\(\int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx\) [252]

Optimal result

Integrand size = 35, antiderivative size = 336 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {1}{8} a^2 \left (12 A c^2+8 B c^2+16 A c d+14 B c d+7 A d^2+6 B d^2\right ) x+\frac {a^2 \left (5 A d \left (c^3-8 c^2 d-20 c d^2-8 d^3\right )-2 B \left (c^4-5 c^3 d+16 c^2 d^2+40 c d^3+18 d^4\right )\right ) \cos (e+f x)}{30 d^2 f}+\frac {a^2 \left (5 A d \left (2 c^2-16 c d-21 d^2\right )-B \left (4 c^3-20 c^2 d+66 c d^2+90 d^3\right )\right ) \cos (e+f x) \sin (e+f x)}{120 d f}+\frac {a^2 \left (5 A (c-8 d) d-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d^2 f}+\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f} \]

[Out]

1/8*a^2*(12*A*c^2+16*A*c*d+7*A*d^2+8*B*c^2+14*B*c*d+6*B*d^2)*x+1/30*a^2*(5*A*d*(c^3-8*c^2*d-20*c*d^2-8*d^3)-2*
B*(c^4-5*c^3*d+16*c^2*d^2+40*c*d^3+18*d^4))*cos(f*x+e)/d^2/f+1/120*a^2*(5*A*d*(2*c^2-16*c*d-21*d^2)-B*(4*c^3-2
0*c^2*d+66*c*d^2+90*d^3))*cos(f*x+e)*sin(f*x+e)/d/f+1/60*a^2*(5*A*(c-8*d)*d-2*B*(c^2-5*c*d+18*d^2))*cos(f*x+e)
*(c+d*sin(f*x+e))^2/d^2/f+1/20*a^2*(2*B*(c-3*d)-5*A*d)*cos(f*x+e)*(c+d*sin(f*x+e))^3/d^2/f-1/5*B*cos(f*x+e)*(a
^2+a^2*sin(f*x+e))*(c+d*sin(f*x+e))^3/d/f

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3055, 3047, 3102, 2832, 2813} \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a^2 \left (5 A d (c-8 d)-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d^2 f}+\frac {1}{8} a^2 x \left (12 A c^2+16 A c d+7 A d^2+8 B c^2+14 B c d+6 B d^2\right )+\frac {a^2 \left (5 A d \left (2 c^2-16 c d-21 d^2\right )-B \left (4 c^3-20 c^2 d+66 c d^2+90 d^3\right )\right ) \sin (e+f x) \cos (e+f x)}{120 d f}+\frac {a^2 \left (5 A d \left (c^3-8 c^2 d-20 c d^2-8 d^3\right )-2 B \left (c^4-5 c^3 d+16 c^2 d^2+40 c d^3+18 d^4\right )\right ) \cos (e+f x)}{30 d^2 f}+\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac {B \cos (e+f x) \left (a^2 \sin (e+f x)+a^2\right ) (c+d \sin (e+f x))^3}{5 d f} \]

[In]

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

[Out]

(a^2*(12*A*c^2 + 8*B*c^2 + 16*A*c*d + 14*B*c*d + 7*A*d^2 + 6*B*d^2)*x)/8 + (a^2*(5*A*d*(c^3 - 8*c^2*d - 20*c*d
^2 - 8*d^3) - 2*B*(c^4 - 5*c^3*d + 16*c^2*d^2 + 40*c*d^3 + 18*d^4))*Cos[e + f*x])/(30*d^2*f) + (a^2*(5*A*d*(2*
c^2 - 16*c*d - 21*d^2) - B*(4*c^3 - 20*c^2*d + 66*c*d^2 + 90*d^3))*Cos[e + f*x]*Sin[e + f*x])/(120*d*f) + (a^2
*(5*A*(c - 8*d)*d - 2*B*(c^2 - 5*c*d + 18*d^2))*Cos[e + f*x]*(c + d*Sin[e + f*x])^2)/(60*d^2*f) + (a^2*(2*B*(c
- 3*d) - 5*A*d)*Cos[e + f*x]*(c + d*Sin[e + f*x])^3)/(20*d^2*f) - (B*Cos[e + f*x]*(a^2 + a^2*Sin[e + f*x])*(c
+ d*Sin[e + f*x])^3)/(5*d*f)

Rule 2813

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(2*a*c +
b*d)*(x/2), x] + (-Simp[(b*c + a*d)*(Cos[e + f*x]/f), x] - Simp[b*d*Cos[e + f*x]*(Sin[e + f*x]/(2*f)), x]) /;
FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0]

Rule 2832

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[1/(m + 1), Int[(a + b*Sin[e + f*x])^(m - 1)*Sim
p[b*d*m + a*c*(m + 1) + (a*d*m + b*c*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[
b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && GtQ[m, 0] && IntegerQ[2*m]

