\(\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx\) [17]

Optimal result

Integrand size = 34, antiderivative size = 182 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {7}{16} a (2 A-B) c^4 x+\frac {7 a (2 A-B) c^4 \cos ^3(e+f x)}{24 f}+\frac {7 a (2 A-B) c^4 \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}+\frac {7 a (2 A-B) \cos ^3(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{40 f} \]

[Out]

7/16*a*(2*A-B)*c^4*x+7/24*a*(2*A-B)*c^4*cos(f*x+e)^3/f+7/16*a*(2*A-B)*c^4*cos(f*x+e)*sin(f*x+e)/f-1/6*a*B*c*co
s(f*x+e)^3*(c-c*sin(f*x+e))^3/f+1/10*a*(2*A-B)*cos(f*x+e)^3*(c^2-c^2*sin(f*x+e))^2/f+7/40*a*(2*A-B)*cos(f*x+e)
^3*(c^4-c^4*sin(f*x+e))/f

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3046, 2939, 2757, 2748, 2715, 8} \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {7 a c^4 (2 A-B) \cos ^3(e+f x)}{24 f}+\frac {7 a (2 A-B) \cos ^3(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{40 f}+\frac {7 a c^4 (2 A-B) \sin (e+f x) \cos (e+f x)}{16 f}+\frac {7}{16} a c^4 x (2 A-B)+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f} \]

[In]

Int[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]

[Out]

(7*a*(2*A - B)*c^4*x)/16 + (7*a*(2*A - B)*c^4*Cos[e + f*x]^3)/(24*f) + (7*a*(2*A - B)*c^4*Cos[e + f*x]*Sin[e +
f*x])/(16*f) - (a*B*c*Cos[e + f*x]^3*(c - c*Sin[e + f*x])^3)/(6*f) + (a*(2*A - B)*Cos[e + f*x]^3*(c^2 - c^2*S
in[e + f*x])^2)/(10*f) + (7*a*(2*A - B)*Cos[e + f*x]^3*(c^4 - c^4*Sin[e + f*x]))/(40*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+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx \\ & = -\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {1}{2} (a (2 A-B) c) \int \cos ^2(e+f x) (c-c \sin (e+f x))^3 \, dx \\ & = -\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}+\frac {1}{10} \left (7 a (2 A-B) c^2\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x))^2 \, dx \\ & = -\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}+\frac {7 a (2 A-B) \cos ^3(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{40 f}+\frac {1}{8} \left (7 a (2 A-B) c^3\right ) \int \cos ^2(e+f x) (c-c \sin (e+f x)) \, dx \\ & = \frac {7 a (2 A-B) c^4 \cos ^3(e+f x)}{24 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}+\frac {7 a (2 A-B) \cos ^3(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{40 f}+\frac {1}{8} \left (7 a (2 A-B) c^4\right ) \int \cos ^2(e+f x) \, dx \\ & = \frac {7 a (2 A-B) c^4 \cos ^3(e+f x)}{24 f}+\frac {7 a (2 A-B) c^4 \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}+\frac {7 a (2 A-B) \cos ^3(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{40 f}+\frac {1}{16} \left (7 a (2 A-B) c^4\right ) \int 1 \, dx \\ & = \frac {7}{16} a (2 A-B) c^4 x+\frac {7 a (2 A-B) c^4 \cos ^3(e+f x)}{24 f}+\frac {7 a (2 A-B) c^4 \cos (e+f x) \sin (e+f x)}{16 f}-\frac {a B c \cos ^3(e+f x) (c-c \sin (e+f x))^3}{6 f}+\frac {a (2 A-B) \cos ^3(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^2}{10 f}+\frac {7 a (2 A-B) \cos ^3(e+f x) \left (c^4-c^4 \sin (e+f x)\right )}{40 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.35 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.77 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {a c^4 \cos (e+f x) \left (272 A-176 B-\frac {210 (2 A-B) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )}{\sqrt {\cos ^2(e+f x)}}+15 (2 A+7 B) \sin (e+f x)-32 (7 A-B) \sin ^2(e+f x)+10 (18 A-17 B) \sin ^3(e+f x)-48 (A-3 B) \sin ^4(e+f x)-40 B \sin ^5(e+f x)\right )}{240 f} \]

[In]

Integrate[(a + a*Sin[e + f*x])*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^4,x]

[Out]

(a*c^4*Cos[e + f*x]*(272*A - 176*B - (210*(2*A - B)*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]])/Sqrt[Cos[e + f*x]^
2] + 15*(2*A + 7*B)*Sin[e + f*x] - 32*(7*A - B)*Sin[e + f*x]^2 + 10*(18*A - 17*B)*Sin[e + f*x]^3 - 48*(A - 3*B
)*Sin[e + f*x]^4 - 40*B*Sin[e + f*x]^5))/(240*f)

