3.39 \(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx\)
Optimal. Leaf size=222 \[ \frac{a^3 c^5 (9 A-2 B) \cos ^7(e+f x)}{56 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{a^3 c^5 (9 A-2 B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 a^3 c^5 (9 A-2 B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 a^3 c^5 (9 A-2 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} a^3 c^5 x (9 A-2 B)-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f} \]
[Out]
(5*a^3*(9*A - 2*B)*c^5*x)/128 + (a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^7)/(56*f) + (5*a^3*(9*A - 2*B)*c^5*Cos[e + f
*x]*Sin[e + f*x])/(128*f) + (5*a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(9*A - 2*B)*c^5
*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*c^3*Cos[e + f*x]^7*(c - c*Sin[e + f*x])^2)/(9*f) + (a^3*(9*A - 2
*B)*Cos[e + f*x]^7*(c^5 - c^5*Sin[e + f*x]))/(72*f)
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Rubi [A] time = 0.322109, antiderivative size = 222, 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} \[ \frac{a^3 c^5 (9 A-2 B) \cos ^7(e+f x)}{56 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{a^3 c^5 (9 A-2 B) \sin (e+f x) \cos ^5(e+f x)}{48 f}+\frac{5 a^3 c^5 (9 A-2 B) \sin (e+f x) \cos ^3(e+f x)}{192 f}+\frac{5 a^3 c^5 (9 A-2 B) \sin (e+f x) \cos (e+f x)}{128 f}+\frac{5}{128} a^3 c^5 x (9 A-2 B)-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f} \]
Antiderivative was successfully verified.
[In]
Int[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]
[Out]
(5*a^3*(9*A - 2*B)*c^5*x)/128 + (a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^7)/(56*f) + (5*a^3*(9*A - 2*B)*c^5*Cos[e + f
*x]*Sin[e + f*x])/(128*f) + (5*a^3*(9*A - 2*B)*c^5*Cos[e + f*x]^3*Sin[e + f*x])/(192*f) + (a^3*(9*A - 2*B)*c^5
*Cos[e + f*x]^5*Sin[e + f*x])/(48*f) - (a^3*B*c^3*Cos[e + f*x]^7*(c - c*Sin[e + f*x])^2)/(9*f) + (a^3*(9*A - 2
*B)*Cos[e + f*x]^7*(c^5 - c^5*Sin[e + f*x]))/(72*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))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^5 \, dx &=\left (a^3 c^3\right ) \int \cos ^6(e+f x) (A+B \sin (e+f x)) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{1}{9} \left (a^3 (9 A-2 B) c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{1}{8} \left (a^3 (9 A-2 B) c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac{a^3 (9 A-2 B) c^5 \cos ^7(e+f x)}{56 f}-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{1}{8} \left (a^3 (9 A-2 B) c^5\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac{a^3 (9 A-2 B) c^5 \cos ^7(e+f x)}{56 f}+\frac{a^3 (9 A-2 B) c^5 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{1}{48} \left (5 a^3 (9 A-2 B) c^5\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac{a^3 (9 A-2 B) c^5 \cos ^7(e+f x)}{56 f}+\frac{5 a^3 (9 A-2 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{a^3 (9 A-2 B) c^5 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{1}{64} \left (5 a^3 (9 A-2 B) c^5\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac{a^3 (9 A-2 B) c^5 \cos ^7(e+f x)}{56 f}+\frac{5 a^3 (9 A-2 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{5 a^3 (9 A-2 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{a^3 (9 A-2 B) c^5 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}+\frac{1}{128} \left (5 a^3 (9 A-2 B) c^5\right ) \int 1 \, dx\\ &=\frac{5}{128} a^3 (9 A-2 B) c^5 x+\frac{a^3 (9 A-2 B) c^5 \cos ^7(e+f x)}{56 f}+\frac{5 a^3 (9 A-2 B) c^5 \cos (e+f x) \sin (e+f x)}{128 f}+\frac{5 a^3 (9 A-2 B) c^5 \cos ^3(e+f x) \sin (e+f x)}{192 f}+\frac{a^3 (9 A-2 B) c^5 \cos ^5(e+f x) \sin (e+f x)}{48 f}-\frac{a^3 B c^3 \cos ^7(e+f x) (c-c \sin (e+f x))^2}{9 f}+\frac{a^3 (9 A-2 B) \cos ^7(e+f x) \left (c^5-c^5 \sin (e+f x)\right )}{72 f}\\ \end{align*}
