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]

5/128*a^3*(9*A-2*B)*c^5*x+1/56*a^3*(9*A-2*B)*c^5*cos(f*x+e)^7/f+5/128*a^3*(9*A-2*B)*c^5*cos(f*x+e)*sin(f*x+e)/
f+5/192*a^3*(9*A-2*B)*c^5*cos(f*x+e)^3*sin(f*x+e)/f+1/48*a^3*(9*A-2*B)*c^5*cos(f*x+e)^5*sin(f*x+e)/f-1/9*a^3*B
*c^3*cos(f*x+e)^7*(c-c*sin(f*x+e))^2/f+1/72*a^3*(9*A-2*B)*cos(f*x+e)^7*(c^5-c^5*sin(f*x+e))/f

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Rubi [A]  time = 0.32, antiderivative size = 222, normalized size of antiderivative = 1.00, 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 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))^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*}

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Mathematica [A]  time = 2.57, 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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fricas [A]  time = 0.50, size = 158, normalized size = 0.71 \[ \frac {896 \, B a^{3} c^{5} \cos \left (f x + e\right )^{9} + 2304 \, {\left (A - B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{7} + 315 \, {\left (9 \, A - 2 \, B\right )} a^{3} c^{5} f x - 21 \, {\left (48 \, {\left (A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{7} - 8 \, {\left (9 \, A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{5} - 10 \, {\left (9 \, A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )^{3} - 15 \, {\left (9 \, A - 2 \, B\right )} a^{3} c^{5} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{8064 \, f} \]

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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giac [A]  time = 0.25, size = 301, normalized size = 1.36 \[ \frac {B a^{3} c^{5} \cos \left (9 \, f x + 9 \, e\right )}{2304 \, f} + \frac {B a^{3} c^{5} \sin \left (6 \, f x + 6 \, e\right )}{96 \, f} + \frac {5}{128} \, {\left (9 \, A a^{3} c^{5} - 2 \, B a^{3} c^{5}\right )} x + \frac {{\left (8 \, A a^{3} c^{5} - B a^{3} c^{5}\right )} \cos \left (7 \, f x + 7 \, e\right )}{1792 \, f} + \frac {{\left (2 \, A a^{3} c^{5} - B a^{3} c^{5}\right )} \cos \left (5 \, f x + 5 \, e\right )}{64 \, f} + \frac {{\left (18 \, A a^{3} c^{5} - 11 \, B a^{3} c^{5}\right )} \cos \left (3 \, f x + 3 \, e\right )}{192 \, f} + \frac {{\left (20 \, A a^{3} c^{5} - 13 \, B a^{3} c^{5}\right )} \cos \left (f x + e\right )}{128 \, f} - \frac {{\left (A a^{3} c^{5} - 2 \, B a^{3} c^{5}\right )} \sin \left (8 \, f x + 8 \, e\right )}{1024 \, f} + \frac {{\left (5 \, A a^{3} c^{5} + 2 \, B a^{3} c^{5}\right )} \sin \left (4 \, f x + 4 \, e\right )}{128 \, f} + \frac {{\left (8 \, A a^{3} c^{5} - B a^{3} c^{5}\right )} \sin \left (2 \, f x + 2 \, e\right )}{32 \, f} \]

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

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maple [B]  time = 0.76, size = 611, normalized size = 2.75 \[ \frac {a^{3} A \,c^{5} \left (f x +e \right )-a^{3} A \,c^{5} \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 )-\frac {2 a^{3} A \,c^{5} \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}+2 a^{3} A \,c^{5} \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^{5} \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}+\frac {B \,a^{3} c^{5} \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}+2 B \,a^{3} c^{5} \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 )-\frac {2 B \,a^{3} c^{5} \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^{5} \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 )-B \,a^{3} c^{5} \cos \left (f x +e \right )-2 a^{3} A \,c^{5} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )+6 B \,a^{3} c^{5} \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 )-2 a^{3} A \,c^{5} \left (-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )+\frac {2 B \,a^{3} c^{5} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 a^{3} A \,c^{5} \cos \left (f x +e \right )-2 B \,a^{3} c^{5} \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))^5,x)

[Out]

1/f*(a^3*A*c^5*(f*x+e)-a^3*A*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)-2/7*a^3*A*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^3*A*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)+6/5*a^3*A*c^5*(8/3
+sin(f*x+e)^4+4/3*sin(f*x+e)^2)*cos(f*x+e)+1/9*B*a^3*c^5*(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)+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*s
in(f*x+e))*cos(f*x+e)+35/128*f*x+35/128*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)-B*a^
3*c^5*cos(f*x+e)-2*a^3*A*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^3*A*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)+2*a^3*A*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))

