) (1+\sin (e+f x))^3-B d (a+a \sin (e+f x))^4-\frac {a^4 (20 A c+15 B c+15 A d+13 B d) \left (30 \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (22+9 \sin (e+f x)+2 \sin ^2(e+f x)\right )\right )}{24 \sqrt {\cos ^2(e+f x)}}\right )}{5 a f} \]

[In]

Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c + d*Sin[e + f*x]),x]

[Out]

(Cos[e + f*x]*(-1/4*(a^4*(5*B*c + 5*A*d - B*d)*(1 + Sin[e + f*x])^3) - B*d*(a + a*Sin[e + f*x])^4 - (a^4*(20*A
*c + 15*B*c + 15*A*d + 13*B*d)*(30*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(22 + 9*Sin[e
+ f*x] + 2*Sin[e + f*x]^2)))/(24*Sqrt[Cos[e + f*x]^2])))/(5*a*f)

Maple [A] (verified)

Time = 1.66 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.83

method	result	size
parallelrisch	\(\frac {\left (\left (\left (3 c +\frac {17 d}{4}\right ) B +A \left (c +3 d \right )\right ) \cos \left (3 f x +3 e \right )+3 \left (4 \left (-c -d \right ) B -3 \left (c +\frac {4 d}{3}\right ) A \right ) \sin \left (2 f x +2 e \right )+\frac {3 \left (B \left (c +3 d \right )+d A \right ) \sin \left (4 f x +4 e \right )}{8}-\frac {3 B d \cos \left (5 f x +5 e \right )}{20}+3 \left (\left (-13 c -\frac {23 d}{2}\right ) B -15 \left (c +\frac {13 d}{15}\right ) A \right ) \cos \left (f x +e \right )+\left (\frac {45}{2} c f x +\frac {39}{2} d f x -36 c -\frac {152}{5} d \right ) B +30 \left (3 \left (-\frac {2}{5}+\frac {f x}{4}\right ) d +c \left (f x -\frac {22}{15}\right )\right ) A \right ) a^{3}}{12 f}\)	\(167\)
parts	\(\frac {\left (A \,a^{3} d +B \,a^{3} c +3 a^{3} d B \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (3 A \,a^{3} c +A \,a^{3} d +B \,a^{3} c \right ) \cos \left (f x +e \right )}{f}-\frac {\left (A \,a^{3} c +3 A \,a^{3} d +3 B \,a^{3} c +3 a^{3} d B \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (3 A \,a^{3} c +3 A \,a^{3} d +3 B \,a^{3} c +a^{3} d B \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+A \,a^{3} c x -\frac {a^{3} d B \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\)	\(236\)
risch	\(\frac {5 A \,a^{3} c x}{2}+\frac {15 A \,a^{3} d x}{8}+\frac {15 B \,a^{3} c x}{8}+\frac {13 B \,a^{3} d x}{8}-\frac {15 a^{3} \cos \left (f x +e \right ) A c}{4 f}-\frac {13 a^{3} \cos \left (f x +e \right ) d A}{4 f}-\frac {13 a^{3} \cos \left (f x +e \right ) B c}{4 f}-\frac {23 a^{3} \cos \left (f x +e \right ) d B}{8 f}-\frac {B \,a^{3} d \cos \left (5 f x +5 e \right )}{80 f}+\frac {\sin \left (4 f x +4 e \right ) A \,a^{3} d}{32 f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{3} c}{32 f}+\frac {3 \sin \left (4 f x +4 e \right ) a^{3} d B}{32 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) A c}{12 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) d A}{4 f}+\frac {a^{3} \cos \left (3 f x +3 e \right ) B c}{4 f}+\frac {17 a^{3} \cos \left (3 f x +3 e \right ) d B}{48 f}-\frac {3 \sin \left (2 f x +2 e \right ) A \,a^{3} c}{4 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{3} d}{f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{3} c}{f}-\frac {\sin \left (2 f x +2 e \right ) a^{3} d B}{f}\)	\(326\)
derivativedivides	\(\frac {-\frac {A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A \,a^{3} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} d B \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 A \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A \,a^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-B \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{3} d B \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-3 A \cos \left (f x +e \right ) a^{3} c +3 A \,a^{3} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} d B \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+A \,a^{3} c \left (f x +e \right )-A \,a^{3} d \cos \left (f x +e \right )-B \cos \left (f x +e \right ) a^{3} c +a^{3} d B \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\)	\(414\)