Mathematica [A] time = 2.52293, size = 232, normalized size = 1.05 \[ \frac{(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^5 (2520 (9 A-2 B) (e+f x)+2016 (8 A-B) \sin (2 (e+f x))+504 (5 A+2 B) \sin (4 (e+f x))-63 (A-2 B) \sin (8 (e+f x))+504 (20 A-13 B) \cos (e+f x)+336 (18 A-11 B) \cos (3 (e+f x))+1008 (2 A-B) \cos (5 (e+f x))+36 (8 A-B) \cos (7 (e+f x))+672 B \sin (6 (e+f x))+28 B \cos (9 (e+f x)))}{64512 f \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^{10} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
[In]
Integrate[(a + a*Sin[e + f*x])^3*(A + B*Sin[e + f*x])*(c - c*Sin[e + f*x])^5,x]
[Out]
((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^5*(2520*(9*A - 2*B)*(e + f*x) + 504*(20*A - 13*B)*Cos[e + f*x] +
336*(18*A - 11*B)*Cos[3*(e + f*x)] + 1008*(2*A - B)*Cos[5*(e + f*x)] + 36*(8*A - B)*Cos[7*(e + f*x)] + 28*B*Co
s[9*(e + f*x)] + 2016*(8*A - B)*Sin[2*(e + f*x)] + 504*(5*A + 2*B)*Sin[4*(e + f*x)] + 672*B*Sin[6*(e + f*x)] -
63*(A - 2*B)*Sin[8*(e + f*x)]))/(64512*f*(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^10*(Cos[(e + f*x)/2] + Sin[(e
+ f*x)/2])^6)
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Maple [B] time = 0.036, size = 611, normalized size = 2.8 \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))^3*(A+B*sin(f*x+e))*(c-c*sin(f*x+e))^5,x)
[Out]
1/f*(A*a^3*c^5*(f*x+e)-2/7*B*a^3*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)-6*B*a^3*
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)+1/9*B*a^3*c^5*(128/35+si
n(f*x+e)^8+8/7*sin(f*x+e)^6+48/35*sin(f*x+e)^4+64/35*sin(f*x+e)^2)*cos(f*x+e)+2*B*a^3*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)+6/5*A*a^3*c^5*(8/3+sin(f
*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-A*a^3*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*s
in(f*x+e))*cos(f*x+e)+35/128*f*x+35/128*e)-2/7*A*a^3*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)+2*A*a^3*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)+2*A*
a^3*c^5*cos(f*x+e)-2*B*a^3*c^5*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e)-2*A*a^3*c^5*(2+sin(f*x+e)^2)*cos(f*x
+e)+6*B*a^3*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)-2*A*a^3*c^5*(-1/2*sin(f*x+e)*cos
(f*x+e)+1/2*f*x+1/2*e)+2/3*B*a^3*c^5*(2+sin(f*x+e)^2)*cos(f*x+e)-B*a^3*c^5*cos(f*x+e))
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Maxima [B] time = 1.03487, size = 833, normalized size = 3.75 \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-c*sin(f*x+e))^5,x, algorithm="maxima")
[Out]
1/322560*(18432*(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^3*c^5 + 12902
4*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c^5 + 645120*(cos(f*x + e)^3 - 3*cos(f*x + e)
)*A*a^3*c^5 - 105*(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))*A*a^3*c^5 + 3360*(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^3*c^5 - 161280*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^3*c^5 + 322560*(f*x + e)*A*a^3*c^5 + 1024*(35*c