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maxima [B]  time = 0.35, size = 617, normalized size = 2.78 \[ \frac {18432 \, {\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 )} A a^{3} c^{5} + 129024 \, {\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^{5} + 645120 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{3} c^{5} - 105 \, {\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^{5} + 3360 \, {\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^{5} - 161280 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{3} c^{5} + 322560 \, {\left (f x + e\right )} A a^{3} c^{5} + 1024 \, {\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^{5} + 18432 \, {\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^{5} - 215040 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{3} c^{5} + 210 \, {\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 )} B a^{3} c^{5} - 10080 \, {\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^{5} + 60480 \, {\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^{5} - 161280 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{3} c^{5} + 645120 \, A a^{3} c^{5} \cos \left (f x + e\right ) - 322560 \, B a^{3} c^{5} \cos \left (f x + e\right )}{322560 \, 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))^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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mupad [B]  time = 14.93, size = 705, normalized size = 3.18 \[ \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}\,\left (4\,A\,a^3\,c^5-2\,B\,a^3\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}\,\left (8\,A\,a^3\,c^5-8\,B\,a^3\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {8\,A\,a^3\,c^5}{7}-\frac {8\,B\,a^3\,c^5}{7}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (32\,A\,a^3\,c^5-4\,B\,a^3\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (24\,A\,a^3\,c^5-24\,B\,a^3\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}\,\left (24\,A\,a^3\,c^5-\frac {16\,B\,a^3\,c^5}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (40\,A\,a^3\,c^5-40\,B\,a^3\,c^5\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {88\,A\,a^3\,c^5}{7}-\frac {32\,B\,a^3\,c^5}{7}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{17}\,\left (\frac {83\,A\,a^3\,c^5}{64}+\frac {5\,B\,a^3\,c^5}{32}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (\frac {149\,A\,a^3\,c^5}{32}+\frac {83\,B\,a^3\,c^5}{16}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{13}\,\left (\frac {149\,A\,a^3\,c^5}{32}+\frac {83\,B\,a^3\,c^5}{16}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (\frac {189\,A\,a^3\,c^5}{32}-\frac {191\,B\,a^3\,c^5}{48}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{15}\,\left (\frac {189\,A\,a^3\,c^5}{32}-\frac {191\,B\,a^3\,c^5}{48}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (\frac {409\,A\,a^3\,c^5}{32}-\frac {145\,B\,a^3\,c^5}{16}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{11}\,\left (\frac {409\,A\,a^3\,c^5}{32}-\frac {145\,B\,a^3\,c^5}{16}\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {83\,A\,a^3\,c^5}{64}+\frac {5\,B\,a^3\,c^5}{32}\right )+\frac {4\,A\,a^3\,c^5}{7}-\frac {22\,B\,a^3\,c^5}{63}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{18}+9\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{16}+36\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{14}+84\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+126\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+84\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+36\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+9\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )}+\frac {5\,a^3\,c^5\,\mathrm {atan}\left (\frac {5\,a^3\,c^5\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (9\,A-2\,B\right )}{64\,\left (\frac {45\,A\,a^3\,c^5}{64}-\frac {5\,B\,a^3\,c^5}{32}\right )}\right )\,\left (9\,A-2\,B\right )}{64\,f} \]

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))^5,x)

[Out]

(tan(e/2 + (f*x)/2)^16*(4*A*a^3*c^5 - 2*B*a^3*c^5) + tan(e/2 + (f*x)/2)^14*(8*A*a^3*c^5 - 8*B*a^3*c^5) + tan(e
/2 + (f*x)/2)^2*((8*A*a^3*c^5)/7 - (8*B*a^3*c^5)/7) + tan(e/2 + (f*x)/2)^8*(32*A*a^3*c^5 - 4*B*a^3*c^5) + tan(
e/2 + (f*x)/2)^6*(24*A*a^3*c^5 - 24*B*a^3*c^5) + tan(e/2 + (f*x)/2)^12*(24*A*a^3*c^5 - (16*B*a^3*c^5)/3) + tan
(e/2 + (f*x)/2)^10*(40*A*a^3*c^5 - 40*B*a^3*c^5) + tan(e/2 + (f*x)/2)^4*((88*A*a^3*c^5)/7 - (32*B*a^3*c^5)/7)
- tan(e/2 + (f*x)/2)^17*((83*A*a^3*c^5)/64 + (5*B*a^3*c^5)/32) + tan(e/2 + (f*x)/2)^5*((149*A*a^3*c^5)/32 + (8
3*B*a^3*c^5)/16) - tan(e/2 + (f*x)/2)^13*((149*A*a^3*c^5)/32 + (83*B*a^3*c^5)/16) + tan(e/2 + (f*x)/2)^3*((189
*A*a^3*c^5)/32 - (191*B*a^3*c^5)/48) - tan(e/2 + (f*x)/2)^15*((189*A*a^3*c^5)/32 - (191*B*a^3*c^5)/48) + tan(e
/2 + (f*x)/2)^7*((409*A*a^3*c^5)/32 - (145*B*a^3*c^5)/16) - tan(e/2 + (f*x)/2)^11*((409*A*a^3*c^5)/32 - (145*B
*a^3*c^5)/16) + tan(e/2 + (f*x)/2)*((83*A*a^3*c^5)/64 + (5*B*a^3*c^5)/32) + (4*A*a^3*c^5)/7 - (22*B*a^3*c^5)/6
3)/(f*(9*tan(e/2 + (f*x)/2)^2 + 36*tan(e/2 + (f*x)/2)^4 + 84*tan(e/2 + (f*x)/2)^6 + 126*tan(e/2 + (f*x)/2)^8 +
126*tan(e/2 + (f*x)/2)^10 + 84*tan(e/2 + (f*x)/2)^12 + 36*tan(e/2 + (f*x)/2)^14 + 9*tan(e/2 + (f*x)/2)^16 + t
an(e/2 + (f*x)/2)^18 + 1)) + (5*a^3*c^5*atan((5*a^3*c^5*tan(e/2 + (f*x)/2)*(9*A - 2*B))/(64*((45*A*a^3*c^5)/64
- (5*B*a^3*c^5)/32)))*(9*A - 2*B))/(64*f)

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sympy [A]  time = 37.38, size = 1753, normalized size = 7.90 \[ \text {result too large to display} \]

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))**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))

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