\(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx\) [253]

Optimal result

Integrand size = 33, antiderivative size = 166 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {1}{8} a^2 (12 A c+8 B c+8 A d+7 B d) x-\frac {a^2 (12 A c+8 B c+8 A d+7 B d) \cos (e+f x)}{6 f}-\frac {a^2 (12 A c+8 B c+8 A d+7 B d) \cos (e+f x) \sin (e+f x)}{24 f}-\frac {(4 B c+4 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f} \]

[Out]

1/8*a^2*(12*A*c+8*A*d+8*B*c+7*B*d)*x-1/6*a^2*(12*A*c+8*A*d+8*B*c+7*B*d)*cos(f*x+e)/f-1/24*a^2*(12*A*c+8*A*d+8*
B*c+7*B*d)*cos(f*x+e)*sin(f*x+e)/f-1/12*(4*A*d+4*B*c-B*d)*cos(f*x+e)*(a+a*sin(f*x+e))^2/f-1/4*B*d*cos(f*x+e)*(
a+a*sin(f*x+e))^3/a/f

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3047, 3102, 2830, 2723} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {a^2 (12 A c+8 A d+8 B c+7 B d) \cos (e+f x)}{6 f}-\frac {a^2 (12 A c+8 A d+8 B c+7 B d) \sin (e+f x) \cos (e+f x)}{24 f}+\frac {1}{8} a^2 x (12 A c+8 A d+8 B c+7 B d)-\frac {(4 A d+4 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^2}{12 f}-\frac {B d \cos (e+f x) (a \sin (e+f x)+a)^3}{4 a f} \]

[In]

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

[Out]

(a^2*(12*A*c + 8*B*c + 8*A*d + 7*B*d)*x)/8 - (a^2*(12*A*c + 8*B*c + 8*A*d + 7*B*d)*Cos[e + f*x])/(6*f) - (a^2*
(12*A*c + 8*B*c + 8*A*d + 7*B*d)*Cos[e + f*x]*Sin[e + f*x])/(24*f) - ((4*B*c + 4*A*d - B*d)*Cos[e + f*x]*(a +
a*Sin[e + f*x])^2)/(12*f) - (B*d*Cos[e + f*x]*(a + a*Sin[e + f*x])^3)/(4*a*f)

Rule 2723

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^2, x_Symbol] :> Simp[(2*a^2 + b^2)*(x/2), x] + (-Simp[2*a*b*(Cos[c
+ d*x]/d), x] - Simp[b^2*Cos[c + d*x]*(Sin[c + d*x]/(2*d)), x]) /; FreeQ[{a, b, c, d}, x]

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))^2 \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))^3}{4 a f}+\frac {\int (a+a \sin (e+f x))^2 (a (4 A c+3 B d)+a (4 B c+4 A d-B d) \sin (e+f x)) \, dx}{4 a} \\ & = -\frac {(4 B c+4 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f}+\frac {1}{12} (12 A c+8 B c+8 A d+7 B d) \int (a+a \sin (e+f x))^2 \, dx \\ & = \frac {1}{8} a^2 (12 A c+8 B c+8 A d+7 B d) x-\frac {a^2 (12 A c+8 B c+8 A d+7 B d) \cos (e+f x)}{6 f}-\frac {a^2 (12 A c+8 B c+8 A d+7 B d) \cos (e+f x) \sin (e+f x)}{24 f}-\frac {(4 B c+4 A d-B d) \cos (e+f x) (a+a \sin (e+f x))^2}{12 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^3}{4 a f} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.43 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.87 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {\cos (e+f x) \left (-\frac {1}{3} a^3 (4 B c+4 A d-B d) (1+\sin (e+f x))^2-B d (a+a \sin (e+f x))^3-\frac {a^3 (12 A c+8 B c+8 A d+7 B d) \left (6 \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} (4+\sin (e+f x))\right )}{6 \sqrt {\cos ^2(e+f x)}}\right )}{4 a f} \]

[In]

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

[Out]

(Cos[e + f*x]*(-1/3*(a^3*(4*B*c + 4*A*d - B*d)*(1 + Sin[e + f*x])^2) - B*d*(a + a*Sin[e + f*x])^3 - (a^3*(12*A
*c + 8*B*c + 8*A*d + 7*B*d)*(6*ArcSin[Sqrt[1 - Sin[e + f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(4 + Sin[e + f*x]
)))/(6*Sqrt[Cos[e + f*x]^2])))/(4*a*f)

Maple [A] (verified)

