3.8.22 \(\int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx\) [722]

Optimal. Leaf size=669 \[ -\frac {b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^5 f}-\frac {d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^5 \left (c^2-d^2\right )^{5/2} f}-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \]

[Out]

-b^3*(10*a^3*b*c*d-4*a*b^3*c*d-20*a^4*d^2-a^2*b^2*(2*c^2-29*d^2)-b^4*(c^2+12*d^2))*arctan((b+a*tan(1/2*f*x+1/2
*e))/(a^2-b^2)^(1/2))/(a^2-b^2)^(5/2)/(-a*d+b*c)^5/f-d^3*(a^2*d^2*(2*c^2+d^2)-a*b*(10*c^3*d-4*c*d^3)+b^2*(20*c
^4-29*c^2*d^2+12*d^4))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/(-a*d+b*c)^5/(c^2-d^2)^(5/2)/f-1/2*d*(
a^4*d^3-b^4*d*(5*c^2-6*d^2)+2*a^2*b^2*d*(4*c^2-5*d^2)-3*a*b^3*c*(c^2-d^2))*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^3
/(c^2-d^2)/f/(c+d*sin(f*x+e))^2+1/2*b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^
2+1/2*b^2*(-7*a^2*d+3*a*b*c+4*b^2*d)*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^2/f/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^2
+3/2*d*(a^5*c*d^4-2*a^3*b^2*c*d^4+a*b^4*c*(c^4-2*c^2*d^2+2*d^4)+b^5*d*(2*c^4-7*c^2*d^2+4*d^4)-a^2*b^3*d*(3*c^4
-12*c^2*d^2+7*d^4)-a^4*b*(3*c^2*d^3-2*d^5))*cos(f*x+e)/(a^2-b^2)^2/(-a*d+b*c)^4/(c^2-d^2)^2/f/(c+d*sin(f*x+e))

________________________________________________________________________________________

Rubi [A]
time = 2.19, antiderivative size = 669, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2881, 3134, 3080, 2739, 632, 210} \begin {gather*} -\frac {d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{5/2} (b c-a d)^5}+\frac {b^2 \left (-7 a^2 d+3 a b c+4 b^2 d\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 (b c-a d)^2 (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {d \left (a^4 d^3+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-b^4 d \left (5 c^2-6 d^2\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right ) (b c-a d)^3 (c+d \sin (e+f x))^2}-\frac {b^3 \left (-20 a^4 d^2+10 a^3 b c d-a^2 b^2 \left (2 c^2-29 d^2\right )-4 a b^3 c d-b^4 \left (c^2+12 d^2\right )\right ) \text {ArcTan}\left (\frac {a \tan \left (\frac {1}{2} (e+f x)\right )+b}{\sqrt {a^2-b^2}}\right )}{f \left (a^2-b^2\right )^{5/2} (b c-a d)^5}+\frac {3 d \left (a^5 c d^4-a^4 b \left (3 c^2 d^3-2 d^5\right )-2 a^3 b^2 c d^4-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right )^2 \left (c^2-d^2\right )^2 (b c-a d)^4 (c+d \sin (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]

[Out]

-((b^3*(10*a^3*b*c*d - 4*a*b^3*c*d - 20*a^4*d^2 - a^2*b^2*(2*c^2 - 29*d^2) - b^4*(c^2 + 12*d^2))*ArcTan[(b + a
*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^5*f)) - (d^3*(a^2*d^2*(2*c^2 + d^2) - a*b*
(10*c^3*d - 4*c*d^3) + b^2*(20*c^4 - 29*c^2*d^2 + 12*d^4))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/(
(b*c - a*d)^5*(c^2 - d^2)^(5/2)*f) - (d*(a^4*d^3 - b^4*d*(5*c^2 - 6*d^2) + 2*a^2*b^2*d*(4*c^2 - 5*d^2) - 3*a*b
^3*c*(c^2 - d^2))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^3*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Co
s[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2) + (b^2*(3*a*b*c - 7*a^
2*d + 4*b^2*d)*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^2*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) + (
3*d*(a^5*c*d^4 - 2*a^3*b^2*c*d^4 + a*b^4*c*(c^4 - 2*c^2*d^2 + 2*d^4) + b^5*d*(2*c^4 - 7*c^2*d^2 + 4*d^4) - a^2
*b^3*d*(3*c^4 - 12*c^2*d^2 + 7*d^4) - a^4*b*(3*c^2*d^3 - 2*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)^2*(b*c - a*d)^4*
(c^2 - d^2)^2*f*(c + d*Sin[e + f*x]))

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[2*(e/d), Subst[Int[1/(a + 2*b*e*x + a*e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] &&
NeQ[a^2 - b^2, 0]

Rule 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2
- b^2))), x] + Dist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])
^n*Simp[a*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m +
n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||
!(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(B*c - A
*d)/(b*c - a*d), Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, A, B}, x] && NeQ[b*c - a*d, 0]
&& NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)), Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin[e + f*x])^n*Simp[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&
LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n]
&&  !IntegerQ[m]) || EqQ[a, 0])))

