3.38 \(\int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, dx\)

Optimal. Leaf size=265 \[ \frac {11 a^3 c^6 (10 A-3 B) \cos ^7(e+f x)}{560 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^5(e+f x)}{480 f}+\frac {11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{384 f}+\frac {11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos (e+f x)}{256 f}+\frac {11}{256} a^3 c^6 x (10 A-3 B)+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f} \]

[Out]

11/256*a^3*(10*A-3*B)*c^6*x+11/560*a^3*(10*A-3*B)*c^6*cos(f*x+e)^7/f+11/256*a^3*(10*A-3*B)*c^6*cos(f*x+e)*sin(
f*x+e)/f+11/384*a^3*(10*A-3*B)*c^6*cos(f*x+e)^3*sin(f*x+e)/f+11/480*a^3*(10*A-3*B)*c^6*cos(f*x+e)^5*sin(f*x+e)
/f-1/10*a^3*B*cos(f*x+e)^7*(c^2-c^2*sin(f*x+e))^3/f+1/90*a^3*(10*A-3*B)*cos(f*x+e)^7*(c^3-c^3*sin(f*x+e))^2/f+
11/720*a^3*(10*A-3*B)*cos(f*x+e)^7*(c^6-c^6*sin(f*x+e))/f

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Rubi [A]  time = 0.39, antiderivative size = 265, normalized size of antiderivative = 1.00, number of steps used = 9, 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} \[ \frac {11 a^3 c^6 (10 A-3 B) \cos ^7(e+f x)}{560 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^5(e+f x)}{480 f}+\frac {11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos ^3(e+f x)}{384 f}+\frac {11 a^3 c^6 (10 A-3 B) \sin (e+f x) \cos (e+f x)}{256 f}+\frac {11}{256} a^3 c^6 x (10 A-3 B)-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f} \]

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])^6,x]

[Out]

(11*a^3*(10*A - 3*B)*c^6*x)/256 + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^7)/(560*f) + (11*a^3*(10*A - 3*B)*c^6*
Cos[e + f*x]*Sin[e + f*x])/(256*f) + (11*a^3*(10*A - 3*B)*c^6*Cos[e + f*x]^3*Sin[e + f*x])/(384*f) + (11*a^3*(
10*A - 3*B)*c^6*Cos[e + f*x]^5*Sin[e + f*x])/(480*f) - (a^3*B*Cos[e + f*x]^7*(c^2 - c^2*Sin[e + f*x])^3)/(10*f
) + (a^3*(10*A - 3*B)*Cos[e + f*x]^7*(c^3 - c^3*Sin[e + f*x])^2)/(90*f) + (11*a^3*(10*A - 3*B)*Cos[e + f*x]^7*
(c^6 - c^6*Sin[e + f*x]))/(720*f)

Rule 8

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

Rubi steps

\begin {align*} \int (a+a \sin (e+f x))^3 (A+B \sin (e+f x)) (c-c \sin (e+f x))^6 \, 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))^3 \, dx\\ &=-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {1}{10} \left (a^3 (10 A-3 B) c^3\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^3 \, dx\\ &=-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {1}{90} \left (11 a^3 (10 A-3 B) c^4\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x))^2 \, dx\\ &=-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {1}{80} \left (11 a^3 (10 A-3 B) c^5\right ) \int \cos ^6(e+f x) (c-c \sin (e+f x)) \, dx\\ &=\frac {11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {1}{80} \left (11 a^3 (10 A-3 B) c^6\right ) \int \cos ^6(e+f x) \, dx\\ &=\frac {11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {1}{96} \left (11 a^3 (10 A-3 B) c^6\right ) \int \cos ^4(e+f x) \, dx\\ &=\frac {11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {1}{128} \left (11 a^3 (10 A-3 B) c^6\right ) \int \cos ^2(e+f x) \, dx\\ &=\frac {11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos (e+f x) \sin (e+f x)}{256 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}+\frac {1}{256} \left (11 a^3 (10 A-3 B) c^6\right ) \int 1 \, dx\\ &=\frac {11}{256} a^3 (10 A-3 B) c^6 x+\frac {11 a^3 (10 A-3 B) c^6 \cos ^7(e+f x)}{560 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos (e+f x) \sin (e+f x)}{256 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^3(e+f x) \sin (e+f x)}{384 f}+\frac {11 a^3 (10 A-3 B) c^6 \cos ^5(e+f x) \sin (e+f x)}{480 f}-\frac {a^3 B \cos ^7(e+f x) \left (c^2-c^2 \sin (e+f x)\right )^3}{10 f}+\frac {a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^3-c^3 \sin (e+f x)\right )^2}{90 f}+\frac {11 a^3 (10 A-3 B) \cos ^7(e+f x) \left (c^6-c^6 \sin (e+f x)\right )}{720 f}\\ \end {align*}

