3.26 \(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx\)

Optimal. Leaf size=229 \[ \frac{3 a^2 c^5 (8 A-3 B) \cos ^5(e+f x)}{80 f}+\frac{3 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{a^2 c^3 (8 A-3 B) \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac{9 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{9}{128} a^2 c^5 x (8 A-3 B)-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f} \]

[Out]

(9*a^2*(8*A - 3*B)*c^5*x)/128 + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^5)/(80*f) + (9*a^2*(8*A - 3*B)*c^5*Cos[e +
f*x]*Sin[e + f*x])/(128*f) + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (a^2*(8*A - 3*B)*c^
3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(56*f) - (a^2*B*c^2*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^3)/(8*f) + (3
*a^2*(8*A - 3*B)*Cos[e + f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(112*f)

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Rubi [A]  time = 0.367915, antiderivative size = 229, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2967, 2860, 2678, 2669, 2635, 8} \[ \frac{3 a^2 c^5 (8 A-3 B) \cos ^5(e+f x)}{80 f}+\frac{3 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{64 f}+\frac{a^2 c^3 (8 A-3 B) \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac{9 a^2 c^5 (8 A-3 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{9}{128} a^2 c^5 x (8 A-3 B)-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]

[Out]

(9*a^2*(8*A - 3*B)*c^5*x)/128 + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^5)/(80*f) + (9*a^2*(8*A - 3*B)*c^5*Cos[e +
f*x]*Sin[e + f*x])/(128*f) + (3*a^2*(8*A - 3*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(64*f) + (a^2*(8*A - 3*B)*c^
3*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^2)/(56*f) - (a^2*B*c^2*Cos[e + f*x]^5*(c - c*Sin[e + f*x])^3)/(8*f) + (3
*a^2*(8*A - 3*B)*Cos[e + f*x]^5*(c^5 - c^5*Sin[e + f*x]))/(112*f)

Rule 2967

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]))

Rule 2860

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] /; Fre
eQ[{a, b, c, d, e, f, g, m, p}, x] && EqQ[a^2 - b^2, 0] && NeQ[m + p + 1, 0]

Rule 2678

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 2669

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> -Simp[(b*(g*Cos[
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 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]

Rubi steps

\begin{align*} \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^3 \, dx\\ &=-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{1}{8} \left (a^2 (8 A-3 B) c^2\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=\frac{a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{1}{56} \left (9 a^2 (8 A-3 B) c^3\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=\frac{a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac{1}{16} \left (3 a^2 (8 A-3 B) c^4\right ) \int \cos ^4(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac{a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac{1}{16} \left (3 a^2 (8 A-3 B) c^5\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac{3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac{a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac{1}{64} \left (9 a^2 (8 A-3 B) c^5\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac{9 a^2 (8 A-3 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac{a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}+\frac{1}{128} \left (9 a^2 (8 A-3 B) c^5\right ) \int 1 \, dx\\ &=\frac{9}{128} a^2 (8 A-3 B) c^5 x+\frac{3 a^2 (8 A-3 B) c^5 \cos ^5(e+f x)}{80 f}+\frac{9 a^2 (8 A-3 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{3 a^2 (8 A-3 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{64 f}+\frac{a^2 (8 A-3 B) c^3 \cos ^5(e+f x) (c-c \sin (e+f x))^2}{56 f}-\frac{a^2 B c^2 \cos ^5(e+f x) (c-c \sin (e+f x))^3}{8 f}+\frac{3 a^2 (8 A-3 B) \cos ^5(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{112 f}\\ \end{align*}

Mathematica [A]  time = 1.93804, size = 219, normalized size = 0.96 \[ \frac{(a \sin (e+f x)+a)^2 (c-c \sin (e+f x))^5 (2520 (8 A-3 B) (e+f x)+560 (19 A-3 B) \sin (2 (e+f x))-280 (2 A-7 B) \sin (4 (e+f x))-560 (A-B) \sin (6 (e+f x))+560 (27 A-17 B) \cos (e+f x)+560 (13 A-7 B) \cos (3 (e+f x))+112 (11 A-B) \cos (5 (e+f x))-80 (A-3 B) \cos (7 (e+f x))-35 B \sin (8 (e+f x)))}{35840 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{10} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Sin[e + f*x])^2*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]