Rubi steps

\begin {align*} \int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^3} \, dx &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}-\frac {\int \frac {-2 \left (a b c-a^2 d+2 b^2 d\right )+b (b c-2 a d) \sin (e+f x)+3 b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)}\\ &=\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {\int \frac {-4 a^3 b c d+7 a b^3 c d+2 a^4 d^2+2 a^2 b^2 \left (c^2-10 d^2\right )+b^4 \left (c^2+12 d^2\right )+b d \left (3 b^3 c-4 a^3 d+a b^2 d\right ) \sin (e+f x)-2 b^2 d \left (3 a b c-7 a^2 d+4 b^2 d\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^2}\\ &=-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {\int \frac {-2 \left (2 a^5 c d^3+2 a^3 b^2 c d \left (3 c^2-5 d^2\right )-2 a b^4 c d \left (3 c^2-4 d^2\right )-6 a^4 b d^2 \left (c^2-d^2\right )-b^5 \left (c^4+11 c^2 d^2-12 d^4\right )-a^2 b^3 \left (2 c^4-23 c^2 d^2+21 d^4\right )\right )-2 d \left (2 a^4 b c d^2-a^5 d^3-2 b^5 c \left (c^2-2 d^2\right )+2 a^3 b^2 d \left (3 c^2-2 d^2\right )-a b^4 d \left (3 c^2-2 d^2\right )-a^2 b^3 c \left (c^2+3 d^2\right )\right ) \sin (e+f x)+2 b d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx}{4 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\int \frac {2 \left (b^6 \left (c^2-d^2\right )^2 \left (c^2+12 d^2\right )+2 a^4 b^2 d^2 \left (6 c^4-14 c^2 d^2+5 d^4\right )-2 a^3 b^3 c d \left (4 c^4-16 c^2 d^2+9 d^4\right )+a b^5 c d \left (5 c^4-18 c^2 d^2+10 d^4\right )-a^5 b \left (8 c^3 d^3-5 c d^5\right )+a^2 b^4 \left (2 c^6-28 c^4 d^2+52 c^2 d^4-23 d^6\right )+a^6 \left (2 c^2 d^4+d^6\right )\right )+2 b d (b c+a d) \left (2 a^2 b^2 c^4+b^4 c^4-10 a^3 b c^3 d+4 a b^3 c^3 d+2 a^4 c^2 d^2+8 a^2 b^2 c^2 d^2-10 b^4 c^2 d^2+4 a^3 b c d^3+2 a b^3 c d^3+a^4 d^4-10 a^2 b^2 d^4+6 b^4 d^4\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx}{4 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2}\\ &=-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{2 \left (a^2-b^2\right )^2 (b c-a d)^5}-\frac {\left (d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (b c-a d)^5 \left (c^2-d^2\right )^2}\\ &=-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a+2 b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right )^2 (b c-a d)^5 f}-\frac {\left (d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right )\right ) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{(b c-a d)^5 \left (c^2-d^2\right )^2 f}\\ &=-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\left (2 b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2-b^2\right )-x^2} \, dx,x,2 b+2 a \tan \left (\frac {1}{2} (e+f x)\right )\right )}{\left (a^2-b^2\right )^2 (b c-a d)^5 f}+\frac {\left (2 d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{(b c-a d)^5 \left (c^2-d^2\right )^2 f}\\ &=-\frac {b^3 \left (10 a^3 b c d-4 a b^3 c d-20 a^4 d^2-a^2 b^2 \left (2 c^2-29 d^2\right )-b^4 \left (c^2+12 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^5 f}-\frac {d^3 \left (a^2 d^2 \left (2 c^2+d^2\right )-a b \left (10 c^3 d-4 c d^3\right )+b^2 \left (20 c^4-29 c^2 d^2+12 d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^5 \left (c^2-d^2\right )^{5/2} f}-\frac {d \left (a^4 d^3-b^4 d \left (5 c^2-6 d^2\right )+2 a^2 b^2 d \left (4 c^2-5 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2}+\frac {b^2 \left (3 a b c-7 a^2 d+4 b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}+\frac {3 d \left (a^5 c d^4-2 a^3 b^2 c d^4+a b^4 c \left (c^4-2 c^2 d^2+2 d^4\right )+b^5 d \left (2 c^4-7 c^2 d^2+4 d^4\right )-a^2 b^3 d \left (3 c^4-12 c^2 d^2+7 d^4\right )-a^4 b \left (3 c^2 d^3-2 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^4 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1815\) vs. \(2(669)=1338\).