Rule 3047

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 3055

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-b)*B*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m - 1)*((c + d*Sin[e + f*x
])^(n + 1)/(d*f*(m + n + 1))), x] + Dist[1/(d*(m + n + 1)), Int[(a + b*Sin[e + f*x])^(m - 1)*(c + d*Sin[e + f*
x])^n*Simp[a*A*d*(m + n + 1) + B*(a*c*(m - 1) + b*d*(n + 1)) + (A*b*d*(m + n + 1) - B*(b*c*m - a*d*(2*m + n)))
*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] &
& NeQ[c^2 - d^2, 0] && GtQ[m, 1/2] &&  !LtQ[n, -1] && IntegerQ[2*m] && (IntegerQ[2*n] || EqQ[c, 0])

Rule 3102

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]

Rubi steps \begin{align*} \text {integral}& = -\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f}+\frac {\int (a+a \sin (e+f x)) (c+d \sin (e+f x))^2 (a (5 A d+B (c+3 d))-a (2 B c-5 A d-6 B d) \sin (e+f x)) \, dx}{5 d} \\ & = -\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (a^2 (5 A d+B (c+3 d))+\left (-a^2 (2 B c-5 A d-6 B d)+a^2 (5 A d+B (c+3 d))\right ) \sin (e+f x)-a^2 (2 B c-5 A d-6 B d) \sin ^2(e+f x)\right ) \, dx}{5 d} \\ & = \frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f}+\frac {\int (c+d \sin (e+f x))^2 \left (-a^2 d (2 B c-35 A d-30 B d)-a^2 \left (5 A (c-8 d) d-2 B \left (c^2-5 c d+18 d^2\right )\right ) \sin (e+f x)\right ) \, dx}{20 d^2} \\ & = \frac {a^2 \left (5 A (c-8 d) d-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d^2 f}+\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f}+\frac {\int (c+d \sin (e+f x)) \left (a^2 d \left (5 A d (19 c+16 d)-B \left (2 c^2-70 c d-72 d^2\right )\right )-a^2 \left (5 A d \left (2 c^2-16 c d-21 d^2\right )-2 B \left (2 c^3-10 c^2 d+33 c d^2+45 d^3\right )\right ) \sin (e+f x)\right ) \, dx}{60 d^2} \\ & = \frac {1}{8} a^2 \left (12 A c^2+8 B c^2+16 A c d+14 B c d+7 A d^2+6 B d^2\right ) x+\frac {a^2 \left (5 A d \left (c^3-8 c^2 d-20 c d^2-8 d^3\right )-2 B \left (c^4-5 c^3 d+16 c^2 d^2+40 c d^3+18 d^4\right )\right ) \cos (e+f x)}{30 d^2 f}+\frac {a^2 \left (5 A d \left (2 c^2-16 c d-21 d^2\right )-B \left (4 c^3-20 c^2 d+66 c d^2+90 d^3\right )\right ) \cos (e+f x) \sin (e+f x)}{120 d f}+\frac {a^2 \left (5 A (c-8 d) d-2 B \left (c^2-5 c d+18 d^2\right )\right ) \cos (e+f x) (c+d \sin (e+f x))^2}{60 d^2 f}+\frac {a^2 (2 B (c-3 d)-5 A d) \cos (e+f x) (c+d \sin (e+f x))^3}{20 d^2 f}-\frac {B \cos (e+f x) \left (a^2+a^2 \sin (e+f x)\right ) (c+d \sin (e+f x))^3}{5 d f} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.00 (sec) , antiderivative size = 296, normalized size of antiderivative = 0.88 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {a^2 \cos (e+f x) \left (60 \left (2 B \left (4 c^2+7 c d+3 d^2\right )+A \left (12 c^2+16 c d+7 d^2\right )\right ) \arcsin \left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )+\sqrt {\cos ^2(e+f x)} \left (480 A c^2+440 B c^2+880 A c d+800 B c d+400 A d^2+378 B d^2-8 \left (10 A d (c+d)+B \left (5 c^2+20 c d+12 d^2\right )\right ) \cos (2 (e+f x))+6 B d^2 \cos (4 (e+f x))+120 A c^2 \sin (e+f x)+240 B c^2 \sin (e+f x)+480 A c d \sin (e+f x)+510 B c d \sin (e+f x)+255 A d^2 \sin (e+f x)+270 B d^2 \sin (e+f x)-30 B c d \sin (3 (e+f x))-15 A d^2 \sin (3 (e+f x))-30 B d^2 \sin (3 (e+f x))\right )\right )}{240 f \sqrt {\cos ^2(e+f x)}} \]

[In]

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

[Out]