Maple [A] (verified)

Time = 2.02 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.65

method	result	size
parallelrisch	\(-\frac {c^{4} a \left (\left (-\frac {65 A}{3}+\frac {35 B}{3}\right ) \cos \left (3 f x +3 e \right )+\left (A -3 B \right ) \cos \left (5 f x +5 e \right )+\left (-20 A -\frac {5 B}{4}\right ) \sin \left (2 f x +2 e \right )+\left (\frac {15 A}{2}-\frac {35 B}{4}\right ) \sin \left (4 f x +4 e \right )+\frac {5 B \sin \left (6 f x +6 e \right )}{12}+\left (-70 A +50 B \right ) \cos \left (f x +e \right )-70 f x A +35 f x B -\frac {272 A}{3}+\frac {176 B}{3}\right )}{80 f}\)	\(119\)
risch	\(\frac {7 a \,c^{4} x A}{8}-\frac {7 a \,c^{4} x B}{16}+\frac {7 c^{4} a \cos \left (f x +e \right ) A}{8 f}-\frac {5 c^{4} a \cos \left (f x +e \right ) B}{8 f}-\frac {B \,c^{4} a \sin \left (6 f x +6 e \right )}{192 f}-\frac {c^{4} a \cos \left (5 f x +5 e \right ) A}{80 f}+\frac {3 c^{4} a \cos \left (5 f x +5 e \right ) B}{80 f}-\frac {3 \sin \left (4 f x +4 e \right ) A \,c^{4} a}{32 f}+\frac {7 \sin \left (4 f x +4 e \right ) B \,c^{4} a}{64 f}+\frac {13 c^{4} a \cos \left (3 f x +3 e \right ) A}{48 f}-\frac {7 c^{4} a \cos \left (3 f x +3 e \right ) B}{48 f}+\frac {\sin \left (2 f x +2 e \right ) A \,c^{4} a}{4 f}+\frac {\sin \left (2 f x +2 e \right ) B \,c^{4} a}{64 f}\)	\(221\)
parts	\(-\frac {\left (-3 A \,c^{4} a +B \,c^{4} a \right ) \cos \left (f x +e \right )}{f}+\frac {\left (-3 A \,c^{4} a +2 B \,c^{4} a \right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (A \,c^{4} a -3 B \,c^{4} a \right ) \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}+\frac {\left (2 A \,c^{4} a -3 B \,c^{4} a \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (2 A \,c^{4} a +2 B \,c^{4} a \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+a \,c^{4} x A +\frac {B \,c^{4} a \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )}{f}\)	\(260\)
derivativedivides	\(\frac {-\frac {A \,c^{4} a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-3 A \,c^{4} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 A \,c^{4} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 A \,c^{4} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 A \,c^{4} a \cos \left (f x +e \right )+B \,c^{4} a \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {3 B \,c^{4} a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+2 B \,c^{4} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 B \,c^{4} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-3 B \,c^{4} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{4} a \left (f x +e \right )-B \,c^{4} a \cos \left (f x +e \right )}{f}\)	\(342\)
default	\(\frac {-\frac {A \,c^{4} a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-3 A \,c^{4} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 A \,c^{4} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 A \,c^{4} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+3 A \,c^{4} a \cos \left (f x +e \right )+B \,c^{4} a \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {3 B \,c^{4} a \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+2 B \,c^{4} a \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-\frac {2 B \,c^{4} a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-3 B \,c^{4} a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+A \,c^{4} a \left (f x +e \right )-B \,c^{4} a \cos \left (f x +e \right )}{f}\)	\(342\)
norman	\(\frac {\left (\frac {7}{8} A \,c^{4} a -\frac {7}{16} B \,c^{4} a \right ) x +\left (\frac {7}{8} A \,c^{4} a -\frac {7}{16} B \,c^{4} a \right ) x \left (\tan ^{12}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {21}{4} A \,c^{4} a -\frac {21}{8} B \,c^{4} a \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {21}{4} A \,c^{4} a -\frac {21}{8} B \,c^{4} a \right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {35}{2} A \,c^{4} a -\frac {35}{4} B \,c^{4} a \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {105}{8} A \,c^{4} a -\frac {105}{16} B \,c^{4} a \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {105}{8} A \,c^{4} a -\frac {105}{16} B \,c^{4} a \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (12 A \,c^{4} a -4 B \,c^{4} a \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {\left (22 A \,c^{4} a -18 B \,c^{4} a \right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {34 A \,c^{4} a -22 B \,c^{4} a}{15 f}+\frac {2 \left (3 A \,c^{4} a -B \,c^{4} a \right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}+\frac {4 \left (17 A \,c^{4} a -11 B \,c^{4} a \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}+\frac {2 \left (19 A \,c^{4} a -17 B \,c^{4} a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 f}+\frac {c^{4} a \left (2 A +7 B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{8 f}-\frac {c^{4} a \left (2 A +7 B \right ) \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{8 f}+\frac {c^{4} a \left (26 A -37 B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {c^{4} a \left (26 A -37 B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {c^{4} a \left (162 A -73 B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}-\frac {c^{4} a \left (162 A -73 B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{24 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{6}}\)	\(536\)