time = 8.65, size = 1815, normalized size = 2.71 \begin {gather*} -\frac {b^3 \left (2 a^2 b^2 c^2+b^4 c^2-10 a^3 b c d+4 a b^3 c d+20 a^4 d^2-29 a^2 b^2 d^2+12 b^4 d^2\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (b \cos \left (\frac {1}{2} (e+f x)\right )+a \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (-b c+a d)^5 f}-\frac {d^3 \left (20 b^2 c^4-10 a b c^3 d+2 a^2 c^2 d^2-29 b^2 c^2 d^2+4 a b c d^3+a^2 d^4+12 b^2 d^4\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (d \cos \left (\frac {1}{2} (e+f x)\right )+c \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^5 \left (c^2-d^2\right )^{5/2} f}+\frac {32 a^2 b^5 c^7 \cos (e+f x)-8 b^7 c^7 \cos (e+f x)-80 a^3 b^4 c^6 d \cos (e+f x)+68 a b^6 c^6 d \cos (e+f x)-92 a^2 b^5 c^5 d^2 \cos (e+f x)+38 b^7 c^5 d^2 \cos (e+f x)+140 a^3 b^4 c^4 d^3 \cos (e+f x)-122 a b^6 c^4 d^3 \cos (e+f x)-80 a^6 b c^3 d^4 \cos (e+f x)+140 a^4 b^3 c^3 d^4 \cos (e+f x)+48 a^2 b^5 c^3 d^4 \cos (e+f x)-72 b^7 c^3 d^4 \cos (e+f x)+32 a^7 c^2 d^5 \cos (e+f x)-92 a^5 b^2 c^2 d^5 \cos (e+f x)+48 a^3 b^4 c^2 d^5 \cos (e+f x)+12 a b^6 c^2 d^5 \cos (e+f x)+68 a^6 b c d^6 \cos (e+f x)-122 a^4 b^3 c d^6 \cos (e+f x)+12 a^2 b^5 c d^6 \cos (e+f x)+36 b^7 c d^6 \cos (e+f x)-8 a^7 d^7 \cos (e+f x)+38 a^5 b^2 d^7 \cos (e+f x)-72 a^3 b^4 d^7 \cos (e+f x)+36 a b^6 d^7 \cos (e+f x)-12 a b^6 c^6 d \cos (3 (e+f x))+28 a^2 b^5 c^5 d^2 \cos (3 (e+f x))-22 b^7 c^5 d^2 \cos (3 (e+f x))+20 a^3 b^4 c^4 d^3 \cos (3 (e+f x))+10 a b^6 c^4 d^3 \cos (3 (e+f x))+20 a^4 b^3 c^3 d^4 \cos (3 (e+f x))-96 a^2 b^5 c^3 d^4 \cos (3 (e+f x))+64 b^7 c^3 d^4 \cos (3 (e+f x))+28 a^5 b^2 c^2 d^5 \cos (3 (e+f x))-96 a^3 b^4 c^2 d^5 \cos (3 (e+f x))+44 a b^6 c^2 d^5 \cos (3 (e+f x))-12 a^6 b c d^6 \cos (3 (e+f x))+10 a^4 b^3 c d^6 \cos (3 (e+f x))+44 a^2 b^5 c d^6 \cos (3 (e+f x))-36 b^7 c d^6 \cos (3 (e+f x))-22 a^5 b^2 d^7 \cos (3 (e+f x))+64 a^3 b^4 d^7 \cos (3 (e+f x))-36 a b^6 d^7 \cos (3 (e+f x))+12 a b^6 c^7 \sin (2 (e+f x))-4 a^2 b^5 c^6 d \sin (2 (e+f x))+16 b^7 c^6 d \sin (2 (e+f x))-80 a^3 b^4 c^5 d^2 \sin (2 (e+f x))+38 a b^6 c^5 d^2 \sin (2 (e+f x))-10 a^2 b^5 c^4 d^3 \sin (2 (e+f x))-20 b^7 c^4 d^3 \sin (2 (e+f x))-80 a^5 b^2 c^3 d^4 \sin (2 (e+f x))+320 a^3 b^4 c^3 d^4 \sin (2 (e+f x))-192 a b^6 c^3 d^4 \sin (2 (e+f x))-4 a^6 b c^2 d^5 \sin (2 (e+f x))-10 a^4 b^3 c^2 d^5 \sin (2 (e+f x))+64 a^2 b^5 c^2 d^5 \sin (2 (e+f x))-26 b^7 c^2 d^5 \sin (2 (e+f x))+12 a^7 c d^6 \sin (2 (e+f x))+38 a^5 b^2 c d^6 \sin (2 (e+f x))-192 a^3 b^4 c d^6 \sin (2 (e+f x))+124 a b^6 c d^6 \sin (2 (e+f x))+16 a^6 b d^7 \sin (2 (e+f x))-20 a^4 b^3 d^7 \sin (2 (e+f x))-26 a^2 b^5 d^7 \sin (2 (e+f x))+24 b^7 d^7 \sin (2 (e+f x))-3 a b^6 c^5 d^2 \sin (4 (e+f x))+9 a^2 b^5 c^4 d^3 \sin (4 (e+f x))-6 b^7 c^4 d^3 \sin (4 (e+f x))+6 a b^6 c^3 d^4 \sin (4 (e+f x))+9 a^4 b^3 c^2 d^5 \sin (4 (e+f x))-36 a^2 b^5 c^2 d^5 \sin (4 (e+f x))+21 b^7 c^2 d^5 \sin (4 (e+f x))-3 a^5 b^2 c d^6 \sin (4 (e+f x))+6 a^3 b^4 c d^6 \sin (4 (e+f x))-6 a b^6 c d^6 \sin (4 (e+f x))-6 a^4 b^3 d^7 \sin (4 (e+f x))+21 a^2 b^5 d^7 \sin (4 (e+f x))-12 b^7 d^7 \sin (4 (e+f x))}{16 \left (a^2-b^2\right )^2 (-b c+a d)^4 \left (c^2-d^2\right )^2 f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sin[e + f*x])^3*(c + d*Sin[e + f*x])^3),x]