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Mathematica [A]  time = 4.31, size = 255, normalized size = 0.96 \[ \frac {(a \sin (e+f x)+a)^3 (c-c \sin (e+f x))^6 (27720 (10 A-3 B) (e+f x)+1260 (144 A-25 B) \sin (2 (e+f x))+2520 (6 A+7 B) \sin (4 (e+f x))-210 (32 A-51 B) \sin (6 (e+f x))-315 (6 A-5 B) \sin (8 (e+f x))+5040 (33 A-19 B) \cos (e+f x)+3360 (29 A-15 B) \cos (3 (e+f x))+10080 (3 A-B) \cos (5 (e+f x))+360 (9 A+5 B) \cos (7 (e+f x))-280 (A-3 B) \cos (9 (e+f x))-126 B \sin (10 (e+f x)))}{645120 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{12} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^6} \]

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])^6,x]

[Out]

((a + a*Sin[e + f*x])^3*(c - c*Sin[e + f*x])^6*(27720*(10*A - 3*B)*(e + f*x) + 5040*(33*A - 19*B)*Cos[e + f*x]
+ 3360*(29*A - 15*B)*Cos[3*(e + f*x)] + 10080*(3*A - B)*Cos[5*(e + f*x)] + 360*(9*A + 5*B)*Cos[7*(e + f*x)] -
280*(A - 3*B)*Cos[9*(e + f*x)] + 1260*(144*A - 25*B)*Sin[2*(e + f*x)] + 2520*(6*A + 7*B)*Sin[4*(e + f*x)] - 2
10*(32*A - 51*B)*Sin[6*(e + f*x)] - 315*(6*A - 5*B)*Sin[8*(e + f*x)] - 126*B*Sin[10*(e + f*x)]))/(645120*f*(Co
s[(e + f*x)/2] - Sin[(e + f*x)/2])^12*(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^6)

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fricas [A]  time = 0.50, size = 181, normalized size = 0.68 \[ -\frac {8960 \, {\left (A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{9} - 46080 \, {\left (A - B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{7} - 3465 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} f x + 21 \, {\left (384 \, B a^{3} c^{6} \cos \left (f x + e\right )^{9} + 48 \, {\left (30 \, A - 41 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{7} - 88 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{5} - 110 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )^{3} - 165 \, {\left (10 \, A - 3 \, B\right )} a^{3} c^{6} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{80640 \, f} \]

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))^6,x, algorithm="fricas")

[Out]

-1/80640*(8960*(A - 3*B)*a^3*c^6*cos(f*x + e)^9 - 46080*(A - B)*a^3*c^6*cos(f*x + e)^7 - 3465*(10*A - 3*B)*a^3
*c^6*f*x + 21*(384*B*a^3*c^6*cos(f*x + e)^9 + 48*(30*A - 41*B)*a^3*c^6*cos(f*x + e)^7 - 88*(10*A - 3*B)*a^3*c^
6*cos(f*x + e)^5 - 110*(10*A - 3*B)*a^3*c^6*cos(f*x + e)^3 - 165*(10*A - 3*B)*a^3*c^6*cos(f*x + e))*sin(f*x +
e))/f