Time = 1.16 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.81

method	result	size
parallelrisch	\(\frac {\left (\left (\left (-6 c -6 d \right ) B -3 A \left (c +2 d \right )\right ) \sin \left (2 f x +2 e \right )+\left (d A +B \left (c +2 d \right )\right ) \cos \left (3 f x +3 e \right )+\frac {3 d B \sin \left (4 f x +4 e \right )}{8}+\left (\left (-21 c -18 d \right ) B -24 \left (c +\frac {7 d}{8}\right ) A \right ) \cos \left (f x +e \right )+\left (12 c f x +\frac {21}{2} d f x -20 c -16 d \right ) B +18 \left (\left (\frac {2 f x}{3}-\frac {10}{9}\right ) d +c \left (f x -\frac {4}{3}\right )\right ) A \right ) a^{2}}{12 f}\)	\(135\)
parts	\(-\frac {\left (A \,a^{2} d +B \,a^{2} c +2 a^{2} d B \right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}-\frac {\left (2 a^{2} A c +A \,a^{2} d +B \,a^{2} c \right ) \cos \left (f x +e \right )}{f}+\frac {\left (a^{2} A c +2 A \,a^{2} d +2 B \,a^{2} c +a^{2} d B \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+A \,a^{2} c x +\frac {a^{2} d B \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}\)	\(176\)
risch	\(\frac {3 A \,a^{2} c x}{2}+A \,a^{2} d x +B \,a^{2} c x +\frac {7 B \,a^{2} d x}{8}-\frac {2 a^{2} \cos \left (f x +e \right ) A c}{f}-\frac {7 a^{2} \cos \left (f x +e \right ) d A}{4 f}-\frac {7 a^{2} \cos \left (f x +e \right ) B c}{4 f}-\frac {3 a^{2} \cos \left (f x +e \right ) d B}{2 f}+\frac {a^{2} d B \sin \left (4 f x +4 e \right )}{32 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) d A}{12 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) B c}{12 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) d B}{6 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} A c}{4 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} d}{2 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{2} c}{2 f}-\frac {\sin \left (2 f x +2 e \right ) a^{2} d B}{2 f}\)	\(248\)
derivativedivides	\(\frac {a^{2} A c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A \,a^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d B \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 a^{2} A c \cos \left (f x +e \right )+2 A \,a^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} d B \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} A c \left (f x +e \right )-A \,a^{2} d \cos \left (f x +e \right )-B \,a^{2} c \cos \left (f x +e \right )+a^{2} d B \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\)	\(278\)
default	\(\frac {a^{2} A c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {A \,a^{2} d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}-\frac {B \,a^{2} c \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} d B \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )-2 a^{2} A c \cos \left (f x +e \right )+2 A \,a^{2} d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+2 B \,a^{2} c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 a^{2} d B \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+a^{2} A c \left (f x +e \right )-A \,a^{2} d \cos \left (f x +e \right )-B \,a^{2} c \cos \left (f x +e \right )+a^{2} d B \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}\)	\(278\)
norman	\(\frac {\left (\frac {3}{2} a^{2} A c +A \,a^{2} d +B \,a^{2} c +\frac {7}{8} a^{2} d B \right ) x +\left (6 a^{2} A c +4 A \,a^{2} d +4 B \,a^{2} c +\frac {7}{2} a^{2} d B \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (6 a^{2} A c +4 A \,a^{2} d +4 B \,a^{2} c +\frac {7}{2} a^{2} d B \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (9 a^{2} A c +6 A \,a^{2} d +6 B \,a^{2} c +\frac {21}{4} a^{2} d B \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {3}{2} a^{2} A c +A \,a^{2} d +B \,a^{2} c +\frac {7}{8} a^{2} d B \right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {12 a^{2} A c +10 A \,a^{2} d +10 B \,a^{2} c +8 a^{2} d B}{3 f}-\frac {2 \left (2 a^{2} A c +A \,a^{2} d +B \,a^{2} c \right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (6 a^{2} A c +5 A \,a^{2} d +5 B \,a^{2} c +4 a^{2} d B \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (18 a^{2} A c +17 A \,a^{2} d +17 B \,a^{2} c +16 a^{2} d B \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a^{2} \left (4 A c +8 d A +8 B c +7 d B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (4 A c +8 d A +8 B c +7 d B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} \left (4 A c +8 d A +8 B c +15 d B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}+\frac {a^{2} \left (4 A c +8 d A +8 B c +15 d B \right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{4}}\)	\(515\)

[In]

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

[Out]

1/12*(((-6*c-6*d)*B-3*A*(c+2*d))*sin(2*f*x+2*e)+(d*A+B*(c+2*d))*cos(3*f*x+3*e)+3/8*d*B*sin(4*f*x+4*e)+((-21*c-
18*d)*B-24*(c+7/8*d)*A)*cos(f*x+e)+(12*c*f*x+21/2*d*f*x-20*c-16*d)*B+18*((2/3*f*x-10/9)*d+c*(f*x-4/3))*A)*a^2/
f