[Out]

-((b^3*(2*a^2*b^2*c^2 + b^4*c^2 - 10*a^3*b*c*d + 4*a*b^3*c*d + 20*a^4*d^2 - 29*a^2*b^2*d^2 + 12*b^4*d^2)*ArcTa
n[(Sec[(e + f*x)/2]*(b*Cos[(e + f*x)/2] + a*Sin[(e + f*x)/2]))/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(-(b*c) +
a*d)^5*f)) - (d^3*(20*b^2*c^4 - 10*a*b*c^3*d + 2*a^2*c^2*d^2 - 29*b^2*c^2*d^2 + 4*a*b*c*d^3 + a^2*d^4 + 12*b^2
*d^4)*ArcTan[(Sec[(e + f*x)/2]*(d*Cos[(e + f*x)/2] + c*Sin[(e + f*x)/2]))/Sqrt[c^2 - d^2]])/((b*c - a*d)^5*(c^
2 - d^2)^(5/2)*f) + (32*a^2*b^5*c^7*Cos[e + f*x] - 8*b^7*c^7*Cos[e + f*x] - 80*a^3*b^4*c^6*d*Cos[e + f*x] + 68
*a*b^6*c^6*d*Cos[e + f*x] - 92*a^2*b^5*c^5*d^2*Cos[e + f*x] + 38*b^7*c^5*d^2*Cos[e + f*x] + 140*a^3*b^4*c^4*d^
3*Cos[e + f*x] - 122*a*b^6*c^4*d^3*Cos[e + f*x] - 80*a^6*b*c^3*d^4*Cos[e + f*x] + 140*a^4*b^3*c^3*d^4*Cos[e +
f*x] + 48*a^2*b^5*c^3*d^4*Cos[e + f*x] - 72*b^7*c^3*d^4*Cos[e + f*x] + 32*a^7*c^2*d^5*Cos[e + f*x] - 92*a^5*b^
2*c^2*d^5*Cos[e + f*x] + 48*a^3*b^4*c^2*d^5*Cos[e + f*x] + 12*a*b^6*c^2*d^5*Cos[e + f*x] + 68*a^6*b*c*d^6*Cos[
e + f*x] - 122*a^4*b^3*c*d^6*Cos[e + f*x] + 12*a^2*b^5*c*d^6*Cos[e + f*x] + 36*b^7*c*d^6*Cos[e + f*x] - 8*a^7*
d^7*Cos[e + f*x] + 38*a^5*b^2*d^7*Cos[e + f*x] - 72*a^3*b^4*d^7*Cos[e + f*x] + 36*a*b^6*d^7*Cos[e + f*x] - 12*
a*b^6*c^6*d*Cos[3*(e + f*x)] + 28*a^2*b^5*c^5*d^2*Cos[3*(e + f*x)] - 22*b^7*c^5*d^2*Cos[3*(e + f*x)] + 20*a^3*
b^4*c^4*d^3*Cos[3*(e + f*x)] + 10*a*b^6*c^4*d^3*Cos[3*(e + f*x)] + 20*a^4*b^3*c^3*d^4*Cos[3*(e + f*x)] - 96*a^
2*b^5*c^3*d^4*Cos[3*(e + f*x)] + 64*b^7*c^3*d^4*Cos[3*(e + f*x)] + 28*a^5*b^2*c^2*d^5*Cos[3*(e + f*x)] - 96*a^
3*b^4*c^2*d^5*Cos[3*(e + f*x)] + 44*a*b^6*c^2*d^5*Cos[3*(e + f*x)] - 12*a^6*b*c*d^6*Cos[3*(e + f*x)] + 10*a^4*
b^3*c*d^6*Cos[3*(e + f*x)] + 44*a^2*b^5*c*d^6*Cos[3*(e + f*x)] - 36*b^7*c*d^6*Cos[3*(e + f*x)] - 22*a^5*b^2*d^
7*Cos[3*(e + f*x)] + 64*a^3*b^4*d^7*Cos[3*(e + f*x)] - 36*a*b^6*d^7*Cos[3*(e + f*x)] + 12*a*b^6*c^7*Sin[2*(e +
f*x)] - 4*a^2*b^5*c^6*d*Sin[2*(e + f*x)] + 16*b^7*c^6*d*Sin[2*(e + f*x)] - 80*a^3*b^4*c^5*d^2*Sin[2*(e + f*x)
] + 38*a*b^6*c^5*d^2*Sin[2*(e + f*x)] - 10*a^2*b^5*c^4*d^3*Sin[2*(e + f*x)] - 20*b^7*c^4*d^3*Sin[2*(e + f*x)]
- 80*a^5*b^2*c^3*d^4*Sin[2*(e + f*x)] + 320*a^3*b^4*c^3*d^4*Sin[2*(e + f*x)] - 192*a*b^6*c^3*d^4*Sin[2*(e + f*
x)] - 4*a^6*b*c^2*d^5*Sin[2*(e + f*x)] - 10*a^4*b^3*c^2*d^5*Sin[2*(e + f*x)] + 64*a^2*b^5*c^2*d^5*Sin[2*(e + f
*x)] - 26*b^7*c^2*d^5*Sin[2*(e + f*x)] + 12*a^7*c*d^6*Sin[2*(e + f*x)] + 38*a^5*b^2*c*d^6*Sin[2*(e + f*x)] - 1
92*a^3*b^4*c*d^6*Sin[2*(e + f*x)] + 124*a*b^6*c*d^6*Sin[2*(e + f*x)] + 16*a^6*b*d^7*Sin[2*(e + f*x)] - 20*a^4*
b^3*d^7*Sin[2*(e + f*x)] - 26*a^2*b^5*d^7*Sin[2*(e + f*x)] + 24*b^7*d^7*Sin[2*(e + f*x)] - 3*a*b^6*c^5*d^2*Sin
[4*(e + f*x)] + 9*a^2*b^5*c^4*d^3*Sin[4*(e + f*x)] - 6*b^7*c^4*d^3*Sin[4*(e + f*x)] + 6*a*b^6*c^3*d^4*Sin[4*(e
+ f*x)] + 9*a^4*b^3*c^2*d^5*Sin[4*(e + f*x)] - 36*a^2*b^5*c^2*d^5*Sin[4*(e + f*x)] + 21*b^7*c^2*d^5*Sin[4*(e
+ f*x)] - 3*a^5*b^2*c*d^6*Sin[4*(e + f*x)] + 6*a^3*b^4*c*d^6*Sin[4*(e + f*x)] - 6*a*b^6*c*d^6*Sin[4*(e + f*x)]
- 6*a^4*b^3*d^7*Sin[4*(e + f*x)] + 21*a^2*b^5*d^7*Sin[4*(e + f*x)] - 12*b^7*d^7*Sin[4*(e + f*x)])/(16*(a^2 -
b^2)^2*(-(b*c) + a*d)^4*(c^2 - d^2)^2*f*(a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^2)

