\(\int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx\) [246]

Optimal result

Integrand size = 31, antiderivative size = 111 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {1}{2} a (B (c+d)+A (2 c+d)) x-\frac {a (3 A (c+d)+B (3 c+d)) \cos (e+f x)}{3 f}-\frac {a (3 B c+3 A d-B d) \cos (e+f x) \sin (e+f x)}{6 f}-\frac {B d \cos (e+f x) (a+a \sin (e+f x))^2}{3 a f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {3047, 3102, 2813} \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {a (3 A (c+d)+B (3 c+d)) \cos (e+f x)}{3 f}-\frac {a (3 A d+3 B c-B d) \sin (e+f x) \cos (e+f x)}{6 f}+\frac {1}{2} a x (A (2 c+d)+B (c+d))-\frac {B d \cos (e+f x) (a \sin (e+f x)+a)^2}{3 a f} \]

[In]

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

[Out]

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

Rule 2813

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

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

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.94 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {a (12 A c f x+6 B c f x+6 A d f x+6 B d f x-3 (4 A (c+d)+B (4 c+3 d)) \cos (e+f x)+B d \cos (3 (e+f x))-3 B c \sin (2 (e+f x))-3 A d \sin (2 (e+f x))-3 B d \sin (2 (e+f x)))}{12 f} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.85

method	result	size
parts	\(\frac {\left (A a d +B a c +d B a \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}-\frac {\left (A a c +A a d +B a c \right ) \cos \left (f x +e \right )}{f}+a A c x -\frac {d B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}\)	\(94\)
parallelrisch	\(-\frac {\left (\left (B \left (c +d \right )+d A \right ) \sin \left (2 f x +2 e \right )-\frac {B d \cos \left (3 f x +3 e \right )}{3}+\left (\left (4 c +3 d \right ) B +4 A \left (c +d \right )\right ) \cos \left (f x +e \right )+\left (-2 c f x -2 d f x +4 c +\frac {8}{3} d \right ) B -4 \left (\left (\frac {f x}{2}-1\right ) d +c \left (f x -1\right )\right ) A \right ) a}{4 f}\)	\(101\)
derivativedivides	\(\frac {-\frac {d B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A a d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+B a c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+d B a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A a c \cos \left (f x +e \right )-A a d \cos \left (f x +e \right )-B a c \cos \left (f x +e \right )+A a c \left (f x +e \right )}{f}\)	\(147\)
default	\(\frac {-\frac {d B a \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A a d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+B a c \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+d B a \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-A a c \cos \left (f x +e \right )-A a d \cos \left (f x +e \right )-B a c \cos \left (f x +e \right )+A a c \left (f x +e \right )}{f}\)	\(147\)
risch	\(a A c x +\frac {A a d x}{2}+\frac {B a c x}{2}+\frac {B a d x}{2}-\frac {a \cos \left (f x +e \right ) A c}{f}-\frac {a \cos \left (f x +e \right ) d A}{f}-\frac {a \cos \left (f x +e \right ) B c}{f}-\frac {3 a \cos \left (f x +e \right ) d B}{4 f}+\frac {B a d \cos \left (3 f x +3 e \right )}{12 f}-\frac {\sin \left (2 f x +2 e \right ) A a d}{4 f}-\frac {\sin \left (2 f x +2 e \right ) B a c}{4 f}-\frac {\sin \left (2 f x +2 e \right ) d B a}{4 f}\)	\(149\)
norman	\(\frac {\left (A a c +\frac {1}{2} A a d +\frac {1}{2} B a c +\frac {1}{2} d B a \right ) x +\left (A a c +\frac {1}{2} A a d +\frac {1}{2} B a c +\frac {1}{2} d B a \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 A a c +\frac {3}{2} A a d +\frac {3}{2} B a c +\frac {3}{2} d B a \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (3 A a c +\frac {3}{2} A a d +\frac {3}{2} B a c +\frac {3}{2} d B a \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\frac {\left (d A +B c +d B \right ) a \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {6 A a c +6 A a d +6 B a c +4 d B a}{3 f}-\frac {\left (2 A a c +2 A a d +2 B a c \right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (4 A a c +4 A a d +4 B a c +4 d B a \right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {\left (d A +B c +d B \right ) a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3}}\)	\(287\)