-1/240*(a^2*Cos[e + f*x]*(60*(2*B*(4*c^2 + 7*c*d + 3*d^2) + A*(12*c^2 + 16*c*d + 7*d^2))*ArcSin[Sqrt[1 - Sin[e
+ f*x]]/Sqrt[2]] + Sqrt[Cos[e + f*x]^2]*(480*A*c^2 + 440*B*c^2 + 880*A*c*d + 800*B*c*d + 400*A*d^2 + 378*B*d^
2 - 8*(10*A*d*(c + d) + B*(5*c^2 + 20*c*d + 12*d^2))*Cos[2*(e + f*x)] + 6*B*d^2*Cos[4*(e + f*x)] + 120*A*c^2*S
in[e + f*x] + 240*B*c^2*Sin[e + f*x] + 480*A*c*d*Sin[e + f*x] + 510*B*c*d*Sin[e + f*x] + 255*A*d^2*Sin[e + f*x
] + 270*B*d^2*Sin[e + f*x] - 30*B*c*d*Sin[3*(e + f*x)] - 15*A*d^2*Sin[3*(e + f*x)] - 30*B*d^2*Sin[3*(e + f*x)]
)))/(f*Sqrt[Cos[e + f*x]^2])

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 222, normalized size of antiderivative = 0.66