[Out]

((a + a*Sin[e + f*x])^2*(c - c*Sin[e + f*x])^5*(2520*(8*A - 3*B)*(e + f*x) + 560*(27*A - 17*B)*Cos[e + f*x] +
560*(13*A - 7*B)*Cos[3*(e + f*x)] + 112*(11*A - B)*Cos[5*(e + f*x)] - 80*(A - 3*B)*Cos[7*(e + f*x)] + 560*(19*
A - 3*B)*Sin[2*(e + f*x)] - 280*(2*A - 7*B)*Sin[4*(e + f*x)] - 560*(A - B)*Sin[6*(e + f*x)] - 35*B*Sin[8*(e +
f*x)]))/(35840*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^4)

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Maple [B]  time = 0.037, size = 569, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x)

[Out]

1/f*(A*a^2*c^5*(f*x+e)-B*a^2*c^5*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/1
6*e)+B*a^2*c^5*(8/3+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+5*B*a^2*c^5*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*
cos(f*x+e)+3/8*f*x+3/8*e)-3/7*B*a^2*c^5*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(f*x+e)^2)*cos(f*x+e)+3*A*a
^2*c^5*(-1/6*(sin(f*x+e)^5+5/4*sin(f*x+e)^3+15/8*sin(f*x+e))*cos(f*x+e)+5/16*f*x+5/16*e)+1/5*A*a^2*c^5*(8/3+si
n(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-5*A*a^2*c^5*(-1/4*(sin(f*x+e)^3+3/2*sin(f*x+e))*cos(f*x+e)+3/8*f*x+3/8
*e)-5/3*A*a^2*c^5*(2+sin(f*x+e)^2)*cos(f*x+e)-B*a^2*c^5*(-1/8*(sin(f*x+e)^7+7/6*sin(f*x+e)^5+35/24*sin(f*x+e)^
3+35/16*sin(f*x+e))*cos(f*x+e)+35/128*f*x+35/128*e)+1/7*A*a^2*c^5*(16/5+sin(f*x+e)^6+6/5*sin(f*x+e)^4+8/5*sin(
f*x+e)^2)*cos(f*x+e)-1/3*B*a^2*c^5*(2+sin(f*x+e)^2)*cos(f*x+e)+3*A*a^2*c^5*cos(f*x+e)-3*B*a^2*c^5*(-1/2*sin(f*
x+e)*cos(f*x+e)+1/2*f*x+1/2*e)+A*a^2*c^5*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-B*a^2*c^5*cos(f*x+e))

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Maxima [B]  time = 1.02262, size = 771, normalized size = 3.37 \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))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x, algorithm="maxima")

[Out]

-1/107520*(3072*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 - 35*cos(f*x + e))*A*a^2*c^5 - 7168*
(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^2*c^5 - 179200*(cos(f*x + e)^3 - 3*cos(f*x + e))*
A*a^2*c^5 - 1680*(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))*A*a^2*c^5 +
16800*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A*a^2*c^5 - 26880*(2*f*x + 2*e - sin(2*f*x + 2*
e))*A*a^2*c^5 - 107520*(f*x + e)*A*a^2*c^5 - 9216*(5*cos(f*x + e)^7 - 21*cos(f*x + e)^5 + 35*cos(f*x + e)^3 -
35*cos(f*x + e))*B*a^2*c^5 - 35840*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^2*c^5 - 35840*
(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*c^5 + 35*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e
) + 168*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*B*a^2*c^5 + 560*(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^2*c^5 - 16800*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))
*B*a^2*c^5 + 80640*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*c^5 - 322560*A*a^2*c^5*cos(f*x + e) + 107520*B*a^2*c
^5*cos(f*x + e))/f