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Maple [A]
time = 20.34, size = 1157, normalized size = 1.73 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(-2*b^3/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)^3*((1/2*b^2*(11*a^4*d^2-16*a^3*b*c*d+5*a^2*b^2*c^2-8*a^2*b^2
*d^2+10*a*b^3*c*d-2*b^4*c^2)/(a^4-2*a^2*b^2+b^4)/a*tan(1/2*f*x+1/2*e)^3+1/2*b*(10*a^6*d^2-14*a^5*b*c*d+4*a^4*b
^2*c^2+13*a^4*b^2*d^2-20*a^3*b^3*c*d+7*a^2*b^4*c^2-14*a^2*b^4*d^2+16*a*b^5*c*d-2*b^6*c^2)/(a^4-2*a^2*b^2+b^4)/
a^2*tan(1/2*f*x+1/2*e)^2+1/2*b^2*(29*a^4*d^2-40*a^3*b*c*d+11*a^2*b^2*c^2-20*a^2*b^2*d^2+22*a*b^3*c*d-2*b^4*c^2
)/a/(a^4-2*a^2*b^2+b^4)*tan(1/2*f*x+1/2*e)+1/2*b*(10*a^4*d^2-14*a^3*b*c*d+4*a^2*b^2*c^2-7*a^2*b^2*d^2+8*a*b^3*
c*d-b^4*c^2)/(a^4-2*a^2*b^2+b^4))/(a*tan(1/2*f*x+1/2*e)^2+2*b*tan(1/2*f*x+1/2*e)+a)^2+1/2*(20*a^4*d^2-10*a^3*b
*c*d+2*a^2*b^2*c^2-29*a^2*b^2*d^2+4*a*b^3*c*d+b^4*c^2+12*b^4*d^2)/(a^4-2*a^2*b^2+b^4)/(a^2-b^2)^(1/2)*arctan(1
/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2)))+2*d^3/(a^3*d^3-3*a^2*b*c*d^2+3*a*b^2*c^2*d-b^3*c^3)/(a*d-b*c
)^2*((1/2*d^2*(5*a^2*c^2*d^2-2*a^2*d^4-16*a*b*c^3*d+10*a*b*c*d^3+11*b^2*c^4-8*b^2*c^2*d^2)/(c^4-2*c^2*d^2+d^4)
/c*tan(1/2*f*x+1/2*e)^3+1/2*d*(4*a^2*c^4*d^2+7*a^2*c^2*d^4-2*a^2*d^6-14*a*b*c^5*d-20*a*b*c^3*d^3+16*a*b*c*d^5+
10*b^2*c^6+13*b^2*c^4*d^2-14*b^2*c^2*d^4)/(c^4-2*c^2*d^2+d^4)/c^2*tan(1/2*f*x+1/2*e)^2+1/2*d^2*(11*a^2*c^2*d^2
-2*a^2*d^4-40*a*b*c^3*d+22*a*b*c*d^3+29*b^2*c^4-20*b^2*c^2*d^2)/(c^4-2*c^2*d^2+d^4)/c*tan(1/2*f*x+1/2*e)+1/2*d
*(4*a^2*c^2*d^2-a^2*d^4-14*a*b*c^3*d+8*a*b*c*d^3+10*b^2*c^4-7*b^2*c^2*d^2)/(c^4-2*c^2*d^2+d^4))/(c*tan(1/2*f*x
+1/2*e)^2+2*d*tan(1/2*f*x+1/2*e)+c)^2+1/2*(2*a^2*c^2*d^2+a^2*d^4-10*a*b*c^3*d+4*a*b*c*d^3+20*b^2*c^4-29*b^2*c^
2*d^2+12*b^2*d^4)/(c^4-2*c^2*d^2+d^4)/(c^2-d^2)^(1/2)*arctan(1/2*(2*c*tan(1/2*f*x+1/2*e)+2*d)/(c^2-d^2)^(1/2))
))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*d^2-4*c^2>0)', see `assume?`
for more de

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))**3/(c+d*sin(f*x+e))**3,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 7128 vs. \(2 (663) = 1326\).
time = 4.51, size = 7128, normalized size = 10.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^3/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