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.87 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {8 \, {\left (B a^{2} c + {\left (A + 2 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )^{3} + 3 \, {\left (4 \, {\left (3 \, A + 2 \, B\right )} a^{2} c + {\left (8 \, A + 7 \, B\right )} a^{2} d\right )} f x - 48 \, {\left ({\left (A + B\right )} a^{2} c + {\left (A + B\right )} a^{2} d\right )} \cos \left (f x + e\right ) + 3 \, {\left (2 \, B a^{2} d \cos \left (f x + e\right )^{3} - {\left (4 \, {\left (A + 2 \, B\right )} a^{2} c + {\left (8 \, A + 9 \, B\right )} a^{2} d\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{24 \, f} \]

[In]

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

[Out]

1/24*(8*(B*a^2*c + (A + 2*B)*a^2*d)*cos(f*x + e)^3 + 3*(4*(3*A + 2*B)*a^2*c + (8*A + 7*B)*a^2*d)*f*x - 48*((A
+ B)*a^2*c + (A + B)*a^2*d)*cos(f*x + e) + 3*(2*B*a^2*d*cos(f*x + e)^3 - (4*(A + 2*B)*a^2*c + (8*A + 9*B)*a^2*
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. 571 vs. \(2 (163) = 326\).

Time = 0.23 (sec) , antiderivative size = 571, normalized size of antiderivative = 3.44 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\begin {cases} \frac {A a^{2} c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {A a^{2} c x \cos ^{2}{\left (e + f x \right )}}{2} + A a^{2} c x - \frac {A a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 A a^{2} c \cos {\left (e + f x \right )}}{f} + A a^{2} d x \sin ^{2}{\left (e + f x \right )} + A a^{2} d x \cos ^{2}{\left (e + f x \right )} - \frac {A a^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {A a^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 A a^{2} d \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {A a^{2} d \cos {\left (e + f x \right )}}{f} + B a^{2} c x \sin ^{2}{\left (e + f x \right )} + B a^{2} c x \cos ^{2}{\left (e + f x \right )} - \frac {B a^{2} c \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {B a^{2} c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {2 B a^{2} c \cos ^{3}{\left (e + f x \right )}}{3 f} - \frac {B a^{2} c \cos {\left (e + f x \right )}}{f} + \frac {3 B a^{2} d x \sin ^{4}{\left (e + f x \right )}}{8} + \frac {3 B a^{2} d x \sin ^{2}{\left (e + f x \right )} \cos ^{2}{\left (e + f x \right )}}{4} + \frac {B a^{2} d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {3 B a^{2} d x \cos ^{4}{\left (e + f x \right )}}{8} + \frac {B a^{2} d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {5 B a^{2} d \sin ^{3}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{8 f} - \frac {2 B a^{2} d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {3 B a^{2} d \sin {\left (e + f x \right )} \cos ^{3}{\left (e + f x \right )}}{8 f} - \frac {B a^{2} d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {4 B a^{2} d \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (c + d \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )^{2} & \text {otherwise} \end {cases} \]

[In]

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e)),x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.61 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c + 96 \, {\left (f x + e\right )} A a^{2} c + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c + 48 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c + 32 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} d + 48 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} d + 64 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} d + 3 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} d + 24 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} d - 192 \, A a^{2} c \cos \left (f x + e\right ) - 96 \, B a^{2} c \cos \left (f x + e\right ) - 96 \, A a^{2} d \cos \left (f x + e\right )}{96 \, f} \]

[In]

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

[Out]

1/96*(24*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*c + 96*(f*x + e)*A*a^2*c + 32*(cos(f*x + e)^3 - 3*cos(f*x + e)
)*B*a^2*c + 48*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*c + 32*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^2*d + 48*(2
*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*d + 64*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*d + 3*(12*f*x + 12*e + sin
(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^2*d + 24*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*d - 192*A*a^2*c*cos(f*
x + e) - 96*B*a^2*c*cos(f*x + e) - 96*A*a^2*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 168, normalized size of antiderivative = 1.01 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {B a^{2} d \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} + \frac {1}{8} \, {\left (12 \, A a^{2} c + 8 \, B a^{2} c + 8 \, A a^{2} d + 7 \, B a^{2} d\right )} x + \frac {{\left (B a^{2} c + A a^{2} d + 2 \, B a^{2} d\right )} \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} - \frac {{\left (8 \, A a^{2} c + 7 \, B a^{2} c + 7 \, A a^{2} d + 6 \, B a^{2} d\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (A a^{2} c + 2 \, B a^{2} c + 2 \, A a^{2} d + 2 \, B a^{2} d\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