os(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 420*cos(f*x + e)^3 + 315*cos(f*x + e))*B*a^3*c^5 + 1
8432*(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^3*c^5 - 215040*(cos(f*x
+ e)^3 - 3*cos(f*x + e))*B*a^3*c^5 + 210*(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^3*c^5 - 10080*(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^3*c^5 + 60480*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3
*c^5 - 161280*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^5 + 645120*A*a^3*c^5*cos(f*x + e) - 322560*B*a^3*c^5*co
s(f*x + e))/f
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Fricas [A] time = 1.82007, size = 381, normalized size = 1.72 \begin{align*} \frac{896 \, B a^{3} c^{5} \cos \left (f x + e\right )^{9} + 2304 \,{\left (A - B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{7} + 315 \,{\left (9 \, A - 2 \, B\right )} a^{3} c^{5} f x - 21 \,{\left (48 \,{\left (A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{7} - 8 \,{\left (9 \, A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{5} - 10 \,{\left (9 \, A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{3} - 15 \,{\left (9 \, A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8064 \, 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-c*sin(f*x+e))^5,x, algorithm="fricas")
[Out]
1/8064*(896*B*a^3*c^5*cos(f*x + e)^9 + 2304*(A - B)*a^3*c^5*cos(f*x + e)^7 + 315*(9*A - 2*B)*a^3*c^5*f*x - 21*
(48*(A - 2*B)*a^3*c^5*cos(f*x + e)^7 - 8*(9*A - 2*B)*a^3*c^5*cos(f*x + e)^5 - 10*(9*A - 2*B)*a^3*c^5*cos(f*x +
e)^3 - 15*(9*A - 2*B)*a^3*c^5*cos(f*x + e))*sin(f*x + e))/f
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Sympy [A] time = 52.2156, size = 1753, normalized size = 7.9 \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-c*sin(f*x+e))**5,x)
[Out]
Piecewise((-35*A*a**3*c**5*x*sin(e + f*x)**8/128 - 35*A*a**3*c**5*x*sin(e + f*x)**6*cos(e + f*x)**2/32 + 5*A*a
**3*c**5*x*sin(e + f*x)**6/8 - 105*A*a**3*c**5*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*A*a**3*c**5*x*sin(e +
f*x)**4*cos(e + f*x)**2/8 - 35*A*a**3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**6/32 + 15*A*a**3*c**5*x*sin(e + f*
x)**2*cos(e + f*x)**4/8 - A*a**3*c**5*x*sin(e + f*x)**2 - 35*A*a**3*c**5*x*cos(e + f*x)**8/128 + 5*A*a**3*c**5
*x*cos(e + f*x)**6/8 - A*a**3*c**5*x*cos(e + f*x)**2 + A*a**3*c**5*x + 93*A*a**3*c**5*sin(e + f*x)**7*cos(e +
f*x)/(128*f) - 2*A*a**3*c**5*sin(e + f*x)**6*cos(e + f*x)/f + 511*A*a**3*c**5*sin(e + f*x)**5*cos(e + f*x)**3/
(384*f) - 11*A*a**3*c**5*sin(e + f*x)**5*cos(e + f*x)/(8*f) - 4*A*a**3*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f
+ 6*A*a**3*c**5*sin(e + f*x)**4*cos(e + f*x)/f + 385*A*a**3*c**5*sin(e + f*x)**3*cos(e + f*x)**5/(384*f) - 5*A
*a**3*c**5*sin(e + f*x)**3*cos(e + f*x)**3/(3*f) - 16*A*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 8*A*
a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**3/f - 6*A*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)/f + 35*A*a**3*c**5*si
n(e + f*x)*cos(e + f*x)**7/(128*f) - 5*A*a**3*c**5*sin(e + f*x)*cos(e + f*x)**5/(8*f) + A*a**3*c**5*sin(e + f*
x)*cos(e + f*x)/f - 32*A*a**3*c**5*cos(e + f*x)**7/(35*f) + 16*A*a**3*c**5*cos(e + f*x)**5/(5*f) - 4*A*a**3*c*
*5*cos(e + f*x)**3/f + 2*A*a**3*c**5*cos(e + f*x)/f + 35*B*a**3*c**5*x*sin(e + f*x)**8/64 + 35*B*a**3*c**5*x*s
in(e + f*x)**6*cos(e + f*x)**2/16 - 15*B*a**3*c**5*x*sin(e + f*x)**6/8 + 105*B*a**3*c**5*x*sin(e + f*x)**4*cos