default	\(\frac {-\frac {A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A \,a^{3} d \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {a^{3} d B \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+3 A \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A \,a^{3} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )-B \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+3 a^{3} d B \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-3 A \cos \left (f x +e \right ) a^{3} c +3 A \,a^{3} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-a^{3} d B \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+A \,a^{3} c \left (f x +e \right )-A \,a^{3} d \cos \left (f x +e \right )-B \cos \left (f x +e \right ) a^{3} c +a^{3} d B \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\)	\(414\)
norman	\(\frac {\left (\frac {5}{2} A \,a^{3} c +\frac {15}{8} A \,a^{3} d +\frac {15}{8} B \,a^{3} c +\frac {13}{8} a^{3} d B \right ) x +\left (25 A \,a^{3} c +\frac {75}{4} A \,a^{3} d +\frac {75}{4} B \,a^{3} c +\frac {65}{4} a^{3} d B \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (25 A \,a^{3} c +\frac {75}{4} A \,a^{3} d +\frac {75}{4} B \,a^{3} c +\frac {65}{4} a^{3} d B \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {5}{2} A \,a^{3} c +\frac {15}{8} A \,a^{3} d +\frac {15}{8} B \,a^{3} c +\frac {13}{8} a^{3} d B \right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{2} A \,a^{3} c +\frac {75}{8} A \,a^{3} d +\frac {75}{8} B \,a^{3} c +\frac {65}{8} a^{3} d B \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{2} A \,a^{3} c +\frac {75}{8} A \,a^{3} d +\frac {75}{8} B \,a^{3} c +\frac {65}{8} a^{3} d B \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {110 A \,a^{3} c +90 A \,a^{3} d +90 B \,a^{3} c +76 a^{3} d B}{15 f}-\frac {\left (6 A \,a^{3} c +2 A \,a^{3} d +2 B \,a^{3} c \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (14 A \,a^{3} c +10 A \,a^{3} d +10 B \,a^{3} c +6 a^{3} d B \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (68 A \,a^{3} c +60 A \,a^{3} d +60 B \,a^{3} c +58 a^{3} d B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (92 A \,a^{3} c +84 A \,a^{3} d +84 B \,a^{3} c +76 a^{3} d B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a^{3} \left (12 A c +15 d A +15 B c +13 d B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{3} \left (12 A c +15 d A +15 B c +13 d B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{3} \left (12 A c +19 d A +19 B c +25 d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{3} \left (12 A c +19 d A +19 B c +25 d B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\)	\(608\)

[In]

int((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/12*(((3*c+17/4*d)*B+A*(c+3*d))*cos(3*f*x+3*e)+3*(4*(-c-d)*B-3*(c+4/3*d)*A)*sin(2*f*x+2*e)+3/8*(B*(c+3*d)+d*A
)*sin(4*f*x+4*e)-3/20*B*d*cos(5*f*x+5*e)+3*((-13*c-23/2*d)*B-15*(c+13/15*d)*A)*cos(f*x+e)+(45/2*c*f*x+39/2*d*f
*x-36*c-152/5*d)*B+30*(3*(-2/5+1/4*f*x)*d+c*(f*x-22/15))*A)*a^3/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.89 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {24 \, B a^{3} d \cos \left (f x + e\right )^{5} - 40 \, {\left ({\left (A + 3 \, B\right )} a^{3} c + {\left (3 \, A + 5 \, B\right )} a^{3} d\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (5 \, {\left (4 \, A + 3 \, B\right )} a^{3} c + {\left (15 \, A + 13 \, B\right )} a^{3} d\right )} f x + 480 \, {\left ({\left (A + B\right )} a^{3} c + {\left (A + B\right )} a^{3} d\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (B a^{3} c + {\left (A + 3 \, B\right )} a^{3} d\right )} \cos \left (f x + e\right )^{3} - {\left ({\left (12 \, A + 17 \, B\right )} a^{3} c + {\left (17 \, A + 19 \, B\right )} a^{3} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="fricas")