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giac [A]  time = 0.27, size = 347, normalized size = 1.31 \[ -\frac {B a^{3} c^{6} \sin \left (10 \, f x + 10 \, e\right )}{5120 \, f} + \frac {11}{256} \, {\left (10 \, A a^{3} c^{6} - 3 \, B a^{3} c^{6}\right )} x - \frac {{\left (A a^{3} c^{6} - 3 \, B a^{3} c^{6}\right )} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac {{\left (9 \, A a^{3} c^{6} + 5 \, B a^{3} c^{6}\right )} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac {{\left (3 \, A a^{3} c^{6} - B a^{3} c^{6}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {{\left (29 \, A a^{3} c^{6} - 15 \, B a^{3} c^{6}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac {{\left (33 \, A a^{3} c^{6} - 19 \, B a^{3} c^{6}\right )} \cos \left (f x + e\right )}{128 \, f} - \frac {{\left (6 \, A a^{3} c^{6} - 5 \, B a^{3} c^{6}\right )} \sin \left (8 \, f x + 8 \, e\right )}{2048 \, f} - \frac {{\left (32 \, A a^{3} c^{6} - 51 \, B a^{3} c^{6}\right )} \sin \left (6 \, f x + 6 \, e\right )}{3072 \, f} + \frac {{\left (6 \, A a^{3} c^{6} + 7 \, B a^{3} c^{6}\right )} \sin \left (4 \, f x + 4 \, e\right )}{256 \, f} + \frac {{\left (144 \, A a^{3} c^{6} - 25 \, B a^{3} c^{6}\right )} \sin \left (2 \, f x + 2 \, e\right )}{512 \, f} \]

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))^6,x, algorithm="giac")

[Out]

-1/5120*B*a^3*c^6*sin(10*f*x + 10*e)/f + 11/256*(10*A*a^3*c^6 - 3*B*a^3*c^6)*x - 1/2304*(A*a^3*c^6 - 3*B*a^3*c
^6)*cos(9*f*x + 9*e)/f + 1/1792*(9*A*a^3*c^6 + 5*B*a^3*c^6)*cos(7*f*x + 7*e)/f + 1/64*(3*A*a^3*c^6 - B*a^3*c^6
)*cos(5*f*x + 5*e)/f + 1/192*(29*A*a^3*c^6 - 15*B*a^3*c^6)*cos(3*f*x + 3*e)/f + 1/128*(33*A*a^3*c^6 - 19*B*a^3
*c^6)*cos(f*x + e)/f - 1/2048*(6*A*a^3*c^6 - 5*B*a^3*c^6)*sin(8*f*x + 8*e)/f - 1/3072*(32*A*a^3*c^6 - 51*B*a^3
*c^6)*sin(6*f*x + 6*e)/f + 1/256*(6*A*a^3*c^6 + 7*B*a^3*c^6)*sin(4*f*x + 4*e)/f + 1/512*(144*A*a^3*c^6 - 25*B*
a^3*c^6)*sin(2*f*x + 2*e)/f

