\(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx\) [43]

Optimal result

Integrand size = 34, antiderivative size = 140 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {1}{8} a^3 (5 A+2 B) c x-\frac {a^3 (5 A+2 B) c \cos ^3(e+f x)}{12 f}+\frac {a^3 (5 A+2 B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}-\frac {(5 A+2 B) c \cos ^3(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{20 f} \]

[Out]

1/8*a^3*(5*A+2*B)*c*x-1/12*a^3*(5*A+2*B)*c*cos(f*x+e)^3/f+1/8*a^3*(5*A+2*B)*c*cos(f*x+e)*sin(f*x+e)/f-1/5*a*B*
c*cos(f*x+e)^3*(a+a*sin(f*x+e))^2/f-1/20*(5*A+2*B)*c*cos(f*x+e)^3*(a^3+a^3*sin(f*x+e))/f

Rubi [A] (verified)

Time = 0.16 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3046, 2939, 2757, 2748, 2715, 8} \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {a^3 c (5 A+2 B) \cos ^3(e+f x)}{12 f}-\frac {c (5 A+2 B) \cos ^3(e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{20 f}+\frac {a^3 c (5 A+2 B) \sin (e+f x) \cos (e+f x)}{8 f}+\frac {1}{8} a^3 c x (5 A+2 B)-\frac {a B c \cos ^3(e+f x) (a \sin (e+f x)+a)^2}{5 f} \]

[In]

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

[Out]

(a^3*(5*A + 2*B)*c*x)/8 - (a^3*(5*A + 2*B)*c*Cos[e + f*x]^3)/(12*f) + (a^3*(5*A + 2*B)*c*Cos[e + f*x]*Sin[e +
f*x])/(8*f) - (a*B*c*Cos[e + f*x]^3*(a + a*Sin[e + f*x])^2)/(5*f) - ((5*A + 2*B)*c*Cos[e + f*x]^3*(a^3 + a^3*S
in[e + f*x]))/(20*f)

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]

Rule 3046

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[a^m*c^m, Int[Cos[e + f*x]^(2*m)*(c + d*Sin[e + f*x])^(n - m)*(A + B
*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 - b^2, 0] && I
ntegerQ[m] &&  !(IntegerQ[n] && ((LtQ[m, 0] && GtQ[n, 0]) || LtQ[0, n, m] || LtQ[m, n, 0]))

Rubi steps \begin{align*} \text {integral}& = (a c) \int \cos ^2(e+f x) (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) \, dx \\ & = -\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}+\frac {1}{5} (a (5 A+2 B) c) \int \cos ^2(e+f x) (a+a \sin (e+f x))^2 \, dx \\ & = -\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}-\frac {(5 A+2 B) c \cos ^3(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{20 f}+\frac {1}{4} \left (a^2 (5 A+2 B) c\right ) \int \cos ^2(e+f x) (a+a \sin (e+f x)) \, dx \\ & = -\frac {a^3 (5 A+2 B) c \cos ^3(e+f x)}{12 f}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}-\frac {(5 A+2 B) c \cos ^3(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{20 f}+\frac {1}{4} \left (a^3 (5 A+2 B) c\right ) \int \cos ^2(e+f x) \, dx \\ & = -\frac {a^3 (5 A+2 B) c \cos ^3(e+f x)}{12 f}+\frac {a^3 (5 A+2 B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}-\frac {(5 A+2 B) c \cos ^3(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{20 f}+\frac {1}{8} \left (a^3 (5 A+2 B) c\right ) \int 1 \, dx \\ & = \frac {1}{8} a^3 (5 A+2 B) c x-\frac {a^3 (5 A+2 B) c \cos ^3(e+f x)}{12 f}+\frac {a^3 (5 A+2 B) c \cos (e+f x) \sin (e+f x)}{8 f}-\frac {a B c \cos ^3(e+f x) (a+a \sin (e+f x))^2}{5 f}-\frac {(5 A+2 B) c \cos ^3(e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{20 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.01 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {a^3 c \cos (e+f x) \left (-30 (5 A+2 B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (-8 (10 A+7 B)+15 (3 A-2 B) \sin (e+f x)+16 (5 A+2 B) \sin ^2(e+f x)+30 (A+2 B) \sin ^3(e+f x)+24 B \sin ^4(e+f x)\right )\right )}{120 f \sqrt {\cos ^2(e+f x)}} \]

[In]

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

[Out]

(a^3*c*Cos[e + f*x]*(-30*(5*A + 2*B)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(-8*(10*A +
7*B) + 15*(3*A - 2*B)*Sin[e + f*x] + 16*(5*A + 2*B)*Sin[e + f*x]^2 + 30*(A + 2*B)*Sin[e + f*x]^3 + 24*B*Sin[e
+ f*x]^4)))/(120*f*Sqrt[Cos[e + f*x]^2])

Maple [A] (verified)

