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

Optimal. Leaf size=201 \[ \frac{a^3 (20 A c+15 A d+15 B c+13 B d) \cos ^3(e+f x)}{60 f}-\frac{a^3 (20 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{5 f}-\frac{3 a^3 (20 A c+15 A d+15 B c+13 B d) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{1}{8} a^3 x (20 A c+15 A d+15 B c+13 B d)-\frac{(5 A d+5 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f} \]

[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)

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Rubi [A]  time = 0.334971, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.242, Rules used = {2968, 3023, 2751, 2645, 2638, 2635, 8, 2633} \[ \frac{a^3 (20 A c+15 A d+15 B c+13 B d) \cos ^3(e+f x)}{60 f}-\frac{a^3 (20 A c+15 A d+15 B c+13 B d) \cos (e+f x)}{5 f}-\frac{3 a^3 (20 A c+15 A d+15 B c+13 B d) \sin (e+f x) \cos (e+f x)}{40 f}+\frac{1}{8} a^3 x (20 A c+15 A d+15 B c+13 B d)-\frac{(5 A d+5 B c-B d) \cos (e+f x) (a \sin (e+f x)+a)^3}{20 f}-\frac{B d \cos (e+f x) (a \sin (e+f x)+a)^4}{5 a f} \]

Antiderivative was successfully verified.

[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 2968

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 3023

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]

Rule 2751

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*Sin
[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 2645

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 2638

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

Rule 2635

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] && Integer
Q[2*n]

Rule 8

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

Rule 2633

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]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c+d \sin (e+f x)) \, dx &=\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 ) \operatorname{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]  time = 1.06431, size = 156, normalized size = 0.78 \[ \frac{\cos (e+f x) \left (-\frac{1}{4} a^4 (5 A d+5 B c-B d) (\sin (e+f x)+1)^3-\frac{a^4 (20 A c+15 A d+15 B c+13 B d) \left (30 \sin ^{-1}\left (\frac{\sqrt{1-\sin (e+f x)}}{\sqrt{2}}\right )+\left (2 \sin ^2(e+f x)+9 \sin (e+f x)+22\right ) \sqrt{\cos ^2(e+f x)}\right )}{24 \sqrt{\cos ^2(e+f x)}}-B d (a \sin (e+f x)+a)^4\right )}{5 a f} \]

Antiderivative was successfully verified.

[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]*(-(a^4*(5*B*c + 5*A*d - B*d)*(1 + Sin[e + f*x])^3)/4 - 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)

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Maple [B]  time = 0.059, size = 414, normalized size = 2.1 \begin{align*}{\frac{1}{f} \left ( -{\frac{A{a}^{3}c \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+A{a}^{3}d \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) +B{a}^{3}c \left ( -{\frac{\cos \left ( fx+e \right ) }{4} \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+{\frac{3\,\sin \left ( fx+e \right ) }{2}} \right ) }+{\frac{3\,fx}{8}}+{\frac{3\,e}{8}} \right ) -{\frac{B{a}^{3}d\cos \left ( fx+e \right ) }{5} \left ({\frac{8}{3}}+ \left ( \sin \left ( fx+e \right ) \right ) ^{4}+{\frac{4\, \left ( \sin \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) }+3\,A{a}^{3}c \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -A{a}^{3}d \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) -B{a}^{3}c \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) +3\,B{a}^{3}d \left ( -1/4\, \left ( \left ( \sin \left ( fx+e \right ) \right ) ^{3}+3/2\,\sin \left ( fx+e \right ) \right ) \cos \left ( fx+e \right ) +3/8\,fx+3/8\,e \right ) -3\,A{a}^{3}c\cos \left ( fx+e \right ) +3\,A{a}^{3}d \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) +3\,B{a}^{3}c \left ( -1/2\,\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) +1/2\,fx+e/2 \right ) -B{a}^{3}d \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) +A{a}^{3}c \left ( fx+e \right ) -A{a}^{3}d\cos \left ( fx+e \right ) -B{a}^{3}c\cos \left ( fx+e \right ) +B{a}^{3}d \left ( -{\frac{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) }{2}}+{\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 0.998088, size = 537, normalized size = 2.67 \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Fricas [A]  time = 2.18467, size = 437, normalized size = 2.17 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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Sympy [A]  time = 5.69284, size = 960, normalized size = 4.78 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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Giac [A]  time = 1.24446, size = 293, normalized size = 1.46 \begin{align*} -\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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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