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maple [B]  time = 0.85, size = 651, normalized size = 2.46 \[ \frac {a^{3} A \,c^{6} \left (f x +e \right )-B \,a^{3} c^{6} \cos \left (f x +e \right )-\frac {8 a^{3} A \,c^{6} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+8 B \,a^{3} c^{6} \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 )+3 a^{3} A \,c^{6} \cos \left (f x +e \right )+\frac {6 B \,a^{3} c^{6} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}+B \,a^{3} c^{6} \left (-\frac {\left (\sin ^{9}\left (f x +e \right )+\frac {9 \left (\sin ^{7}\left (f x +e \right )\right )}{8}+\frac {21 \left (\sin ^{5}\left (f x +e \right )\right )}{16}+\frac {105 \left (\sin ^{3}\left (f x +e \right )\right )}{64}+\frac {315 \sin \left (f x +e \right )}{128}\right ) \cos \left (f x +e \right )}{10}+\frac {63 f x}{256}+\frac {63 e}{256}\right )+\frac {B \,a^{3} c^{6} \left (\frac {128}{35}+\sin ^{8}\left (f x +e \right )+\frac {8 \left (\sin ^{6}\left (f x +e \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (f x +e \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (f x +e \right )\right )}{35}\right ) \cos \left (f x +e \right )}{3}-\frac {8 B \,a^{3} c^{6} \left (\frac {16}{5}+\sin ^{6}\left (f x +e \right )+\frac {6 \left (\sin ^{4}\left (f x +e \right )\right )}{5}+\frac {8 \left (\sin ^{2}\left (f x +e \right )\right )}{5}\right ) \cos \left (f x +e \right )}{7}-6 B \,a^{3} c^{6} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )-\frac {a^{3} A \,c^{6} \left (\frac {128}{35}+\sin ^{8}\left (f x +e \right )+\frac {8 \left (\sin ^{6}\left (f x +e \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (f x +e \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (f x +e \right )\right )}{35}\right ) \cos \left (f x +e \right )}{9}-3 a^{3} A \,c^{6} \left (-\frac {\left (\sin ^{7}\left (f x +e \right )+\frac {7 \left (\sin ^{5}\left (f x +e \right )\right )}{6}+\frac {35 \left (\sin ^{3}\left (f x +e \right )\right )}{24}+\frac {35 \sin \left (f x +e \right )}{16}\right ) \cos \left (f x +e \right )}{8}+\frac {35 f x}{128}+\frac {35 e}{128}\right )+8 a^{3} A \,c^{6} \left (-\frac {\left (\sin ^{5}\left (f x +e \right )+\frac {5 \left (\sin ^{3}\left (f x +e \right )\right )}{4}+\frac {15 \sin \left (f x +e \right )}{8}\right ) \cos \left (f x +e \right )}{6}+\frac {5 f x}{16}+\frac {5 e}{16}\right )+\frac {6 a^{3} A \,c^{6} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5}-6 a^{3} A \,c^{6} \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 )-3 B \,a^{3} c^{6} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f} \]

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))^6,x)

[Out]

1/f*(a^3*A*c^6*(f*x+e)-B*a^3*c^6*cos(f*x+e)-8/3*a^3*A*c^6*(2+sin(f*x+e)^2)*cos(f*x+e)+8*B*a^3*c^6*(-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^3*A*c^6*cos(f*x+e)+6/5*B*a^3*c^6*(8/3+sin(f*x+e)^4+4/3*s
in(f*x+e)^2)*cos(f*x+e)+B*a^3*c^6*(-1/10*(sin(f*x+e)^9+9/8*sin(f*x+e)^7+21/16*sin(f*x+e)^5+105/64*sin(f*x+e)^3
+315/128*sin(f*x+e))*cos(f*x+e)+63/256*f*x+63/256*e)+1/3*B*a^3*c^6*(128/35+sin(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)-8/7*B*a^3*c^6*(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^6*(-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*a^3*A*c^6*(128/35+sin(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)-3*a^3*A*c^
6*(-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)+8*
a^3*A*c^6*(-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)+6/5*a^3*A*c^6*(8/3
+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)-6*a^3*A*c^6*(-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*B*a^3*c^6*(-1/2*sin(f*x+e)*cos(f*x+e)+1/2*f*x+1/2*e))