[In]

int((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

-1/80*c^4*a*((-65/3*A+35/3*B)*cos(3*f*x+3*e)+(A-3*B)*cos(5*f*x+5*e)+(-20*A-5/4*B)*sin(2*f*x+2*e)+(15/2*A-35/4*
B)*sin(4*f*x+4*e)+5/12*B*sin(6*f*x+6*e)+(-70*A+50*B)*cos(f*x+e)-70*f*x*A+35*f*x*B-272/3*A+176/3*B)/f

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.68 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=-\frac {48 \, {\left (A - 3 \, B\right )} a c^{4} \cos \left (f x + e\right )^{5} - 320 \, {\left (A - B\right )} a c^{4} \cos \left (f x + e\right )^{3} - 105 \, {\left (2 \, A - B\right )} a c^{4} f x + 5 \, {\left (8 \, B a c^{4} \cos \left (f x + e\right )^{5} + 2 \, {\left (18 \, A - 25 \, B\right )} a c^{4} \cos \left (f x + e\right )^{3} - 21 \, {\left (2 \, A - B\right )} a c^{4} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{240 \, f} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

-1/240*(48*(A - 3*B)*a*c^4*cos(f*x + e)^5 - 320*(A - B)*a*c^4*cos(f*x + e)^3 - 105*(2*A - B)*a*c^4*f*x + 5*(8*
B*a*c^4*cos(f*x + e)^5 + 2*(18*A - 25*B)*a*c^4*cos(f*x + e)^3 - 21*(2*A - B)*a*c^4*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. 853 vs. \(2 (163) = 326\).

Time = 0.44 (sec) , antiderivative size = 853, normalized size of antiderivative = 4.69 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\begin {cases} - \frac {9 A a c^{4} x \sin ^{4}{\left (e + f x \right )}}{8} - \frac {9 A a c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + A a c^{4} x \sin ^{2}{\left (e + f x \right )} - \frac {9 A a c^{4} x \cos ^{4}{\left (e + f x \right )}}{8} + A a c^{4} x \cos ^{2}{\left (e + f x \right )} + A a c^{4} x - \frac {A a c^{4} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {15 A a c^{4} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {4 A a c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {2 A a c^{4} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} + \frac {9 A a c^{4} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {A a c^{4} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {8 A a c^{4} \cos ^{5}{\left (e + f x \right )}}{15 f} - \frac {4 A a c^{4} \cos ^{3}{\left (e + f x \right )}}{3 f} + \frac {3 A a c^{4} \cos {\left (e + f x \right )}}{f} + \frac {5 B a c^{4} x \sin ^{6}{\left (e + f x \right )}}{16} + \frac {15 B a c^{4} x \sin ^{4}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{16} + \frac {3 B a c^{4} x \sin ^{4}{\left (e + f x \right )}}{4} + \frac {15 B a c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{4}{\left (e + f x \right )}}{16} + \frac {3 B a c^{4} x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{2} - \frac {3 B a c^{4} x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {5 B a c^{4} x \cos ^{6}{\left (e + f x \right )}}{16} + \frac {3 B a c^{4} x \cos ^{4}{\left (e + f x \right )}}{4} - \frac {3 B a c^{4} x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {11 B a c^{4} \sin ^{5}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{16 f} + \frac {3 B a c^{4} \sin ^{4}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 B a c^{4} \sin ^{3}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{6 f} - \frac {5 B a c^{4} \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{4 f} + \frac {4 B a c^{4} \sin ^{2}{\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{f} - \frac {2 B a c^{4} \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {5 B a c^{4} \sin {\left (e + f x \right )} \cos ^{5}{\left (e + f x \right )}}{16 f} - \frac {3 B a c^{4} \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{4 f} + \frac {3 B a c^{4} \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} + \frac {8 B a c^{4} \cos ^{5}{\left (e + f x \right )}}{5 f} - \frac {4 B a c^{4} \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a c^{4} \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 ) \left (- c \sin {\left (e \right )} + c\right )^{4} & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**4,x)