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Fricas [A]  time = 1.68156, size = 381, normalized size = 1.66 \begin{align*} -\frac{640 \,{\left (A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{7} - 3584 \,{\left (A - B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{5} - 315 \,{\left (8 \, A - 3 \, B\right )} a^{2} c^{5} f x + 35 \,{\left (16 \, B a^{2} c^{5} \cos \left (f x + e\right )^{7} + 8 \,{\left (8 \, A - 11 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{5} - 6 \,{\left (8 \, A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )^{3} - 9 \,{\left (8 \, A - 3 \, B\right )} a^{2} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4480 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x, algorithm="fricas")

[Out]

-1/4480*(640*(A - 3*B)*a^2*c^5*cos(f*x + e)^7 - 3584*(A - B)*a^2*c^5*cos(f*x + e)^5 - 315*(8*A - 3*B)*a^2*c^5*
f*x + 35*(16*B*a^2*c^5*cos(f*x + e)^7 + 8*(8*A - 11*B)*a^2*c^5*cos(f*x + e)^5 - 6*(8*A - 3*B)*a^2*c^5*cos(f*x
+ e)^3 - 9*(8*A - 3*B)*a^2*c^5*cos(f*x + e))*sin(f*x + e))/f

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Sympy [A]  time = 40.2389, size = 1586, normalized size = 6.93 \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))**2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))**5,x)

[Out]

Piecewise((15*A*a**2*c**5*x*sin(e + f*x)**6/16 + 45*A*a**2*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/16 - 15*A*a*
*2*c**5*x*sin(e + f*x)**4/8 + 45*A*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/16 - 15*A*a**2*c**5*x*sin(e + f
*x)**2*cos(e + f*x)**2/4 + A*a**2*c**5*x*sin(e + f*x)**2/2 + 15*A*a**2*c**5*x*cos(e + f*x)**6/16 - 15*A*a**2*c
**5*x*cos(e + f*x)**4/8 + A*a**2*c**5*x*cos(e + f*x)**2/2 + A*a**2*c**5*x + A*a**2*c**5*sin(e + f*x)**6*cos(e
+ f*x)/f - 33*A*a**2*c**5*sin(e + f*x)**5*cos(e + f*x)/(16*f) + 2*A*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)**3/
f + A*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)/f - 5*A*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(2*f) + 25*A*a*
*2*c**5*sin(e + f*x)**3*cos(e + f*x)/(8*f) + 8*A*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 4*A*a**2*c*
*5*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - 5*A*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)/f - 15*A*a**2*c**5*sin(e
+ f*x)*cos(e + f*x)**5/(16*f) + 15*A*a**2*c**5*sin(e + f*x)*cos(e + f*x)**3/(8*f) - A*a**2*c**5*sin(e + f*x)*
cos(e + f*x)/(2*f) + 16*A*a**2*c**5*cos(e + f*x)**7/(35*f) + 8*A*a**2*c**5*cos(e + f*x)**5/(15*f) - 10*A*a**2*
c**5*cos(e + f*x)**3/(3*f) + 3*A*a**2*c**5*cos(e + f*x)/f - 35*B*a**2*c**5*x*sin(e + f*x)**8/128 - 35*B*a**2*c
**5*x*sin(e + f*x)**6*cos(e + f*x)**2/32 - 5*B*a**2*c**5*x*sin(e + f*x)**6/16 - 105*B*a**2*c**5*x*sin(e + f*x)
**4*cos(e + f*x)**4/64 - 15*B*a**2*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/16 + 15*B*a**2*c**5*x*sin(e + f*x)**
4/8 - 35*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**6/32 - 15*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/1
6 + 15*B*a**2*c**5*x*sin(e + f*x)**2*cos(e + f*x)**2/4 - 3*B*a**2*c**5*x*sin(e + f*x)**2/2 - 35*B*a**2*c**5*x*
cos(e + f*x)**8/128 - 5*B*a**2*c**5*x*cos(e + f*x)**6/16 + 15*B*a**2*c**5*x*cos(e + f*x)**4/8 - 3*B*a**2*c**5*
x*cos(e + f*x)**2/2 + 93*B*a**2*c**5*sin(e + f*x)**7*cos(e + f*x)/(128*f) - 3*B*a**2*c**5*sin(e + f*x)**6*cos(
e + f*x)/f + 511*B*a**2*c**5*sin(e + f*x)**5*cos(e + f*x)**3/(384*f) + 11*B*a**2*c**5*sin(e + f*x)**5*cos(e +
f*x)/(16*f) - 6*B*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f + 5*B*a**2*c**5*sin(e + f*x)**4*cos(e + f*x)/f +
385*B*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**5/(384*f) + 5*B*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(6*f)
- 25*B*a**2*c**5*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 24*B*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) +
20*B*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - B*a**2*c**5*sin(e + f*x)**2*cos(e + f*x)/f + 35*B*a**2*
c**5*sin(e + f*x)*cos(e + f*x)**7/(128*f) + 5*B*a**2*c**5*sin(e + f*x)*cos(e + f*x)**5/(16*f) - 15*B*a**2*c**5
*sin(e + f*x)*cos(e + f*x)**3/(8*f) + 3*B*a**2*c**5*sin(e + f*x)*cos(e + f*x)/(2*f) - 48*B*a**2*c**5*cos(e + f
*x)**7/(35*f) + 8*B*a**2*c**5*cos(e + f*x)**5/(3*f) - 2*B*a**2*c**5*cos(e + f*x)**3/(3*f) - B*a**2*c**5*cos(e
+ f*x)/f, Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**2*(-c*sin(e) + c)**5, True))