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maxima [B]  time = 0.36, size = 661, normalized size = 2.49 \[ -\frac {2048 \, {\left (35 \, \cos \left (f x + e\right )^{9} - 180 \, \cos \left (f x + e\right )^{7} + 378 \, \cos \left (f x + e\right )^{5} - 420 \, \cos \left (f x + e\right )^{3} + 315 \, \cos \left (f x + e\right )\right )} A a^{3} c^{6} - 258048 \, {\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 )} A a^{3} c^{6} - 1720320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c^{6} + 630 \, {\left (128 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 840 \, f x + 840 \, e + 3 \, \sin \left (8 \, f x + 8 \, e\right ) + 168 \, \sin \left (4 \, f x + 4 \, e\right ) - 768 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{6} - 26880 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{6} + 120960 \, {\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} c^{6} - 645120 \, {\left (f x + e\right )} A a^{3} c^{6} - 6144 \, {\left (35 \, \cos \left (f x + e\right )^{9} - 180 \, \cos \left (f x + e\right )^{7} + 378 \, \cos \left (f x + e\right )^{5} - 420 \, \cos \left (f x + e\right )^{3} + 315 \, \cos \left (f x + e\right )\right )} B a^{3} c^{6} - 147456 \, {\left (5 \, \cos \left (f x + e\right )^{7} - 21 \, \cos \left (f x + e\right )^{5} + 35 \, \cos \left (f x + e\right )^{3} - 35 \, \cos \left (f x + e\right )\right )} B a^{3} c^{6} - 258048 \, {\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} c^{6} + 63 \, {\left (32 \, \sin \left (2 \, f x + 2 \, e\right )^{5} - 640 \, \sin \left (2 \, f x + 2 \, e\right )^{3} - 2520 \, f x - 2520 \, e - 25 \, \sin \left (8 \, f x + 8 \, e\right ) - 600 \, \sin \left (4 \, f x + 4 \, e\right ) + 2560 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{6} + 20160 \, {\left (4 \, \sin \left (2 \, f x + 2 \, e\right )^{3} + 60 \, f x + 60 \, e + 9 \, \sin \left (4 \, f x + 4 \, e\right ) - 48 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{6} - 161280 \, {\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^{6} + 483840 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{6} - 1935360 \, A a^{3} c^{6} \cos \left (f x + e\right ) + 645120 \, B a^{3} c^{6} \cos \left (f x + e\right )}{645120 \, f} \]

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))^6,x, algorithm="maxima")

[Out]

-1/645120*(2048*(35*cos(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))*A*a^3*c^6 - 258048*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*A*a^3*c^6 - 1720320*(cos(f
*x + e)^3 - 3*cos(f*x + e))*A*a^3*c^6 + 630*(128*sin(2*f*x + 2*e)^3 + 840*f*x + 840*e + 3*sin(8*f*x + 8*e) + 1
68*sin(4*f*x + 4*e) - 768*sin(2*f*x + 2*e))*A*a^3*c^6 - 26880*(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^6 + 120960*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*A
*a^3*c^6 - 645120*(f*x + e)*A*a^3*c^6 - 6144*(35*cos(f*x + e)^9 - 180*cos(f*x + e)^7 + 378*cos(f*x + e)^5 - 42
0*cos(f*x + e)^3 + 315*cos(f*x + e))*B*a^3*c^6 - 147456*(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^6 - 258048*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3 + 15*cos(f*x + e))*B*a^3*c^6 +
63*(32*sin(2*f*x + 2*e)^5 - 640*sin(2*f*x + 2*e)^3 - 2520*f*x - 2520*e - 25*sin(8*f*x + 8*e) - 600*sin(4*f*x
+ 4*e) + 2560*sin(2*f*x + 2*e))*B*a^3*c^6 + 20160*(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^6 - 161280*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^3*c^6 + 4
83840*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^3*c^6 - 1935360*A*a^3*c^6*cos(f*x + e) + 645120*B*a^3*c^6*cos(f*x +
e))/f

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mupad [B]  time = 14.87, size = 812, normalized size = 3.06 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*sin(e + f*x))*(a + a*sin(e + f*x))^3*(c - c*sin(e + f*x))^6,x)

[Out]