[Out]

Piecewise((-9*A*a*c**4*x*sin(e + f*x)**4/8 - 9*A*a*c**4*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + A*a*c**4*x*sin(e
+ f*x)**2 - 9*A*a*c**4*x*cos(e + f*x)**4/8 + A*a*c**4*x*cos(e + f*x)**2 + A*a*c**4*x - A*a*c**4*sin(e + f*x)*
*4*cos(e + f*x)/f + 15*A*a*c**4*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 4*A*a*c**4*sin(e + f*x)**2*cos(e + f*x)**
3/(3*f) - 2*A*a*c**4*sin(e + f*x)**2*cos(e + f*x)/f + 9*A*a*c**4*sin(e + f*x)*cos(e + f*x)**3/(8*f) - A*a*c**4
*sin(e + f*x)*cos(e + f*x)/f - 8*A*a*c**4*cos(e + f*x)**5/(15*f) - 4*A*a*c**4*cos(e + f*x)**3/(3*f) + 3*A*a*c*
*4*cos(e + f*x)/f + 5*B*a*c**4*x*sin(e + f*x)**6/16 + 15*B*a*c**4*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 3*B*a
*c**4*x*sin(e + f*x)**4/4 + 15*B*a*c**4*x*sin(e + f*x)**2*cos(e + f*x)**4/16 + 3*B*a*c**4*x*sin(e + f*x)**2*co
s(e + f*x)**2/2 - 3*B*a*c**4*x*sin(e + f*x)**2/2 + 5*B*a*c**4*x*cos(e + f*x)**6/16 + 3*B*a*c**4*x*cos(e + f*x)
**4/4 - 3*B*a*c**4*x*cos(e + f*x)**2/2 - 11*B*a*c**4*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 3*B*a*c**4*sin(e +
f*x)**4*cos(e + f*x)/f - 5*B*a*c**4*sin(e + f*x)**3*cos(e + f*x)**3/(6*f) - 5*B*a*c**4*sin(e + f*x)**3*cos(e +
f*x)/(4*f) + 4*B*a*c**4*sin(e + f*x)**2*cos(e + f*x)**3/f - 2*B*a*c**4*sin(e + f*x)**2*cos(e + f*x)/f - 5*B*a
*c**4*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 3*B*a*c**4*sin(e + f*x)*cos(e + f*x)**3/(4*f) + 3*B*a*c**4*sin(e +
f*x)*cos(e + f*x)/(2*f) + 8*B*a*c**4*cos(e + f*x)**5/(5*f) - 4*B*a*c**4*cos(e + f*x)**3/(3*f) - B*a*c**4*cos(
e + f*x)/f, Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)*(-c*sin(e) + c)**4, True))

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.85 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=-\frac {64 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} A a c^{4} - 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a c^{4} + 90 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{4} - 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a c^{4} - 960 \, {\left (f x + e\right )} A a c^{4} - 192 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a c^{4} - 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a c^{4} - 5 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{4} - 60 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{4} + 720 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c^{4} - 2880 \, A a c^{4} \cos \left (f x + e\right ) + 960 \, B a c^{4} \cos \left (f x + e\right )}{960 \, f} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm="maxima")

[Out]

-1/960*(64*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a*c^4 - 640*(cos(f*x + e)^3 - 3*cos(f*x
+ e))*A*a*c^4 + 90*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a*c^4 - 480*(2*f*x + 2*e - sin(2*
f*x + 2*e))*A*a*c^4 - 960*(f*x + e)*A*a*c^4 - 192*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a
*c^4 - 640*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a*c^4 - 5*(4*sin(2*f*x + 2*e)^3 + 60*f*x + 60*e + 9*sin(4*f*x +
4*e) - 48*sin(2*f*x + 2*e))*B*a*c^4 - 60*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a*c^4 + 72
0*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a*c^4 - 2880*A*a*c^4*cos(f*x + e) + 960*B*a*c^4*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.98 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=-\frac {B a c^{4} \sin \left (6 \, f x + 6 \, e\right )}{192 \, f} + \frac {7}{16} \, {\left (2 \, A a c^{4} - B a c^{4}\right )} x - \frac {{\left (A a c^{4} - 3 \, B a c^{4}\right )} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {{\left (13 \, A a c^{4} - 7 \, B a c^{4}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} + \frac {{\left (7 \, A a c^{4} - 5 \, B a c^{4}\right )} \cos \left (f x + e\right )}{8 \, f} - \frac {{\left (6 \, A a c^{4} - 7 \, B a c^{4}\right )} \sin \left (4 \, f x + 4 \, e\right )}{64 \, f} + \frac {{\left (16 \, A a c^{4} + B a c^{4}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \]