[Out]

-1/120*(24*B*a^3*d*cos(f*x + e)^5 - 40*((A + 3*B)*a^3*c + (3*A + 5*B)*a^3*d)*cos(f*x + e)^3 - 15*(5*(4*A + 3*B
)*a^3*c + (15*A + 13*B)*a^3*d)*f*x + 480*((A + B)*a^3*c + (A + B)*a^3*d)*cos(f*x + e) - 15*(2*(B*a^3*c + (A +
3*B)*a^3*d)*cos(f*x + e)^3 - ((12*A + 17*B)*a^3*c + (17*A + 19*B)*a^3*d)*cos(f*x + e))*sin(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 960 vs. \(2 (201) = 402\).

Time = 0.33 (sec) , antiderivative size = 960, normalized size of antiderivative = 4.78 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x)

[Out]

Piecewise((3*A*a**3*c*x*sin(e + f*x)**2/2 + 3*A*a**3*c*x*cos(e + f*x)**2/2 + A*a**3*c*x - A*a**3*c*sin(e + f*x
)**2*cos(e + f*x)/f - 3*A*a**3*c*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a**3*c*cos(e + f*x)**3/(3*f) - 3*A*a**3
*c*cos(e + f*x)/f + 3*A*a**3*d*x*sin(e + f*x)**4/8 + 3*A*a**3*d*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*A*a**3
*d*x*sin(e + f*x)**2/2 + 3*A*a**3*d*x*cos(e + f*x)**4/8 + 3*A*a**3*d*x*cos(e + f*x)**2/2 - 5*A*a**3*d*sin(e +
f*x)**3*cos(e + f*x)/(8*f) - 3*A*a**3*d*sin(e + f*x)**2*cos(e + f*x)/f - 3*A*a**3*d*sin(e + f*x)*cos(e + f*x)*
*3/(8*f) - 3*A*a**3*d*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a**3*d*cos(e + f*x)**3/f - A*a**3*d*cos(e + f*x)/f
+ 3*B*a**3*c*x*sin(e + f*x)**4/8 + 3*B*a**3*c*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + 3*B*a**3*c*x*sin(e + f*x)
**2/2 + 3*B*a**3*c*x*cos(e + f*x)**4/8 + 3*B*a**3*c*x*cos(e + f*x)**2/2 - 5*B*a**3*c*sin(e + f*x)**3*cos(e + f
*x)/(8*f) - 3*B*a**3*c*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*a**3*c*sin(e + f*x)*cos(e + f*x)**3/(8*f) - 3*B*a*
*3*c*sin(e + f*x)*cos(e + f*x)/(2*f) - 2*B*a**3*c*cos(e + f*x)**3/f - B*a**3*c*cos(e + f*x)/f + 9*B*a**3*d*x*s
in(e + f*x)**4/8 + 9*B*a**3*d*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + B*a**3*d*x*sin(e + f*x)**2/2 + 9*B*a**3*d*
x*cos(e + f*x)**4/8 + B*a**3*d*x*cos(e + f*x)**2/2 - B*a**3*d*sin(e + f*x)**4*cos(e + f*x)/f - 15*B*a**3*d*sin
(e + f*x)**3*cos(e + f*x)/(8*f) - 4*B*a**3*d*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 3*B*a**3*d*sin(e + f*x)**
2*cos(e + f*x)/f - 9*B*a**3*d*sin(e + f*x)*cos(e + f*x)**3/(8*f) - B*a**3*d*sin(e + f*x)*cos(e + f*x)/(2*f) -
8*B*a**3*d*cos(e + f*x)**5/(15*f) - 2*B*a**3*d*cos(e + f*x)**3/f, Ne(f, 0)), (x*(A + B*sin(e))*(c + d*sin(e))*
(a*sin(e) + a)**3, True))

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 398 vs. \(2 (189) = 378\).