(e + f*x)**4/32 - 45*B*a**3*c**5*x*sin(e + f*x)**4*cos(e + f*x)**2/8 + 9*B*a**3*c**5*x*sin(e + f*x)**4/4 + 35*
B*a**3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**6/16 - 45*B*a**3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**4/8 + 9*B*a*
*3*c**5*x*sin(e + f*x)**2*cos(e + f*x)**2/2 - B*a**3*c**5*x*sin(e + f*x)**2 + 35*B*a**3*c**5*x*cos(e + f*x)**8
/64 - 15*B*a**3*c**5*x*cos(e + f*x)**6/8 + 9*B*a**3*c**5*x*cos(e + f*x)**4/4 - B*a**3*c**5*x*cos(e + f*x)**2 +
B*a**3*c**5*sin(e + f*x)**8*cos(e + f*x)/f - 93*B*a**3*c**5*sin(e + f*x)**7*cos(e + f*x)/(64*f) + 8*B*a**3*c*
*5*sin(e + f*x)**6*cos(e + f*x)**3/(3*f) - 2*B*a**3*c**5*sin(e + f*x)**6*cos(e + f*x)/f - 511*B*a**3*c**5*sin(
e + f*x)**5*cos(e + f*x)**3/(192*f) + 33*B*a**3*c**5*sin(e + f*x)**5*cos(e + f*x)/(8*f) + 16*B*a**3*c**5*sin(e
+ f*x)**4*cos(e + f*x)**5/(5*f) - 4*B*a**3*c**5*sin(e + f*x)**4*cos(e + f*x)**3/f - 385*B*a**3*c**5*sin(e + f
*x)**3*cos(e + f*x)**5/(192*f) + 5*B*a**3*c**5*sin(e + f*x)**3*cos(e + f*x)**3/f - 15*B*a**3*c**5*sin(e + f*x)
**3*cos(e + f*x)/(4*f) + 64*B*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)**7/(35*f) - 16*B*a**3*c**5*sin(e + f*x)**
2*cos(e + f*x)**5/(5*f) + 2*B*a**3*c**5*sin(e + f*x)**2*cos(e + f*x)/f - 35*B*a**3*c**5*sin(e + f*x)*cos(e + f
*x)**7/(64*f) + 15*B*a**3*c**5*sin(e + f*x)*cos(e + f*x)**5/(8*f) - 9*B*a**3*c**5*sin(e + f*x)*cos(e + f*x)**3
/(4*f) + B*a**3*c**5*sin(e + f*x)*cos(e + f*x)/f + 128*B*a**3*c**5*cos(e + f*x)**9/(315*f) - 32*B*a**3*c**5*co
s(e + f*x)**7/(35*f) + 4*B*a**3*c**5*cos(e + f*x)**3/(3*f) - B*a**3*c**5*cos(e + f*x)/f, Ne(f, 0)), (x*(A + B*
sin(e))*(a*sin(e) + a)**3*(-c*sin(e) + c)**5, True))
________________________________________________________________________________________
Giac [A] time = 1.34458, size = 406, normalized size = 1.83 \begin{align*} \frac{B a^{3} c^{5} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac{B a^{3} c^{5} \sin \left (6 \, f x + 6 \, e\right )}{96 \, f} + \frac{5}{128} \,{\left (9 \, A a^{3} c^{5} - 2 \, B a^{3} c^{5}\right )} x + \frac{{\left (8 \, A a^{3} c^{5} - B a^{3} c^{5}\right )} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac{{\left (2 \, A a^{3} c^{5} - B a^{3} c^{5}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac{{\left (18 \, A a^{3} c^{5} - 11 \, B a^{3} c^{5}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac{{\left (20 \, A a^{3} c^{5} - 13 \, B a^{3} c^{5}\right )} \cos \left (f x + e\right )}{128 \, f} - \frac{{\left (A a^{3} c^{5} - 2 \, B a^{3} c^{5}\right )} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac{{\left (5 \, A a^{3} c^{5} + 2 \, B a^{3} c^{5}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac{{\left (8 \, A a^{3} c^{5} - B a^{3} c^{5}\right )} \sin \left (2 \, f x + 2 \, e\right )}{32 \, 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-c*sin(f*x+e))^5,x, algorithm="giac")
[Out]
1/2304*B*a^3*c^5*cos(9*f*x + 9*e)/f + 1/96*B*a^3*c^5*sin(6*f*x + 6*e)/f + 5/128*(9*A*a^3*c^5 - 2*B*a^3*c^5)*x
+ 1/1792*(8*A*a^3*c^5 - B*a^3*c^5)*cos(7*f*x + 7*e)/f + 1/64*(2*A*a^3*c^5 - B*a^3*c^5)*cos(5*f*x + 5*e)/f + 1/
192*(18*A*a^3*c^5 - 11*B*a^3*c^5)*cos(3*f*x + 3*e)/f + 1/128*(20*A*a^3*c^5 - 13*B*a^3*c^5)*cos(f*x + e)/f - 1/
1024*(A*a^3*c^5 - 2*B*a^3*c^5)*sin(8*f*x + 8*e)/f + 1/128*(5*A*a^3*c^5 + 2*B*a^3*c^5)*sin(4*f*x + 4*e)/f + 1/3
2*(8*A*a^3*c^5 - B*a^3*c^5)*sin(2*f*x + 2*e)/f