method	result	size
parallelrisch	\(\frac {\left (\left (\left (-3 B -3 A \right ) d^{2}-6 d c \left (A +B \right )-\frac {3 c^{2} \left (A +2 B \right )}{2}\right ) \sin \left (2 f x +2 e \right )+\left (\left (\frac {9 B}{8}+A \right ) d^{2}+c \left (A +2 B \right ) d +\frac {B \,c^{2}}{2}\right ) \cos \left (3 f x +3 e \right )+\frac {3 \left (\left (A +2 B \right ) d +2 B c \right ) d \sin \left (4 f x +4 e \right )}{16}-\frac {3 B \,d^{2} \cos \left (5 f x +5 e \right )}{40}+\left (\left (-9 A -\frac {33 B}{4}\right ) d^{2}-21 \left (A +\frac {6 B}{7}\right ) c d -12 c^{2} \left (A +\frac {7 B}{8}\right )\right ) \cos \left (f x +e \right )+\left (-\frac {36}{5} B +\frac {21}{4} f x A +\frac {9}{2} f x B -8 A \right ) d^{2}+12 c \left (f x A +\frac {7}{8} f x B -\frac {5}{3} A -\frac {4}{3} B \right ) d +9 c^{2} \left (f x A +\frac {2}{3} f x B -\frac {4}{3} A -\frac {10}{9} B \right )\right ) a^{2}}{6 f}\)	\(222\)
parts	\(\frac {\left (A \,a^{2} d^{2}+2 B \,a^{2} c d +2 B \,a^{2} d^{2}\right ) \left (-\frac {\left (\sin ^{3}\left (f x +e \right )+\frac {3 \sin \left (f x +e \right )}{2}\right ) \cos \left (f x +e \right )}{4}+\frac {3 f x}{8}+\frac {3 e}{8}\right )}{f}-\frac {\left (2 A \,a^{2} c^{2}+2 A \,a^{2} c d +B \,a^{2} c^{2}\right ) \cos \left (f x +e \right )}{f}-\frac {\left (2 A \,a^{2} c d +2 A \,a^{2} d^{2}+B \,a^{2} c^{2}+4 B \,a^{2} c d +B \,a^{2} d^{2}\right ) \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3 f}+\frac {\left (A \,a^{2} c^{2}+4 A \,a^{2} c d +A \,a^{2} d^{2}+2 B \,a^{2} c^{2}+2 B \,a^{2} c d \right ) \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )}{f}+a^{2} A \,c^{2} x -\frac {B \,a^{2} d^{2} \left (\frac {8}{3}+\sin ^{4}\left (f x +e \right )+\frac {4 \left (\sin ^{2}\left (f x +e \right )\right )}{3}\right ) \cos \left (f x +e \right )}{5 f}\)	\(280\)
risch	\(-\frac {7 a^{2} \cos \left (f x +e \right ) A c d}{2 f}-\frac {3 a^{2} \cos \left (f x +e \right ) c d B}{f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{2} c d}{16 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) A c d}{6 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) c d B}{3 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} c d}{f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{2} c d}{f}+\frac {7 A \,a^{2} d^{2} x}{8}+B \,a^{2} c^{2} x +\frac {3 B \,a^{2} d^{2} x}{4}+2 A \,a^{2} c d x +\frac {7 B \,a^{2} c d x}{4}-\frac {2 a^{2} \cos \left (f x +e \right ) A \,c^{2}}{f}-\frac {3 a^{2} \cos \left (f x +e \right ) A \,d^{2}}{2 f}-\frac {7 a^{2} \cos \left (f x +e \right ) B \,c^{2}}{4 f}-\frac {11 a^{2} \cos \left (f x +e \right ) d^{2} B}{8 f}-\frac {B \,a^{2} d^{2} \cos \left (5 f x +5 e \right )}{80 f}+\frac {\sin \left (4 f x +4 e \right ) A \,a^{2} d^{2}}{32 f}+\frac {\sin \left (4 f x +4 e \right ) B \,a^{2} d^{2}}{16 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) A \,d^{2}}{6 f}+\frac {a^{2} \cos \left (3 f x +3 e \right ) B \,c^{2}}{12 f}+\frac {3 a^{2} \cos \left (3 f x +3 e \right ) d^{2} B}{16 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} c^{2}}{4 f}-\frac {\sin \left (2 f x +2 e \right ) A \,a^{2} d^{2}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{2} c^{2}}{2 f}-\frac {\sin \left (2 f x +2 e \right ) B \,a^{2} d^{2}}{2 f}+\frac {3 a^{2} A \,c^{2} x}{2}\)	\(475\)
derivativedivides	\(\frac {A \,a^{2} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 A \,a^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A \,a^{2} d^{2} \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 )-\frac {B \,a^{2} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 B \,a^{2} c d \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 )-\frac {B \,a^{2} d^{2} \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}-2 A \,a^{2} c^{2} \cos \left (f x +e \right )+4 A \,a^{2} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 A \,a^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 B \,a^{2} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 B \,a^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 B \,a^{2} d^{2} \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 )+A \,a^{2} c^{2} \left (f x +e \right )-2 A \,a^{2} c d \cos \left (f x +e \right )+A \,a^{2} d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c^{2} \cos \left (f x +e \right )+2 B \,a^{2} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {B \,a^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(496\)
default	\(\frac {A \,a^{2} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 A \,a^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+A \,a^{2} d^{2} \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 )-\frac {B \,a^{2} c^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 B \,a^{2} c d \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 )-\frac {B \,a^{2} d^{2} \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}-2 A \,a^{2} c^{2} \cos \left (f x +e \right )+4 A \,a^{2} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {2 A \,a^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 B \,a^{2} c^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {4 B \,a^{2} c d \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}+2 B \,a^{2} d^{2} \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 )+A \,a^{2} c^{2} \left (f x +e \right )-2 A \,a^{2} c d \cos \left (f x +e \right )+A \,a^{2} d^{2} \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-B \,a^{2} c^{2} \cos \left (f x +e \right )+2 B \,a^{2} c d \left (-\frac {\cos \left (f x +e \right ) \sin \left (f x +e \right )}{2}+\frac {f x}{2}+\frac {e}{2}\right )-\frac {B \,a^{2} d^{2} \left (2+\sin ^{2}\left (f x +e \right )\right ) \cos \left (f x +e \right )}{3}}{f}\)	\(496\)
norman	\(\frac {\left (\frac {3}{2} A \,a^{2} c^{2}+2 A \,a^{2} c d +\frac {7}{8} A \,a^{2} d^{2}+B \,a^{2} c^{2}+\frac {7}{4} B \,a^{2} c d +\frac {3}{4} B \,a^{2} d^{2}\right ) x +\left (15 A \,a^{2} c^{2}+20 A \,a^{2} c d +\frac {35}{4} A \,a^{2} d^{2}+10 B \,a^{2} c^{2}+\frac {35}{2} B \,a^{2} c d +\frac {15}{2} B \,a^{2} d^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (15 A \,a^{2} c^{2}+20 A \,a^{2} c d +\frac {35}{4} A \,a^{2} d^{2}+10 B \,a^{2} c^{2}+\frac {35}{2} B \,a^{2} c d +\frac {15}{2} B \,a^{2} d^{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {3}{2} A \,a^{2} c^{2}+2 A \,a^{2} c d +\frac {7}{8} A \,a^{2} d^{2}+B \,a^{2} c^{2}+\frac {7}{4} B \,a^{2} c d +\frac {3}{4} B \,a^{2} d^{2}\right ) x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} A \,a^{2} c^{2}+10 A \,a^{2} c d +\frac {35}{8} A \,a^{2} d^{2}+5 B \,a^{2} c^{2}+\frac {35}{4} B \,a^{2} c d +\frac {15}{4} B \,a^{2} d^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+\left (\frac {15}{2} A \,a^{2} c^{2}+10 A \,a^{2} c d +\frac {35}{8} A \,a^{2} d^{2}+5 B \,a^{2} c^{2}+\frac {35}{4} B \,a^{2} c d +\frac {15}{4} B \,a^{2} d^{2}\right ) x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {60 A \,a^{2} c^{2}+100 A \,a^{2} c d +40 A \,a^{2} d^{2}+50 B \,a^{2} c^{2}+80 B \,a^{2} c d +36 B \,a^{2} d^{2}}{15 f}-\frac {\left (4 A \,a^{2} c^{2}+4 A \,a^{2} c d +2 B \,a^{2} c^{2}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (8 A \,a^{2} c^{2}+12 A \,a^{2} c d +4 A \,a^{2} d^{2}+6 B \,a^{2} c^{2}+8 B \,a^{2} c d +2 B \,a^{2} d^{2}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f}-\frac {2 \left (36 A \,a^{2} c^{2}+64 A \,a^{2} c d +28 A \,a^{2} d^{2}+32 B \,a^{2} c^{2}+56 B \,a^{2} c d +30 B \,a^{2} d^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {\left (48 A \,a^{2} c^{2}+88 A \,a^{2} c d +40 A \,a^{2} d^{2}+44 B \,a^{2} c^{2}+80 B \,a^{2} c d +36 B \,a^{2} d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 f}-\frac {a^{2} \left (4 A \,c^{2}+16 A c d +7 A \,d^{2}+8 B \,c^{2}+14 c d B +6 d^{2} B \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{4 f}+\frac {a^{2} \left (4 A \,c^{2}+16 A c d +7 A \,d^{2}+8 B \,c^{2}+14 c d B +6 d^{2} B \right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{4 f}-\frac {a^{2} \left (4 A \,c^{2}+16 A c d +11 A \,d^{2}+8 B \,c^{2}+22 c d B +14 d^{2} B \right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}+\frac {a^{2} \left (4 A \,c^{2}+16 A c d +11 A \,d^{2}+8 B \,c^{2}+22 c d B +14 d^{2} B \right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{5}}\)	\(923\)