[In]

integrate((a+a*sin(f*x+e))*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^4,x, algorithm="giac")

[Out]

-1/192*B*a*c^4*sin(6*f*x + 6*e)/f + 7/16*(2*A*a*c^4 - B*a*c^4)*x - 1/80*(A*a*c^4 - 3*B*a*c^4)*cos(5*f*x + 5*e)
/f + 1/48*(13*A*a*c^4 - 7*B*a*c^4)*cos(3*f*x + 3*e)/f + 1/8*(7*A*a*c^4 - 5*B*a*c^4)*cos(f*x + e)/f - 1/64*(6*A
*a*c^4 - 7*B*a*c^4)*sin(4*f*x + 4*e)/f + 1/64*(16*A*a*c^4 + B*a*c^4)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 15.30 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.49 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c-c \sin (e+f x))^4 \, dx=\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {A\,a\,c^4}{4}+\frac {7\,B\,a\,c^4}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (6\,A\,a\,c^4-2\,B\,a\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (12\,A\,a\,c^4-4\,B\,a\,c^4\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {A\,a\,c^4}{4}+\frac {7\,B\,a\,c^4}{8}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (22\,A\,a\,c^4-18\,B\,a\,c^4\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {13\,A\,a\,c^4}{2}-\frac {37\,B\,a\,c^4}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {13\,A\,a\,c^4}{2}-\frac {37\,B\,a\,c^4}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {38\,A\,a\,c^4}{5}-\frac {34\,B\,a\,c^4}{5}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (\frac {68\,A\,a\,c^4}{3}-\frac {44\,B\,a\,c^4}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {27\,A\,a\,c^4}{4}-\frac {73\,B\,a\,c^4}{24}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (\frac {27\,A\,a\,c^4}{4}-\frac {73\,B\,a\,c^4}{24}\right )+\frac {34\,A\,a\,c^4}{15}-\frac {22\,B\,a\,c^4}{15}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+20\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+15\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {7\,a\,c^4\,\mathrm {atan}\left (\frac {7\,a\,c^4\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,A-B\right )}{8\,\left (\frac {7\,A\,a\,c^4}{4}-\frac {7\,B\,a\,c^4}{8}\right )}\right )\,\left (2\,A-B\right )}{8\,f} \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))*(c - c*sin(e + f*x))^4,x)

[Out]

(tan(e/2 + (f*x)/2)*((A*a*c^4)/4 + (7*B*a*c^4)/8) + tan(e/2 + (f*x)/2)^10*(6*A*a*c^4 - 2*B*a*c^4) + tan(e/2 +
(f*x)/2)^4*(12*A*a*c^4 - 4*B*a*c^4) - tan(e/2 + (f*x)/2)^11*((A*a*c^4)/4 + (7*B*a*c^4)/8) + tan(e/2 + (f*x)/2)
^8*(22*A*a*c^4 - 18*B*a*c^4) + tan(e/2 + (f*x)/2)^5*((13*A*a*c^4)/2 - (37*B*a*c^4)/4) - tan(e/2 + (f*x)/2)^7*(
(13*A*a*c^4)/2 - (37*B*a*c^4)/4) + tan(e/2 + (f*x)/2)^2*((38*A*a*c^4)/5 - (34*B*a*c^4)/5) + tan(e/2 + (f*x)/2)
^6*((68*A*a*c^4)/3 - (44*B*a*c^4)/3) + tan(e/2 + (f*x)/2)^3*((27*A*a*c^4)/4 - (73*B*a*c^4)/24) - tan(e/2 + (f*
x)/2)^9*((27*A*a*c^4)/4 - (73*B*a*c^4)/24) + (34*A*a*c^4)/15 - (22*B*a*c^4)/15)/(f*(6*tan(e/2 + (f*x)/2)^2 + 1
5*tan(e/2 + (f*x)/2)^4 + 20*tan(e/2 + (f*x)/2)^6 + 15*tan(e/2 + (f*x)/2)^8 + 6*tan(e/2 + (f*x)/2)^10 + tan(e/2
+ (f*x)/2)^12 + 1)) + (7*a*c^4*atan((7*a*c^4*tan(e/2 + (f*x)/2)*(2*A - B))/(8*((7*A*a*c^4)/4 - (7*B*a*c^4)/8)
))*(2*A - B))/(8*f)