Time = 0.22 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.98 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c + 480 \, {\left (f x + e\right )} A a^{3} c + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} d + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} d + 360 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} d - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} d + 480 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} d + 45 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} d + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} d - 1440 \, A a^{3} c \cos \left (f x + e\right ) - 480 \, B a^{3} c \cos \left (f x + e\right ) - 480 \, A a^{3} d \cos \left (f x + e\right )}{480 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="maxima")

[Out]

1/480*(160*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c + 360*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*c + 480*(f*x
+ e)*A*a^3*c + 480*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^3*c + 15*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2
*f*x + 2*e))*B*a^3*c + 360*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c + 480*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*
a^3*d + 15*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^3*d + 360*(2*f*x + 2*e - sin(2*f*x + 2*
e))*A*a^3*d - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*d + 480*(cos(f*x + e)^3 - 3*co
s(f*x + e))*B*a^3*d + 45*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*d + 120*(2*f*x + 2*e -
sin(2*f*x + 2*e))*B*a^3*d - 1440*A*a^3*c*cos(f*x + e) - 480*B*a^3*c*cos(f*x + e) - 480*A*a^3*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.05 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {B a^{3} d \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {1}{8} \, {\left (20 \, A a^{3} c + 15 \, B a^{3} c + 15 \, A a^{3} d + 13 \, B a^{3} d\right )} x + \frac {{\left (4 \, A a^{3} c + 12 \, B a^{3} c + 12 \, A a^{3} d + 17 \, B a^{3} d\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (30 \, A a^{3} c + 26 \, B a^{3} c + 26 \, A a^{3} d + 23 \, B a^{3} d\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (B a^{3} c + A a^{3} d + 3 \, B a^{3} d\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (3 \, A a^{3} c + 4 \, B a^{3} c + 4 \, A a^{3} d + 4 \, B a^{3} d\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^3*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x, algorithm="giac")

[Out]