[In]

int((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/6*(((-3*B-3*A)*d^2-6*d*c*(A+B)-3/2*c^2*(A+2*B))*sin(2*f*x+2*e)+((9/8*B+A)*d^2+c*(A+2*B)*d+1/2*B*c^2)*cos(3*f
*x+3*e)+3/16*((A+2*B)*d+2*B*c)*d*sin(4*f*x+4*e)-3/40*B*d^2*cos(5*f*x+5*e)+((-9*A-33/4*B)*d^2-21*(A+6/7*B)*c*d-
12*c^2*(A+7/8*B))*cos(f*x+e)+(-36/5*B+21/4*f*x*A+9/2*f*x*B-8*A)*d^2+12*c*(f*x*A+7/8*f*x*B-5/3*A-4/3*B)*d+9*c^2
*(f*x*A+2/3*f*x*B-4/3*A-10/9*B))*a^2/f

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 245, normalized size of antiderivative = 0.73 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {24 \, B a^{2} d^{2} \cos \left (f x + e\right )^{5} - 40 \, {\left (B a^{2} c^{2} + 2 \, {\left (A + 2 \, B\right )} a^{2} c d + {\left (2 \, A + 3 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - 15 \, {\left (4 \, {\left (3 \, A + 2 \, B\right )} a^{2} c^{2} + 2 \, {\left (8 \, A + 7 \, B\right )} a^{2} c d + {\left (7 \, A + 6 \, B\right )} a^{2} d^{2}\right )} f x + 240 \, {\left ({\left (A + B\right )} a^{2} c^{2} + 2 \, {\left (A + B\right )} a^{2} c d + {\left (A + B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right ) - 15 \, {\left (2 \, {\left (2 \, B a^{2} c d + {\left (A + 2 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )^{3} - {\left (4 \, {\left (A + 2 \, B\right )} a^{2} c^{2} + 2 \, {\left (8 \, A + 9 \, B\right )} a^{2} c d + {\left (9 \, A + 10 \, B\right )} a^{2} d^{2}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{120 \, f} \]

[In]

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

[Out]

-1/120*(24*B*a^2*d^2*cos(f*x + e)^5 - 40*(B*a^2*c^2 + 2*(A + 2*B)*a^2*c*d + (2*A + 3*B)*a^2*d^2)*cos(f*x + e)^
3 - 15*(4*(3*A + 2*B)*a^2*c^2 + 2*(8*A + 7*B)*a^2*c*d + (7*A + 6*B)*a^2*d^2)*f*x + 240*((A + B)*a^2*c^2 + 2*(A
+ B)*a^2*c*d + (A + B)*a^2*d^2)*cos(f*x + e) - 15*(2*(2*B*a^2*c*d + (A + 2*B)*a^2*d^2)*cos(f*x + e)^3 - (4*(A
+ 2*B)*a^2*c^2 + 2*(8*A + 9*B)*a^2*c*d + (9*A + 10*B)*a^2*d^2)*cos(f*x + e))*sin(f*x + e))/f

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1129 vs. \(2 (330) = 660\).

Time = 0.35 (sec) , antiderivative size = 1129, normalized size of antiderivative = 3.36 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\text {Too large to display} \]

[In]

integrate((a+a*sin(f*x+e))**2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**2,x)

[Out]