[In]

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

[Out]

(A*a*d+B*a*c+B*a*d)/f*(-1/2*cos(f*x+e)*sin(f*x+e)+1/2*f*x+1/2*e)-(A*a*c+A*a*d+B*a*c)/f*cos(f*x+e)+a*A*c*x-1/3*
d*B*a/f*(2+sin(f*x+e)^2)*cos(f*x+e)

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.76 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {2 \, B a d \cos \left (f x + e\right )^{3} + 3 \, {\left ({\left (2 \, A + B\right )} a c + {\left (A + B\right )} a d\right )} f x - 3 \, {\left (B a c + {\left (A + B\right )} a d\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 6 \, {\left ({\left (A + B\right )} a c + {\left (A + B\right )} a d\right )} \cos \left (f x + e\right )}{6 \, f} \]

[In]

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

[Out]

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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 277 vs. \(2 (100) = 200\).

Time = 0.14 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.50 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\begin {cases} A a c x - \frac {A a c \cos {\left (e + f x \right )}}{f} + \frac {A a d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {A a d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {A a d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {A a d \cos {\left (e + f x \right )}}{f} + \frac {B a c x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {B a c x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {B a c \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {B a c \cos {\left (e + f x \right )}}{f} + \frac {B a d x \sin ^{2}{\left (e + f x \right )}}{2} + \frac {B a d x \cos ^{2}{\left (e + f x \right )}}{2} - \frac {B a d \sin ^{2}{\left (e + f x \right )} \cos {\left (e + f x \right )}}{f} - \frac {B a d \sin {\left (e + f x \right )} \cos {\left (e + f x \right )}}{2 f} - \frac {2 B a d \cos ^{3}{\left (e + f x \right )}}{3 f} & \text {for}\: f \neq 0 \\x \left (A + B \sin {\left (e \right )}\right ) \left (c + d \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right ) & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.29 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {12 \, {\left (f x + e\right )} A a c + 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a c + 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a d + 4 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a d + 3 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a d - 12 \, A a c \cos \left (f x + e\right ) - 12 \, B a c \cos \left (f x + e\right ) - 12 \, A a d \cos \left (f x + e\right )}{12 \, f} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=\frac {B a d \cos \left (3 \, f x + 3 \, e\right )}{12 \, f} + \frac {1}{2} \, {\left (2 \, A a c + B a c + A a d + B a d\right )} x - \frac {{\left (4 \, A a c + 4 \, B a c + 4 \, A a d + 3 \, B a d\right )} \cos \left (f x + e\right )}{4 \, f} - \frac {{\left (B a c + A a d + B a d\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 12.65 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.21 \[ \int (a+a \sin (e+f x)) (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx=-\frac {\frac {3\,A\,a\,d\,\sin \left (2\,e+2\,f\,x\right )}{2}-\frac {B\,a\,d\,\cos \left (3\,e+3\,f\,x\right )}{2}+\frac {3\,B\,a\,c\,\sin \left (2\,e+2\,f\,x\right )}{2}+\frac {3\,B\,a\,d\,\sin \left (2\,e+2\,f\,x\right )}{2}+6\,A\,a\,c\,\cos \left (e+f\,x\right )+6\,A\,a\,d\,\cos \left (e+f\,x\right )+6\,B\,a\,c\,\cos \left (e+f\,x\right )+\frac {9\,B\,a\,d\,\cos \left (e+f\,x\right )}{2}-6\,A\,a\,c\,f\,x-3\,A\,a\,d\,f\,x-3\,B\,a\,c\,f\,x-3\,B\,a\,d\,f\,x}{6\,f} \]

[In]

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

[Out]

-((3*A*a*d*sin(2*e + 2*f*x))/2 - (B*a*d*cos(3*e + 3*f*x))/2 + (3*B*a*c*sin(2*e + 2*f*x))/2 + (3*B*a*d*sin(2*e
+ 2*f*x))/2 + 6*A*a*c*cos(e + f*x) + 6*A*a*d*cos(e + f*x) + 6*B*a*c*cos(e + f*x) + (9*B*a*d*cos(e + f*x))/2 -
6*A*a*c*f*x - 3*A*a*d*f*x - 3*B*a*c*f*x - 3*B*a*d*f*x)/(6*f)