Time = 1.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.69

method	result	size
parallelrisch	\(\frac {\left (\left (-\frac {2 A}{3}-\frac {5 B}{12}\right ) \cos \left (3 f x +3 e \right )+\left (-\frac {A}{8}-\frac {B}{4}\right ) \sin \left (4 f x +4 e \right )+\frac {\cos \left (5 f x +5 e \right ) B}{20}+A \sin \left (2 f x +2 e \right )+\left (-2 A -\frac {3 B}{2}\right ) \cos \left (f x +e \right )+\frac {5 f x A}{2}+f x B -\frac {8 A}{3}-\frac {28 B}{15}\right ) c \,a^{3}}{4 f}\)	\(97\)
risch	\(\frac {5 a^{3} c x A}{8}+\frac {a^{3} c x B}{4}-\frac {a^{3} c \cos \left (f x +e \right ) A}{2 f}-\frac {3 a^{3} c \cos \left (f x +e \right ) B}{8 f}+\frac {B \,a^{3} c \cos \left (5 f x +5 e \right )}{80 f}-\frac {\sin \left (4 f x +4 e \right ) A \,a^{3} c}{32 f}-\frac {\sin \left (4 f x +4 e \right ) B \,a^{3} c}{16 f}-\frac {a^{3} c \cos \left (3 f x +3 e \right ) A}{6 f}-\frac {5 a^{3} c \cos \left (3 f x +3 e \right ) B}{48 f}+\frac {A \,a^{3} c \sin \left (2 f x +2 e \right )}{4 f}\)	\(164\)
parts	\(\frac {\left (-A \,a^{3} c -2 B \,a^{3} c \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 A \,a^{3} c +B \,a^{3} c \right ) \cos \left (f x +e \right )}{f}+a^{3} c x A +\frac {2 A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {2 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+\frac {B \,a^{3} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\)	\(180\)
derivativedivides	\(\frac {-A \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+\frac {B \,a^{3} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 A \cos \left (f x +e \right ) a^{3} c +2 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{3} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{3} c}{f}\)	\(208\)
default	\(\frac {-A \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )+\frac {2 A \,a^{3} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+\frac {B \,a^{3} c \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-2 B \,a^{3} c \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 A \cos \left (f x +e \right ) a^{3} c +2 B \,a^{3} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,a^{3} c \left (f x +e \right )-B \cos \left (f x +e \right ) a^{3} c}{f}\)	\(208\)
norman	\(\frac {\left (\frac {5}{8} A \,a^{3} c +\frac {1}{4} B \,a^{3} c \right ) x +\left (\frac {5}{8} A \,a^{3} c +\frac {1}{4} B \,a^{3} c \right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{4} A \,a^{3} c +\frac {5}{2} B \,a^{3} c \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{4} A \,a^{3} c +\frac {5}{2} B \,a^{3} c \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{8} A \,a^{3} c +\frac {5}{4} B \,a^{3} c \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {25}{8} A \,a^{3} c +\frac {5}{4} B \,a^{3} c \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {20 A \,a^{3} c +14 B \,a^{3} c}{15 f}-\frac {\left (4 A \,a^{3} c +2 B \,a^{3} c \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (4 A \,a^{3} c +4 B \,a^{3} c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (8 A \,a^{3} c +2 B \,a^{3} c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (8 A \,a^{3} c +8 B \,a^{3} c \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {a^{3} c \left (3 A -2 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}-\frac {a^{3} c \left (3 A -2 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{3} c \left (6 B +7 A \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}-\frac {a^{3} c \left (6 B +7 A \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\)	\(425\)

[In]

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

[Out]

1/4*((-2/3*A-5/12*B)*cos(3*f*x+3*e)+(-1/8*A-1/4*B)*sin(4*f*x+4*e)+1/20*cos(5*f*x+5*e)*B+A*sin(2*f*x+2*e)+(-2*A
-3/2*B)*cos(f*x+e)+5/2*f*x*A+f*x*B-8/3*A-28/15*B)*c*a^3/f

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.71 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {24 \, B a^{3} c \cos \left (f x + e\right )^{5} - 80 \, {\left (A + B\right )} a^{3} c \cos \left (f x + e\right )^{3} + 15 \, {\left (5 \, A + 2 \, B\right )} a^{3} c f x - 15 \, {\left (2 \, {\left (A + 2 \, B\right )} a^{3} c \cos \left (f x + e\right )^{3} - {\left (5 \, A + 2 \, B\right )} a^{3} c \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]

[In]

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

[Out]

1/120*(24*B*a^3*c*cos(f*x + e)^5 - 80*(A + B)*a^3*c*cos(f*x + e)^3 + 15*(5*A + 2*B)*a^3*c*f*x - 15*(2*(A + 2*B
)*a^3*c*cos(f*x + e)^3 - (5*A + 2*B)*a^3*c*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. 486 vs. \(2 (128) = 256\).