Piecewise((A*a**2*c**2*x*sin(e + f*x)**2/2 + A*a**2*c**2*x*cos(e + f*x)**2/2 + A*a**2*c**2*x - A*a**2*c**2*sin
(e + f*x)*cos(e + f*x)/(2*f) - 2*A*a**2*c**2*cos(e + f*x)/f + 2*A*a**2*c*d*x*sin(e + f*x)**2 + 2*A*a**2*c*d*x*
cos(e + f*x)**2 - 2*A*a**2*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 2*A*a**2*c*d*sin(e + f*x)*cos(e + f*x)/f - 4*A
*a**2*c*d*cos(e + f*x)**3/(3*f) - 2*A*a**2*c*d*cos(e + f*x)/f + 3*A*a**2*d**2*x*sin(e + f*x)**4/8 + 3*A*a**2*d
**2*x*sin(e + f*x)**2*cos(e + f*x)**2/4 + A*a**2*d**2*x*sin(e + f*x)**2/2 + 3*A*a**2*d**2*x*cos(e + f*x)**4/8
+ A*a**2*d**2*x*cos(e + f*x)**2/2 - 5*A*a**2*d**2*sin(e + f*x)**3*cos(e + f*x)/(8*f) - 2*A*a**2*d**2*sin(e + f
*x)**2*cos(e + f*x)/f - 3*A*a**2*d**2*sin(e + f*x)*cos(e + f*x)**3/(8*f) - A*a**2*d**2*sin(e + f*x)*cos(e + f*
x)/(2*f) - 4*A*a**2*d**2*cos(e + f*x)**3/(3*f) + B*a**2*c**2*x*sin(e + f*x)**2 + B*a**2*c**2*x*cos(e + f*x)**2
- B*a**2*c**2*sin(e + f*x)**2*cos(e + f*x)/f - B*a**2*c**2*sin(e + f*x)*cos(e + f*x)/f - 2*B*a**2*c**2*cos(e
+ f*x)**3/(3*f) - B*a**2*c**2*cos(e + f*x)/f + 3*B*a**2*c*d*x*sin(e + f*x)**4/4 + 3*B*a**2*c*d*x*sin(e + f*x)*
*2*cos(e + f*x)**2/2 + B*a**2*c*d*x*sin(e + f*x)**2 + 3*B*a**2*c*d*x*cos(e + f*x)**4/4 + B*a**2*c*d*x*cos(e +
f*x)**2 - 5*B*a**2*c*d*sin(e + f*x)**3*cos(e + f*x)/(4*f) - 4*B*a**2*c*d*sin(e + f*x)**2*cos(e + f*x)/f - 3*B*
a**2*c*d*sin(e + f*x)*cos(e + f*x)**3/(4*f) - B*a**2*c*d*sin(e + f*x)*cos(e + f*x)/f - 8*B*a**2*c*d*cos(e + f*
x)**3/(3*f) + 3*B*a**2*d**2*x*sin(e + f*x)**4/4 + 3*B*a**2*d**2*x*sin(e + f*x)**2*cos(e + f*x)**2/2 + 3*B*a**2
*d**2*x*cos(e + f*x)**4/4 - B*a**2*d**2*sin(e + f*x)**4*cos(e + f*x)/f - 5*B*a**2*d**2*sin(e + f*x)**3*cos(e +
f*x)/(4*f) - 4*B*a**2*d**2*sin(e + f*x)**2*cos(e + f*x)**3/(3*f) - B*a**2*d**2*sin(e + f*x)**2*cos(e + f*x)/f
- 3*B*a**2*d**2*sin(e + f*x)*cos(e + f*x)**3/(4*f) - 8*B*a**2*d**2*cos(e + f*x)**5/(15*f) - 2*B*a**2*d**2*cos
(e + f*x)**3/(3*f), Ne(f, 0)), (x*(A + B*sin(e))*(c + d*sin(e))**2*(a*sin(e) + a)**2, True))

Maxima [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.42 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c^{2} + 480 \, {\left (f x + e\right )} A a^{2} c^{2} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c^{2} + 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c^{2} + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} c d + 480 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} c d + 640 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} c d + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c d + 240 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} c d + 320 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} A a^{2} d^{2} + 15 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} d^{2} + 120 \, {\left (2 \, f x + 2 \, e - \sin \left (2 \, f x + 2 \, e\right )\right )} A a^{2} d^{2} - 32 \, {\left (3 \, \cos \left (f x + e\right )^{5} - 10 \, \cos \left (f x + e\right )^{3} + 15 \, \cos \left (f x + e\right )\right )} B a^{2} d^{2} + 160 \, {\left (\cos \left (f x + e\right )^{3} - 3 \, \cos \left (f x + e\right )\right )} B a^{2} d^{2} + 30 \, {\left (12 \, f x + 12 \, e + \sin \left (4 \, f x + 4 \, e\right ) - 8 \, \sin \left (2 \, f x + 2 \, e\right )\right )} B a^{2} d^{2} - 960 \, A a^{2} c^{2} \cos \left (f x + e\right ) - 480 \, B a^{2} c^{2} \cos \left (f x + e\right ) - 960 \, A a^{2} c d \cos \left (f x + e\right )}{480 \, f} \]

[In]

integrate((a+a*sin(f*x+e))^2*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

1/480*(120*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*c^2 + 480*(f*x + e)*A*a^2*c^2 + 160*(cos(f*x + e)^3 - 3*cos(
f*x + e))*B*a^2*c^2 + 240*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*a^2*c^2 + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A
*a^2*c*d + 480*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*c*d + 640*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*c*d +
30*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^2*c*d + 240*(2*f*x + 2*e - sin(2*f*x + 2*e))*B*
a^2*c*d + 320*(cos(f*x + e)^3 - 3*cos(f*x + e))*A*a^2*d^2 + 15*(12*f*x + 12*e + sin(4*f*x + 4*e) - 8*sin(2*f*x
+ 2*e))*A*a^2*d^2 + 120*(2*f*x + 2*e - sin(2*f*x + 2*e))*A*a^2*d^2 - 32*(3*cos(f*x + e)^5 - 10*cos(f*x + e)^3
+ 15*cos(f*x + e))*B*a^2*d^2 + 160*(cos(f*x + e)^3 - 3*cos(f*x + e))*B*a^2*d^2 + 30*(12*f*x + 12*e + sin(4*f*
x + 4*e) - 8*sin(2*f*x + 2*e))*B*a^2*d^2 - 960*A*a^2*c^2*cos(f*x + e) - 480*B*a^2*c^2*cos(f*x + e) - 960*A*a^2
*c*d*cos(f*x + e))/f