-1/80*B*a^3*d*cos(5*f*x + 5*e)/f + 1/8*(20*A*a^3*c + 15*B*a^3*c + 15*A*a^3*d + 13*B*a^3*d)*x + 1/48*(4*A*a^3*c
+ 12*B*a^3*c + 12*A*a^3*d + 17*B*a^3*d)*cos(3*f*x + 3*e)/f - 1/8*(30*A*a^3*c + 26*B*a^3*c + 26*A*a^3*d + 23*B
*a^3*d)*cos(f*x + e)/f + 1/32*(B*a^3*c + A*a^3*d + 3*B*a^3*d)*sin(4*f*x + 4*e)/f - 1/4*(3*A*a^3*c + 4*B*a^3*c
+ 4*A*a^3*d + 4*B*a^3*d)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 14.93 (sec) , antiderivative size = 550, normalized size of antiderivative = 2.74 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {a^3\,\mathrm {atan}\left (\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (20\,A\,c+15\,A\,d+15\,B\,c+13\,B\,d\right )}{4\,\left (5\,A\,a^3\,c+\frac {15\,A\,a^3\,d}{4}+\frac {15\,B\,a^3\,c}{4}+\frac {13\,B\,a^3\,d}{4}\right )}\right )\,\left (20\,A\,c+15\,A\,d+15\,B\,c+13\,B\,d\right )}{4\,f}-\frac {a^3\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (20\,A\,c+15\,A\,d+15\,B\,c+13\,B\,d\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (6\,A\,a^3\,c+\frac {19\,A\,a^3\,d}{2}+\frac {19\,B\,a^3\,c}{2}+\frac {25\,B\,a^3\,d}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (3\,A\,a^3\,c+\frac {15\,A\,a^3\,d}{4}+\frac {15\,B\,a^3\,c}{4}+\frac {13\,B\,a^3\,d}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (6\,A\,a^3\,c+\frac {19\,A\,a^3\,d}{2}+\frac {19\,B\,a^3\,c}{2}+\frac {25\,B\,a^3\,d}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (28\,A\,a^3\,c+20\,A\,a^3\,d+20\,B\,a^3\,c+12\,B\,a^3\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {92\,A\,a^3\,c}{3}+28\,A\,a^3\,d+28\,B\,a^3\,c+\frac {76\,B\,a^3\,d}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {136\,A\,a^3\,c}{3}+40\,A\,a^3\,d+40\,B\,a^3\,c+\frac {116\,B\,a^3\,d}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (6\,A\,a^3\,c+2\,A\,a^3\,d+2\,B\,a^3\,c\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,A\,a^3\,c+\frac {15\,A\,a^3\,d}{4}+\frac {15\,B\,a^3\,c}{4}+\frac {13\,B\,a^3\,d}{4}\right )+\frac {22\,A\,a^3\,c}{3}+6\,A\,a^3\,d+6\,B\,a^3\,c+\frac {76\,B\,a^3\,d}{15}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c + d*sin(e + f*x)),x)

[Out]

(a^3*atan((a^3*tan(e/2 + (f*x)/2)*(20*A*c + 15*A*d + 15*B*c + 13*B*d))/(4*(5*A*a^3*c + (15*A*a^3*d)/4 + (15*B*
a^3*c)/4 + (13*B*a^3*d)/4)))*(20*A*c + 15*A*d + 15*B*c + 13*B*d))/(4*f) - (a^3*(atan(tan(e/2 + (f*x)/2)) - (f*
x)/2)*(20*A*c + 15*A*d + 15*B*c + 13*B*d))/(4*f) - (tan(e/2 + (f*x)/2)^3*(6*A*a^3*c + (19*A*a^3*d)/2 + (19*B*a
^3*c)/2 + (25*B*a^3*d)/2) - tan(e/2 + (f*x)/2)^9*(3*A*a^3*c + (15*A*a^3*d)/4 + (15*B*a^3*c)/4 + (13*B*a^3*d)/4
) - tan(e/2 + (f*x)/2)^7*(6*A*a^3*c + (19*A*a^3*d)/2 + (19*B*a^3*c)/2 + (25*B*a^3*d)/2) + tan(e/2 + (f*x)/2)^6
*(28*A*a^3*c + 20*A*a^3*d + 20*B*a^3*c + 12*B*a^3*d) + tan(e/2 + (f*x)/2)^2*((92*A*a^3*c)/3 + 28*A*a^3*d + 28*
B*a^3*c + (76*B*a^3*d)/3) + tan(e/2 + (f*x)/2)^4*((136*A*a^3*c)/3 + 40*A*a^3*d + 40*B*a^3*c + (116*B*a^3*d)/3)
+ tan(e/2 + (f*x)/2)^8*(6*A*a^3*c + 2*A*a^3*d + 2*B*a^3*c) + tan(e/2 + (f*x)/2)*(3*A*a^3*c + (15*A*a^3*d)/4 +
(15*B*a^3*c)/4 + (13*B*a^3*d)/4) + (22*A*a^3*c)/3 + 6*A*a^3*d + 6*B*a^3*c + (76*B*a^3*d)/15)/(f*(5*tan(e/2 +
(f*x)/2)^2 + 10*tan(e/2 + (f*x)/2)^4 + 10*tan(e/2 + (f*x)/2)^6 + 5*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (f*x)/2)^1
0 + 1))