(tan(e/2 + (f*x)/2)^18*(6*A*a^3*c^6 - 2*B*a^3*c^6) + tan(e/2 + (f*x)/2)^16*(22*A*a^3*c^6 - 18*B*a^3*c^6) + tan
(e/2 + (f*x)/2)^8*(84*A*a^3*c^6 - 28*B*a^3*c^6) + tan(e/2 + (f*x)/2)^14*((136*A*a^3*c^6)/3 - 8*B*a^3*c^6) + ta
n(e/2 + (f*x)/2)^4*((136*A*a^3*c^6)/7 - (24*B*a^3*c^6)/7) + tan(e/2 + (f*x)/2)^10*(116*A*a^3*c^6 - 60*B*a^3*c^
6) - tan(e/2 + (f*x)/2)^19*((73*A*a^3*c^6)/64 + (33*B*a^3*c^6)/128) + tan(e/2 + (f*x)/2)^2*((202*A*a^3*c^6)/63
- (58*B*a^3*c^6)/21) + tan(e/2 + (f*x)/2)^12*((328*A*a^3*c^6)/3 - 72*B*a^3*c^6) + tan(e/2 + (f*x)/2)^7*((341*
A*a^3*c^6)/16 + (333*B*a^3*c^6)/32) - tan(e/2 + (f*x)/2)^13*((341*A*a^3*c^6)/16 + (333*B*a^3*c^6)/32) + tan(e/
2 + (f*x)/2)^6*((456*A*a^3*c^6)/7 - (344*B*a^3*c^6)/7) + tan(e/2 + (f*x)/2)^5*((449*A*a^3*c^6)/48 - (577*B*a^3
*c^6)/160) - tan(e/2 + (f*x)/2)^15*((449*A*a^3*c^6)/48 - (577*B*a^3*c^6)/160) + tan(e/2 + (f*x)/2)^3*((2117*A*
a^3*c^6)/192 - (705*B*a^3*c^6)/128) - tan(e/2 + (f*x)/2)^17*((2117*A*a^3*c^6)/192 - (705*B*a^3*c^6)/128) + tan
(e/2 + (f*x)/2)^9*((699*A*a^3*c^6)/32 - (2749*B*a^3*c^6)/64) - tan(e/2 + (f*x)/2)^11*((699*A*a^3*c^6)/32 - (27
49*B*a^3*c^6)/64) + tan(e/2 + (f*x)/2)*((73*A*a^3*c^6)/64 + (33*B*a^3*c^6)/128) + (58*A*a^3*c^6)/63 - (10*B*a^
3*c^6)/21)/(f*(10*tan(e/2 + (f*x)/2)^2 + 45*tan(e/2 + (f*x)/2)^4 + 120*tan(e/2 + (f*x)/2)^6 + 210*tan(e/2 + (f
*x)/2)^8 + 252*tan(e/2 + (f*x)/2)^10 + 210*tan(e/2 + (f*x)/2)^12 + 120*tan(e/2 + (f*x)/2)^14 + 45*tan(e/2 + (f
*x)/2)^16 + 10*tan(e/2 + (f*x)/2)^18 + tan(e/2 + (f*x)/2)^20 + 1)) + (11*a^3*c^6*atan((11*a^3*c^6*tan(e/2 + (f
*x)/2)*(10*A - 3*B))/(128*((55*A*a^3*c^6)/64 - (33*B*a^3*c^6)/128)))*(10*A - 3*B))/(128*f)

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sympy [A]  time = 52.76, size = 1948, normalized size = 7.35 \[ \text {result too large to display} \]

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))**6,x)

[Out]