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 306, normalized size of antiderivative = 0.91 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=-\frac {B a^{2} d^{2} \cos \left (5 \, f x + 5 \, e\right )}{80 \, f} + \frac {1}{8} \, {\left (12 \, A a^{2} c^{2} + 8 \, B a^{2} c^{2} + 16 \, A a^{2} c d + 14 \, B a^{2} c d + 7 \, A a^{2} d^{2} + 6 \, B a^{2} d^{2}\right )} x + \frac {{\left (4 \, B a^{2} c^{2} + 8 \, A a^{2} c d + 16 \, B a^{2} c d + 8 \, A a^{2} d^{2} + 9 \, B a^{2} d^{2}\right )} \cos \left (3 \, f x + 3 \, e\right )}{48 \, f} - \frac {{\left (16 \, A a^{2} c^{2} + 14 \, B a^{2} c^{2} + 28 \, A a^{2} c d + 24 \, B a^{2} c d + 12 \, A a^{2} d^{2} + 11 \, B a^{2} d^{2}\right )} \cos \left (f x + e\right )}{8 \, f} + \frac {{\left (2 \, B a^{2} c d + A a^{2} d^{2} + 2 \, B a^{2} d^{2}\right )} \sin \left (4 \, f x + 4 \, e\right )}{32 \, f} - \frac {{\left (A a^{2} c^{2} + 2 \, B a^{2} c^{2} + 4 \, A a^{2} c d + 4 \, B a^{2} c d + 2 \, A a^{2} d^{2} + 2 \, B a^{2} d^{2}\right )} \sin \left (2 \, f x + 2 \, e\right )}{4 \, f} \]

[In]

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

[Out]

-1/80*B*a^2*d^2*cos(5*f*x + 5*e)/f + 1/8*(12*A*a^2*c^2 + 8*B*a^2*c^2 + 16*A*a^2*c*d + 14*B*a^2*c*d + 7*A*a^2*d
^2 + 6*B*a^2*d^2)*x + 1/48*(4*B*a^2*c^2 + 8*A*a^2*c*d + 16*B*a^2*c*d + 8*A*a^2*d^2 + 9*B*a^2*d^2)*cos(3*f*x +
3*e)/f - 1/8*(16*A*a^2*c^2 + 14*B*a^2*c^2 + 28*A*a^2*c*d + 24*B*a^2*c*d + 12*A*a^2*d^2 + 11*B*a^2*d^2)*cos(f*x
+ e)/f + 1/32*(2*B*a^2*c*d + A*a^2*d^2 + 2*B*a^2*d^2)*sin(4*f*x + 4*e)/f - 1/4*(A*a^2*c^2 + 2*B*a^2*c^2 + 4*A
*a^2*c*d + 4*B*a^2*c*d + 2*A*a^2*d^2 + 2*B*a^2*d^2)*sin(2*f*x + 2*e)/f

Mupad [B] (verification not implemented)