((2*a^2*b^5*c^2 + b^7*c^2 - 10*a^3*b^4*c*d + 4*a*b^6*c*d + 20*a^4*b^3*d^2 - 29*a^2*b^5*d^2 + 12*b^7*d^2)*(pi*f
loor(1/2*(f*x + e)/pi + 1/2)*sgn(a) + arctan((a*tan(1/2*f*x + 1/2*e) + b)/sqrt(a^2 - b^2)))/((a^4*b^5*c^5 - 2*
a^2*b^7*c^5 + b^9*c^5 - 5*a^5*b^4*c^4*d + 10*a^3*b^6*c^4*d - 5*a*b^8*c^4*d + 10*a^6*b^3*c^3*d^2 - 20*a^4*b^5*c
^3*d^2 + 10*a^2*b^7*c^3*d^2 - 10*a^7*b^2*c^2*d^3 + 20*a^5*b^4*c^2*d^3 - 10*a^3*b^6*c^2*d^3 + 5*a^8*b*c*d^4 - 1
0*a^6*b^3*c*d^4 + 5*a^4*b^5*c*d^4 - a^9*d^5 + 2*a^7*b^2*d^5 - a^5*b^4*d^5)*sqrt(a^2 - b^2)) - (20*b^2*c^4*d^3
- 10*a*b*c^3*d^4 + 2*a^2*c^2*d^5 - 29*b^2*c^2*d^5 + 4*a*b*c*d^6 + a^2*d^7 + 12*b^2*d^7)*(pi*floor(1/2*(f*x + e
)/pi + 1/2)*sgn(c) + arctan((c*tan(1/2*f*x + 1/2*e) + d)/sqrt(c^2 - d^2)))/((b^5*c^9 - 5*a*b^4*c^8*d + 10*a^2*
b^3*c^7*d^2 - 2*b^5*c^7*d^2 - 10*a^3*b^2*c^6*d^3 + 10*a*b^4*c^6*d^3 + 5*a^4*b*c^5*d^4 - 20*a^2*b^3*c^5*d^4 + b
^5*c^5*d^4 - a^5*c^4*d^5 + 20*a^3*b^2*c^4*d^5 - 5*a*b^4*c^4*d^5 - 10*a^4*b*c^3*d^6 + 10*a^2*b^3*c^3*d^6 + 2*a^
5*c^2*d^7 - 10*a^3*b^2*c^2*d^7 + 5*a^4*b*c*d^8 - a^5*d^9)*sqrt(c^2 - d^2)) + (5*a^3*b^6*c^9*tan(1/2*f*x + 1/2*
e)^7 - 2*a*b^8*c^9*tan(1/2*f*x + 1/2*e)^7 - 11*a^4*b^5*c^8*d*tan(1/2*f*x + 1/2*e)^7 + 8*a^2*b^7*c^8*d*tan(1/2*
f*x + 1/2*e)^7 - 10*a^3*b^6*c^7*d^2*tan(1/2*f*x + 1/2*e)^7 + 4*a*b^8*c^7*d^2*tan(1/2*f*x + 1/2*e)^7 + 22*a^4*b
^5*c^6*d^3*tan(1/2*f*x + 1/2*e)^7 - 16*a^2*b^7*c^6*d^3*tan(1/2*f*x + 1/2*e)^7 + 5*a^3*b^6*c^5*d^4*tan(1/2*f*x
+ 1/2*e)^7 - 2*a*b^8*c^5*d^4*tan(1/2*f*x + 1/2*e)^7 - 11*a^8*b*c^4*d^5*tan(1/2*f*x + 1/2*e)^7 + 22*a^6*b^3*c^4
*d^5*tan(1/2*f*x + 1/2*e)^7 - 22*a^4*b^5*c^4*d^5*tan(1/2*f*x + 1/2*e)^7 + 8*a^2*b^7*c^4*d^5*tan(1/2*f*x + 1/2*
e)^7 + 5*a^9*c^3*d^6*tan(1/2*f*x + 1/2*e)^7 - 10*a^7*b^2*c^3*d^6*tan(1/2*f*x + 1/2*e)^7 + 5*a^5*b^4*c^3*d^6*ta
n(1/2*f*x + 1/2*e)^7 + 8*a^8*b*c^2*d^7*tan(1/2*f*x + 1/2*e)^7 - 16*a^6*b^3*c^2*d^7*tan(1/2*f*x + 1/2*e)^7 + 8*
a^4*b^5*c^2*d^7*tan(1/2*f*x + 1/2*e)^7 - 2*a^9*c*d^8*tan(1/2*f*x + 1/2*e)^7 + 4*a^7*b^2*c*d^8*tan(1/2*f*x + 1/
2*e)^7 - 2*a^5*b^4*c*d^8*tan(1/2*f*x + 1/2*e)^7 + 4*a^4*b^5*c^9*tan(1/2*f*x + 1/2*e)^6 + 7*a^2*b^7*c^9*tan(1/2
*f*x + 1/2*e)^6 - 2*b^9*c^9*tan(1/2*f*x + 1/2*e)^6 - 10*a^5*b^4*c^8*d*tan(1/2*f*x + 1/2*e)^6 + 7*a^3*b^6*c^8*d
*tan(1/2*f*x + 1/2*e)^6 + 6*a*b^8*c^8*d*tan(1/2*f*x + 1/2*e)^6 - 52*a^4*b^5*c^7*d^2*tan(1/2*f*x + 1/2*e)^6 + 1
8*a^2*b^7*c^7*d^2*tan(1/2*f*x + 1/2*e)^6 + 4*b^9*c^7*d^2*tan(1/2*f*x + 1/2*e)^6 + 20*a^5*b^4*c^6*d^3*tan(1/2*f
*x + 1/2*e)^6 - 14*a^3*b^6*c^6*d^3*tan(1/2*f*x + 1/2*e)^6 - 12*a*b^8*c^6*d^3*tan(1/2*f*x + 1/2*e)^6 - 10*a^8*b