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

[Out]

1/32*B*a^2*d*sin(4*f*x + 4*e)/f + 1/8*(12*A*a^2*c + 8*B*a^2*c + 8*A*a^2*d + 7*B*a^2*d)*x + 1/12*(B*a^2*c + A*a
^2*d + 2*B*a^2*d)*cos(3*f*x + 3*e)/f - 1/4*(8*A*a^2*c + 7*B*a^2*c + 7*A*a^2*d + 6*B*a^2*d)*cos(f*x + e)/f - 1/
4*(A*a^2*c + 2*B*a^2*c + 2*A*a^2*d + 2*B*a^2*d)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 14.99 (sec) , antiderivative size = 492, normalized size of antiderivative = 2.96 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (12\,A\,c+8\,A\,d+8\,B\,c+7\,B\,d\right )}{4\,\left (3\,A\,a^2\,c+2\,A\,a^2\,d+2\,B\,a^2\,c+\frac {7\,B\,a^2\,d}{4}\right )}\right )\,\left (12\,A\,c+8\,A\,d+8\,B\,c+7\,B\,d\right )}{4\,f}-\frac {a^2\,\left (\mathrm {atan}\left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )-\frac {f\,x}{2}\right )\,\left (12\,A\,c+8\,A\,d+8\,B\,c+7\,B\,d\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (A\,a^2\,c+2\,A\,a^2\,d+2\,B\,a^2\,c+\frac {15\,B\,a^2\,d}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (A\,a^2\,c+2\,A\,a^2\,d+2\,B\,a^2\,c+\frac {7\,B\,a^2\,d}{4}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (A\,a^2\,c+2\,A\,a^2\,d+2\,B\,a^2\,c+\frac {15\,B\,a^2\,d}{4}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (12\,A\,a^2\,c+10\,A\,a^2\,d+10\,B\,a^2\,c+8\,B\,a^2\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (12\,A\,a^2\,c+\frac {34\,A\,a^2\,d}{3}+\frac {34\,B\,a^2\,c}{3}+\frac {32\,B\,a^2\,d}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,A\,a^2\,c+2\,A\,a^2\,d+2\,B\,a^2\,c\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a^2\,c+2\,A\,a^2\,d+2\,B\,a^2\,c+\frac {7\,B\,a^2\,d}{4}\right )+4\,A\,a^2\,c+\frac {10\,A\,a^2\,d}{3}+\frac {10\,B\,a^2\,c}{3}+\frac {8\,B\,a^2\,d}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^2*(c + d*sin(e + f*x)),x)

[Out]

(a^2*atan((a^2*tan(e/2 + (f*x)/2)*(12*A*c + 8*A*d + 8*B*c + 7*B*d))/(4*(3*A*a^2*c + 2*A*a^2*d + 2*B*a^2*c + (7
*B*a^2*d)/4)))*(12*A*c + 8*A*d + 8*B*c + 7*B*d))/(4*f) - (a^2*(atan(tan(e/2 + (f*x)/2)) - (f*x)/2)*(12*A*c + 8
*A*d + 8*B*c + 7*B*d))/(4*f) - (tan(e/2 + (f*x)/2)^3*(A*a^2*c + 2*A*a^2*d + 2*B*a^2*c + (15*B*a^2*d)/4) - tan(
e/2 + (f*x)/2)^7*(A*a^2*c + 2*A*a^2*d + 2*B*a^2*c + (7*B*a^2*d)/4) - tan(e/2 + (f*x)/2)^5*(A*a^2*c + 2*A*a^2*d
+ 2*B*a^2*c + (15*B*a^2*d)/4) + tan(e/2 + (f*x)/2)^4*(12*A*a^2*c + 10*A*a^2*d + 10*B*a^2*c + 8*B*a^2*d) + tan
(e/2 + (f*x)/2)^2*(12*A*a^2*c + (34*A*a^2*d)/3 + (34*B*a^2*c)/3 + (32*B*a^2*d)/3) + tan(e/2 + (f*x)/2)^6*(4*A*
a^2*c + 2*A*a^2*d + 2*B*a^2*c) + tan(e/2 + (f*x)/2)*(A*a^2*c + 2*A*a^2*d + 2*B*a^2*c + (7*B*a^2*d)/4) + 4*A*a^
2*c + (10*A*a^2*d)/3 + (10*B*a^2*c)/3 + (8*B*a^2*d)/3)/(f*(4*tan(e/2 + (f*x)/2)^2 + 6*tan(e/2 + (f*x)/2)^4 + 4
*tan(e/2 + (f*x)/2)^6 + tan(e/2 + (f*x)/2)^8 + 1))