Piecewise((-105*A*a**3*c**6*x*sin(e + f*x)**8/128 - 105*A*a**3*c**6*x*sin(e + f*x)**6*cos(e + f*x)**2/32 + 5*A
*a**3*c**6*x*sin(e + f*x)**6/2 - 315*A*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**4/64 + 15*A*a**3*c**6*x*sin(e
+ f*x)**4*cos(e + f*x)**2/2 - 9*A*a**3*c**6*x*sin(e + f*x)**4/4 - 105*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f
*x)**6/32 + 15*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**4/2 - 9*A*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)*
*2/2 - 105*A*a**3*c**6*x*cos(e + f*x)**8/128 + 5*A*a**3*c**6*x*cos(e + f*x)**6/2 - 9*A*a**3*c**6*x*cos(e + f*x
)**4/4 + A*a**3*c**6*x - A*a**3*c**6*sin(e + f*x)**8*cos(e + f*x)/f + 279*A*a**3*c**6*sin(e + f*x)**7*cos(e +
f*x)/(128*f) - 8*A*a**3*c**6*sin(e + f*x)**6*cos(e + f*x)**3/(3*f) + 511*A*a**3*c**6*sin(e + f*x)**5*cos(e + f
*x)**3/(128*f) - 11*A*a**3*c**6*sin(e + f*x)**5*cos(e + f*x)/(2*f) - 16*A*a**3*c**6*sin(e + f*x)**4*cos(e + f*
x)**5/(5*f) + 6*A*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)/f + 385*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**5/(
128*f) - 20*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**3/(3*f) + 15*A*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/(4
*f) - 64*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**7/(35*f) + 8*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**3/f
- 8*A*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)/f + 105*A*a**3*c**6*sin(e + f*x)*cos(e + f*x)**7/(128*f) - 5*A*a*
*3*c**6*sin(e + f*x)*cos(e + f*x)**5/(2*f) + 9*A*a**3*c**6*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 128*A*a**3*c**
6*cos(e + f*x)**9/(315*f) + 16*A*a**3*c**6*cos(e + f*x)**5/(5*f) - 16*A*a**3*c**6*cos(e + f*x)**3/(3*f) + 3*A*
a**3*c**6*cos(e + f*x)/f + 63*B*a**3*c**6*x*sin(e + f*x)**10/256 + 315*B*a**3*c**6*x*sin(e + f*x)**8*cos(e + f
*x)**2/256 + 315*B*a**3*c**6*x*sin(e + f*x)**6*cos(e + f*x)**4/128 - 15*B*a**3*c**6*x*sin(e + f*x)**6/8 + 315*
B*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**6/128 - 45*B*a**3*c**6*x*sin(e + f*x)**4*cos(e + f*x)**2/8 + 3*B*a
**3*c**6*x*sin(e + f*x)**4 + 315*B*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**8/256 - 45*B*a**3*c**6*x*sin(e +
f*x)**2*cos(e + f*x)**4/8 + 6*B*a**3*c**6*x*sin(e + f*x)**2*cos(e + f*x)**2 - 3*B*a**3*c**6*x*sin(e + f*x)**2/
2 + 63*B*a**3*c**6*x*cos(e + f*x)**10/256 - 15*B*a**3*c**6*x*cos(e + f*x)**6/8 + 3*B*a**3*c**6*x*cos(e + f*x)*
*4 - 3*B*a**3*c**6*x*cos(e + f*x)**2/2 - 193*B*a**3*c**6*sin(e + f*x)**9*cos(e + f*x)/(256*f) + 3*B*a**3*c**6*
sin(e + f*x)**8*cos(e + f*x)/f - 237*B*a**3*c**6*sin(e + f*x)**7*cos(e + f*x)**3/(128*f) + 8*B*a**3*c**6*sin(e
+ f*x)**6*cos(e + f*x)**3/f - 8*B*a**3*c**6*sin(e + f*x)**6*cos(e + f*x)/f - 21*B*a**3*c**6*sin(e + f*x)**5*c
os(e + f*x)**5/(10*f) + 33*B*a**3*c**6*sin(e + f*x)**5*cos(e + f*x)/(8*f) + 48*B*a**3*c**6*sin(e + f*x)**4*cos
(e + f*x)**5/(5*f) - 16*B*a**3*c**6*sin(e + f*x)**4*cos(e + f*x)**3/f + 6*B*a**3*c**6*sin(e + f*x)**4*cos(e +
f*x)/f - 147*B*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)**7/(128*f) + 5*B*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)*
*3/f - 5*B*a**3*c**6*sin(e + f*x)**3*cos(e + f*x)/f + 192*B*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**7/(35*f) -
64*B*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**5/(5*f) + 8*B*a**3*c**6*sin(e + f*x)**2*cos(e + f*x)**3/f - 63*B
*a**3*c**6*sin(e + f*x)*cos(e + f*x)**9/(256*f) + 15*B*a**3*c**6*sin(e + f*x)*cos(e + f*x)**5/(8*f) - 3*B*a**3
*c**6*sin(e + f*x)*cos(e + f*x)**3/f + 3*B*a**3*c**6*sin(e + f*x)*cos(e + f*x)/(2*f) + 128*B*a**3*c**6*cos(e +
f*x)**9/(105*f) - 128*B*a**3*c**6*cos(e + f*x)**7/(35*f) + 16*B*a**3*c**6*cos(e + f*x)**5/(5*f) - B*a**3*c**6
*cos(e + f*x)/f, Ne(f, 0)), (x*(A + B*sin(e))*(a*sin(e) + a)**3*(-c*sin(e) + c)**6, True))

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