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Giac [A]  time = 1.24666, size = 375, normalized size = 1.64 \begin{align*} -\frac{B a^{2} c^{5} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac{9}{128} \,{\left (8 \, A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} x - \frac{{\left (A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} \cos \left (7 \, f x + 7 \, e\right )}{448 \, f} + \frac{{\left (11 \, A a^{2} c^{5} - B a^{2} c^{5}\right )} \cos \left (5 \, f x + 5 \, e\right )}{320 \, f} + \frac{{\left (13 \, A a^{2} c^{5} - 7 \, B a^{2} c^{5}\right )} \cos \left (3 \, f x + 3 \, e\right )}{64 \, f} + \frac{{\left (27 \, A a^{2} c^{5} - 17 \, B a^{2} c^{5}\right )} \cos \left (f x + e\right )}{64 \, f} - \frac{{\left (A a^{2} c^{5} - B a^{2} c^{5}\right )} \sin \left (6 \, f x + 6 \, e\right )}{64 \, f} - \frac{{\left (2 \, A a^{2} c^{5} - 7 \, B a^{2} c^{5}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac{{\left (19 \, A a^{2} c^{5} - 3 \, B a^{2} c^{5}\right )} \sin \left (2 \, f x + 2 \, e\right )}{64 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x, algorithm="giac")

[Out]

-1/1024*B*a^2*c^5*sin(8*f*x + 8*e)/f + 9/128*(8*A*a^2*c^5 - 3*B*a^2*c^5)*x - 1/448*(A*a^2*c^5 - 3*B*a^2*c^5)*c
os(7*f*x + 7*e)/f + 1/320*(11*A*a^2*c^5 - B*a^2*c^5)*cos(5*f*x + 5*e)/f + 1/64*(13*A*a^2*c^5 - 7*B*a^2*c^5)*co
s(3*f*x + 3*e)/f + 1/64*(27*A*a^2*c^5 - 17*B*a^2*c^5)*cos(f*x + e)/f - 1/64*(A*a^2*c^5 - B*a^2*c^5)*sin(6*f*x
+ 6*e)/f - 1/128*(2*A*a^2*c^5 - 7*B*a^2*c^5)*sin(4*f*x + 4*e)/f + 1/64*(19*A*a^2*c^5 - 3*B*a^2*c^5)*sin(2*f*x
+ 2*e)/f