*c^5*d^4*tan(1/2*f*x + 1/2*e)^6 + 20*a^6*b^3*c^5*d^4*tan(1/2*f*x + 1/2*e)^6 + 82*a^4*b^5*c^5*d^4*tan(1/2*f*x +
1/2*e)^6 - 57*a^2*b^7*c^5*d^4*tan(1/2*f*x + 1/2*e)^6 - 2*b^9*c^5*d^4*tan(1/2*f*x + 1/2*e)^6 + 4*a^9*c^4*d^5*t
an(1/2*f*x + 1/2*e)^6 - 52*a^7*b^2*c^4*d^5*tan(1/2*f*x + 1/2*e)^6 + 82*a^5*b^4*c^4*d^5*tan(1/2*f*x + 1/2*e)^6
- 37*a^3*b^6*c^4*d^5*tan(1/2*f*x + 1/2*e)^6 + 6*a*b^8*c^4*d^5*tan(1/2*f*x + 1/2*e)^6 + 7*a^8*b*c^3*d^6*tan(1/2
*f*x + 1/2*e)^6 - 14*a^6*b^3*c^3*d^6*tan(1/2*f*x + 1/2*e)^6 - 37*a^4*b^5*c^3*d^6*tan(1/2*f*x + 1/2*e)^6 + 32*a
^2*b^7*c^3*d^6*tan(1/2*f*x + 1/2*e)^6 + 7*a^9*c^2*d^7*tan(1/2*f*x + 1/2*e)^6 + 18*a^7*b^2*c^2*d^7*tan(1/2*f*x
+ 1/2*e)^6 - 57*a^5*b^4*c^2*d^7*tan(1/2*f*x + 1/2*e)^6 + 32*a^3*b^6*c^2*d^7*tan(1/2*f*x + 1/2*e)^6 + 6*a^8*b*c
*d^8*tan(1/2*f*x + 1/2*e)^6 - 12*a^6*b^3*c*d^8*tan(1/2*f*x + 1/2*e)^6 + 6*a^4*b^5*c*d^8*tan(1/2*f*x + 1/2*e)^6
- 2*a^9*d^9*tan(1/2*f*x + 1/2*e)^6 + 4*a^7*b^2*d^9*tan(1/2*f*x + 1/2*e)^6 - 2*a^5*b^4*d^9*tan(1/2*f*x + 1/2*e
)^6 + 21*a^3*b^6*c^9*tan(1/2*f*x + 1/2*e)^5 - 6*a*b^8*c^9*tan(1/2*f*x + 1/2*e)^5 - 35*a^4*b^5*c^8*d*tan(1/2*f*
x + 1/2*e)^5 + 64*a^2*b^7*c^8*d*tan(1/2*f*x + 1/2*e)^5 - 8*b^9*c^8*d*tan(1/2*f*x + 1/2*e)^5 - 40*a^5*b^4*c^7*d
^2*tan(1/2*f*x + 1/2*e)^5 - 74*a^3*b^6*c^7*d^2*tan(1/2*f*x + 1/2*e)^5 + 60*a*b^8*c^7*d^2*tan(1/2*f*x + 1/2*e)^
5 + 26*a^4*b^5*c^6*d^3*tan(1/2*f*x + 1/2*e)^5 - 96*a^2*b^7*c^6*d^3*tan(1/2*f*x + 1/2*e)^5 + 16*b^9*c^6*d^3*tan
(1/2*f*x + 1/2*e)^5 - 40*a^7*b^2*c^5*d^4*tan(1/2*f*x + 1/2*e)^5 + 160*a^5*b^4*c^5*d^4*tan(1/2*f*x + 1/2*e)^5 +
45*a^3*b^6*c^5*d^4*tan(1/2*f*x + 1/2*e)^5 - 102*a*b^8*c^5*d^4*tan(1/2*f*x + 1/2*e)^5 - 35*a^8*b*c^4*d^5*tan(1
/2*f*x + 1/2*e)^5 + 26*a^6*b^3*c^4*d^5*tan(1/2*f*x + 1/2*e)^5 + 106*a^4*b^5*c^4*d^5*tan(1/2*f*x + 1/2*e)^5 - 4
4*a^2*b^7*c^4*d^5*tan(1/2*f*x + 1/2*e)^5 - 8*b^9*c^4*d^5*tan(1/2*f*x + 1/2*e)^5 + 21*a^9*c^3*d^6*tan(1/2*f*x +
1/2*e)^5 - 74*a^7*b^2*c^3*d^6*tan(1/2*f*x + 1/2*e)^5 + 45*a^5*b^4*c^3*d^6*tan(1/2*f*x + 1/2*e)^5 - 64*a^3*b^6
*c^3*d^6*tan(1/2*f*x + 1/2*e)^5 + 48*a*b^8*c^3*d^6*tan(1/2*f*x + 1/2*e)^5 + 64*a^8*b*c^2*d^7*tan(1/2*f*x + 1/2
*e)^5 - 96*a^6*b^3*c^2*d^7*tan(1/2*f*x + 1/2*e)^5 - 44*a^4*b^5*c^2*d^7*tan(1/2*f*x + 1/2*e)^5 + 64*a^2*b^7*c^2
*d^7*tan(1/2*f*x + 1/2*e)^5 - 6*a^9*c*d^8*tan(1/2*f*x + 1/2*e)^5 + 60*a^7*b^2*c*d^8*tan(1/2*f*x + 1/2*e)^5 - 1
02*a^5*b^4*c*d^8*tan(1/2*f*x + 1/2*e)^5 + 48*a^3*b^6*c*d^8*tan(1/2*f*x + 1/2*e)^5 - 8*a^8*b*d^9*tan(1/2*f*x +
1/2*e)^5 + 16*a^6*b^3*d^9*tan(1/2*f*x + 1/2*e)^...