Time = 15.55 (sec) , antiderivative size = 765, normalized size of antiderivative = 2.28 \[ \int (a+a \sin (e+f x))^2 (A+B \sin (e+f x)) (c+d \sin (e+f x))^2 \, dx=\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (12\,A\,c^2+7\,A\,d^2+8\,B\,c^2+6\,B\,d^2+16\,A\,c\,d+14\,B\,c\,d\right )}{4\,\left (3\,A\,a^2\,c^2+\frac {7\,A\,a^2\,d^2}{4}+2\,B\,a^2\,c^2+\frac {3\,B\,a^2\,d^2}{2}+4\,A\,a^2\,c\,d+\frac {7\,B\,a^2\,c\,d}{2}\right )}\right )\,\left (12\,A\,c^2+7\,A\,d^2+8\,B\,c^2+6\,B\,d^2+16\,A\,c\,d+14\,B\,c\,d\right )}{4\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (4\,A\,a^2\,c^2+2\,B\,a^2\,c^2+4\,A\,a^2\,c\,d\right )+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (A\,a^2\,c^2+\frac {7\,A\,a^2\,d^2}{4}+2\,B\,a^2\,c^2+\frac {3\,B\,a^2\,d^2}{2}+4\,A\,a^2\,c\,d+\frac {7\,B\,a^2\,c\,d}{2}\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^9\,\left (A\,a^2\,c^2+\frac {7\,A\,a^2\,d^2}{4}+2\,B\,a^2\,c^2+\frac {3\,B\,a^2\,d^2}{2}+4\,A\,a^2\,c\,d+\frac {7\,B\,a^2\,c\,d}{2}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,A\,a^2\,c^2+\frac {11\,A\,a^2\,d^2}{2}+4\,B\,a^2\,c^2+7\,B\,a^2\,d^2+8\,A\,a^2\,c\,d+11\,B\,a^2\,c\,d\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (2\,A\,a^2\,c^2+\frac {11\,A\,a^2\,d^2}{2}+4\,B\,a^2\,c^2+7\,B\,a^2\,d^2+8\,A\,a^2\,c\,d+11\,B\,a^2\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (16\,A\,a^2\,c^2+8\,A\,a^2\,d^2+12\,B\,a^2\,c^2+4\,B\,a^2\,d^2+24\,A\,a^2\,c\,d+16\,B\,a^2\,c\,d\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (16\,A\,a^2\,c^2+\frac {40\,A\,a^2\,d^2}{3}+\frac {44\,B\,a^2\,c^2}{3}+12\,B\,a^2\,d^2+\frac {88\,A\,a^2\,c\,d}{3}+\frac {80\,B\,a^2\,c\,d}{3}\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (24\,A\,a^2\,c^2+\frac {56\,A\,a^2\,d^2}{3}+\frac {64\,B\,a^2\,c^2}{3}+20\,B\,a^2\,d^2+\frac {128\,A\,a^2\,c\,d}{3}+\frac {112\,B\,a^2\,c\,d}{3}\right )+4\,A\,a^2\,c^2+\frac {8\,A\,a^2\,d^2}{3}+\frac {10\,B\,a^2\,c^2}{3}+\frac {12\,B\,a^2\,d^2}{5}+\frac {20\,A\,a^2\,c\,d}{3}+\frac {16\,B\,a^2\,c\,d}{3}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+10\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+5\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+1\right )} \]

[In]

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

[Out]

(a^2*atan((a^2*tan(e/2 + (f*x)/2)*(12*A*c^2 + 7*A*d^2 + 8*B*c^2 + 6*B*d^2 + 16*A*c*d + 14*B*c*d))/(4*(3*A*a^2*
c^2 + (7*A*a^2*d^2)/4 + 2*B*a^2*c^2 + (3*B*a^2*d^2)/2 + 4*A*a^2*c*d + (7*B*a^2*c*d)/2)))*(12*A*c^2 + 7*A*d^2 +
8*B*c^2 + 6*B*d^2 + 16*A*c*d + 14*B*c*d))/(4*f) - (tan(e/2 + (f*x)/2)^8*(4*A*a^2*c^2 + 2*B*a^2*c^2 + 4*A*a^2*
c*d) + tan(e/2 + (f*x)/2)*(A*a^2*c^2 + (7*A*a^2*d^2)/4 + 2*B*a^2*c^2 + (3*B*a^2*d^2)/2 + 4*A*a^2*c*d + (7*B*a^
2*c*d)/2) - tan(e/2 + (f*x)/2)^9*(A*a^2*c^2 + (7*A*a^2*d^2)/4 + 2*B*a^2*c^2 + (3*B*a^2*d^2)/2 + 4*A*a^2*c*d +
(7*B*a^2*c*d)/2) + tan(e/2 + (f*x)/2)^3*(2*A*a^2*c^2 + (11*A*a^2*d^2)/2 + 4*B*a^2*c^2 + 7*B*a^2*d^2 + 8*A*a^2*
c*d + 11*B*a^2*c*d) - tan(e/2 + (f*x)/2)^7*(2*A*a^2*c^2 + (11*A*a^2*d^2)/2 + 4*B*a^2*c^2 + 7*B*a^2*d^2 + 8*A*a
^2*c*d + 11*B*a^2*c*d) + tan(e/2 + (f*x)/2)^6*(16*A*a^2*c^2 + 8*A*a^2*d^2 + 12*B*a^2*c^2 + 4*B*a^2*d^2 + 24*A*
a^2*c*d + 16*B*a^2*c*d) + tan(e/2 + (f*x)/2)^2*(16*A*a^2*c^2 + (40*A*a^2*d^2)/3 + (44*B*a^2*c^2)/3 + 12*B*a^2*
d^2 + (88*A*a^2*c*d)/3 + (80*B*a^2*c*d)/3) + tan(e/2 + (f*x)/2)^4*(24*A*a^2*c^2 + (56*A*a^2*d^2)/3 + (64*B*a^2
*c^2)/3 + 20*B*a^2*d^2 + (128*A*a^2*c*d)/3 + (112*B*a^2*c*d)/3) + 4*A*a^2*c^2 + (8*A*a^2*d^2)/3 + (10*B*a^2*c^
2)/3 + (12*B*a^2*d^2)/5 + (20*A*a^2*c*d)/3 + (16*B*a^2*c*d)/3)/(f*(5*tan(e/2 + (f*x)/2)^2 + 10*tan(e/2 + (f*x)
/2)^4 + 10*tan(e/2 + (f*x)/2)^6 + 5*tan(e/2 + (f*x)/2)^8 + tan(e/2 + (f*x)/2)^10 + 1))