Time = 0.30 (sec) , antiderivative size = 486, normalized size of antiderivative = 3.47 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\begin {cases} - \frac {3 A a^{3} c x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {3 A a^{3} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} - \frac {3 A a^{3} c x \cos ^{4}{\left (e + f x \right )}}{8} + A a^{3} c x + \frac {5 A a^{3} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} + \frac {2 A a^{3} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {3 A a^{3} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} + \frac {4 A a^{3} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 A a^{3} c \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{3} c x \sin ^{4}{\left (e + f x \right )}}{4} - \frac {3 B a^{3} c x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} + B a^{3} c x \sin ^{2}{\left (e + f x \right )} - \frac {3 B a^{3} c x \cos ^{4}{\left (e + f x \right )}}{4} + B a^{3} c x \cos ^{2}{\left (e + f x \right )} + \frac {B a^{3} c \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {5 B a^{3} c \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} + \frac {4 B a^{3} c \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 B a^{3} c \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} - \frac {B a^{3} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {8 B a^{3} c \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {B a^{3} c \cos {\left (e + f x \right )}}{f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{3} \left (- c \sin {\left (e \right )} + c\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.43 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=-\frac {320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c - 480 \, {\left (f x + e\right )} A a^{3} c - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{3} c + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c - 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c + 960 \, A a^{3} c \cos \left (f x + e\right ) + 480 \, B a^{3} c \cos \left (f x + e\right )}{480 \, f} \]

[In]

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

[Out]

-1/480*(320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^3*c + 15*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2
*e))*A*a^3*c - 480*(f*x + e)*A*a^3*c - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c + 3
0*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c - 240*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3
*c + 960*A*a^3*c*cos(f*x + e) + 480*B*a^3*c*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.44 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.00 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {B a^{3} c \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {A a^{3} c \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} + \frac {1}{8} \, {\left (5 \, A a^{3} c + 2 \, B a^{3} c\right )} x - \frac {{\left (8 \, A a^{3} c + 5 \, B a^{3} c\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (4 \, A a^{3} c + 3 \, B a^{3} c\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {{\left (A a^{3} c + 2 \, B a^{3} c\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} \]

[In]

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

[Out]

1/80*B*a^3*c*cos(5*f*x + 5*e)/f + 1/4*A*a^3*c*sin(2*f*x + 2*e)/f + 1/8*(5*A*a^3*c + 2*B*a^3*c)*x - 1/48*(8*A*a
^3*c + 5*B*a^3*c)*cos(3*f*x + 3*e)/f - 1/8*(4*A*a^3*c + 3*B*a^3*c)*cos(f*x + e)/f - 1/32*(A*a^3*c + 2*B*a^3*c)
*sin(4*f*x + 4*e)/f

Mupad [B] (verification not implemented)

Time = 14.34 (sec) , antiderivative size = 390, normalized size of antiderivative = 2.79 \[ \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x)) \, dx=\frac {a^3\,c\,\mathrm {atan}\left (\frac {a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (5\,A+2\,B\right )}{4\,\left (\frac {5\,A\,a^3\,c}{4}+\frac {B\,a^3\,c}{2}\right )}\right )\,\left (5\,A+2\,B\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,A\,a^3\,c+2\,B\,a^3\,c\right )-\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {3\,A\,a^3\,c}{4}-\frac {B\,a^3\,c}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {7\,A\,a^3\,c}{2}+3\,B\,a^3\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {7\,A\,a^3\,c}{2}+3\,B\,a^3\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {3\,A\,a^3\,c}{4}-\frac {B\,a^3\,c}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (8\,A\,a^3\,c+8\,B\,a^3\,c\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^3\,c}{3}+\frac {8\,B\,a^3\,c}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {16\,A\,a^3\,c}{3}+\frac {4\,B\,a^3\,c}{3}\right )+\frac {4\,A\,a^3\,c}{3}+\frac {14\,B\,a^3\,c}{15}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}-\frac {a^3\,c\,\left (5\,A+2\,B\right )\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )}{4\,f} \]

[In]

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

[Out]

(a^3*c*atan((a^3*c*tan(e/2 + (f*x)/2)*(5*A + 2*B))/(4*((5*A*a^3*c)/4 + (B*a^3*c)/2)))*(5*A + 2*B))/(4*f) - (ta
n(e/2 + (f*x)/2)^8*(4*A*a^3*c + 2*B*a^3*c) - tan(e/2 + (f*x)/2)*((3*A*a^3*c)/4 - (B*a^3*c)/2) - tan(e/2 + (f*x
)/2)^3*((7*A*a^3*c)/2 + 3*B*a^3*c) + tan(e/2 + (f*x)/2)^7*((7*A*a^3*c)/2 + 3*B*a^3*c) + tan(e/2 + (f*x)/2)^9*(
(3*A*a^3*c)/4 - (B*a^3*c)/2) + tan(e/2 + (f*x)/2)^6*(8*A*a^3*c + 8*B*a^3*c) + tan(e/2 + (f*x)/2)^2*((8*A*a^3*c
)/3 + (8*B*a^3*c)/3) + tan(e/2 + (f*x)/2)^4*((16*A*a^3*c)/3 + (4*B*a^3*c)/3) + (4*A*a^3*c)/3 + (14*B*a^3*c)/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 + ta
n(e/2 + (f*x)/2)^10 + 1)) - (a^3*c*(5*A + 2*B)*(atan(tan(e/2 + (f*x)/2)) - (f*x)/2))/(4*f)