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Mupad [B]
time = 80.30, size = 2500, normalized size = 3.74 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*sin(e + f*x))^3*(c + d*sin(e + f*x))^3),x)

[Out]

(atan((((((4*a^24*d^24 + 4*b^24*c^24 + 16*a^2*b^22*c^24 + 16*a^4*b^20*c^24 - 1152*a^10*b^14*d^24 + 5568*a^12*b
^12*d^24 - 10568*a^14*b^10*d^24 + 9460*a^16*b^8*d^24 - 3560*a^18*b^6*d^24 + 136*a^20*b^4*d^24 + 76*a^22*b^2*d^
24 + 16*a^24*c^2*d^22 + 16*a^24*c^4*d^20 - 1152*b^24*c^10*d^14 + 5568*b^24*c^12*d^12 - 10568*b^24*c^14*d^10 +
9460*b^24*c^16*d^8 - 3560*b^24*c^18*d^6 + 136*b^24*c^20*d^4 + 76*b^24*c^22*d^2 + 11520*a*b^23*c^9*d^15 - 56448
*a*b^23*c^11*d^13 + 109456*a*b^23*c^13*d^11 - 101240*a*b^23*c^15*d^9 + 40720*a*b^23*c^17*d^7 - 2960*a*b^23*c^1
9*d^5 - 536*a*b^23*c^21*d^3 - 176*a^3*b^21*c^23*d - 320*a^5*b^19*c^23*d + 11520*a^9*b^15*c*d^23 - 56448*a^11*b
^13*c*d^23 + 109456*a^13*b^11*c*d^23 - 101240*a^15*b^9*c*d^23 + 40720*a^17*b^7*c*d^23 - 2960*a^19*b^5*c*d^23 -
536*a^21*b^3*c*d^23 - 176*a^23*b*c^3*d^21 - 320*a^23*b*c^5*d^19 - 51840*a^2*b^22*c^8*d^16 + 263808*a^2*b^22*c
^10*d^14 - 541208*a^2*b^22*c^12*d^12 + 547088*a^2*b^22*c^14*d^10 - 263320*a^2*b^22*c^16*d^8 + 44120*a^2*b^22*c
^18*d^6 - 1564*a^2*b^22*c^20*d^4 - 196*a^2*b^22*c^22*d^2 + 138240*a^3*b^21*c^7*d^17 - 758400*a^3*b^21*c^9*d^15
+ 1720736*a^3*b^21*c^11*d^13 - 2002728*a^3*b^21*c^13*d^11 + 1210560*a^3*b^21*c^15*d^9 - 335040*a^3*b^21*c^17*
d^7 + 37680*a^3*b^21*c^19*d^5 - 288*a^3*b^21*c^21*d^3 - 241920*a^4*b^20*c^6*d^18 + 1512000*a^4*b^20*c^8*d^16 -
3975688*a^4*b^20*c^10*d^14 + 5501328*a^4*b^20*c^12*d^12 - 4147952*a^4*b^20*c^14*d^10 + 1586920*a^4*b^20*c^16*
d^8 - 276020*a^4*b^20*c^18*d^6 + 21124*a^4*b^20*c^20*d^4 + 176*a^4*b^20*c^22*d^2 + 290304*a^5*b^19*c^5*d^19 -
2232576*a^5*b^19*c^7*d^17 + 7078256*a^5*b^19*c^9*d^15 - 11781560*a^5*b^19*c^11*d^13 + 10875200*a^5*b^19*c^13*d
^11 - 5365072*a^5*b^19*c^15*d^9 + 1310168*a^5*b^19*c^17*d^7 - 170968*a^5*b^19*c^19*d^5 + 8160*a^5*b^19*c^21*d^
3 - 241920*a^6*b^18*c^4*d^20 + 2532096*a^6*b^18*c^6*d^18 - 9955992*a^6*b^18*c^8*d^16 + 20019440*a^6*b^18*c^10*
d^14 - 22419600*a^6*b^18*c^12*d^12 + 13887520*a^6*b^18*c^14*d^10 - 4506428*a^6*b^18*c^16*d^8 + 793756*a^6*b^18
*c^18*d^6 - 72240*a^6*b^18*c^20*d^4 + 3040*a^6*b^18*c^22*d^2 + 138240*a^7*b^17*c^3*d^21 - 2232576*a^7*b^17*c^5
*d^19 + 11150016*a^7*b^17*c^7*d^17 - 27336616*a^7*b^17*c^9*d^15 + 37153600*a^7*b^17*c^11*d^13 - 28461040*a^7*b
^17*c^13*d^11 + 11779808*a^7*b^17*c^15*d^9 - 2621008*a^7*b^17*c^17*d^7 + 336688*a^7*b^17*c^19*d^5 - 17920*a^7*
b^17*c^21*d^3 - 51840*a^8*b^16*c^2*d^22 + 1512000*a^8*b^16*c^4*d^20 - 9955992*a^8*b^16*c^6*d^18 + 30289656*a^8
*b^16*c^8*d^16 - 50137600*a^8*b^16*c^10*d^14 + 46972560*a^8*b^16*c^12*d^12 - 24199280*a^8*b^16*c^14*d^10 + 666
1036*a^8*b^16*c^16*d^8 - 1058448*a^8*b^16*c^18*d^6 + 72560*a^8*b^16*c^20*d^4 - 758400*a^9*b^15*c^3*d^21 + 7078
256*a^9*b^15*c^5*d^19 - 27336616*a^9*b^15*c^7*d^17 + 55383904*a^9*b^15*c^9*d^15 - 63124080*a^9*b^15*c^11*d^13
+ 39987520*a^9*b^15*c^13*d^11 - 13462088*a^9*b^15*c^15*d^9 + 2478528*a^9*b^15*c^17*d^7 - 212032*a^9*b^15*c^19*
d^5 + 263808*a^10*b^14*c^2*d^22 - 3975688*a^10*b^14*c^4*d^20 + 20019440*a^10*b^14*c^6*d^18 - 50137600*a^10*b^1
4*c^8*d^16 + 69593872*a^10*b^14*c^10*d^14 - 53854288*a^10*b^14*c^12*d^12 + 21989928*a^10*b^14*c^14*d^10 - 4591
360*a^10*b^14*c^16*d^8 + 460480*a^10*b^14*c^18*d^6 + 1720736*a^11*b^13*c^3*d^21 - 11781560*a^11*b^13*c^5*d^19
+ 37153600*a^11*b^13*c^7*d^17 - 63124080*a^11*b^13*c^9*d^15 + 59445728*a^11*b^13*c^11*d^13 - 29358696*a^11*b^1
3*c^13*d^11 + 6995840*a^11*b^13*c^15*d^9 - 762560*a^11*b^13*c^17*d^7 - 541208*a^12*b^12*c^2*d^22 + 5501328*a^1
2*b^12*c^4*d^20 - 22419600*a^12*b^12*c^6*d^18 + 46972560*a^12*b^12*c^8*d^16 - 53854288*a^12*b^12*c^10*d^14 + 3
2294808*a^12*b^12*c^12*d^12 - 8958208*a^12*b^12*c^14*d^10 + 999040*a^12*b^12*c^16*d^8 - 2002728*a^13*b^11*c^3*
d^21 + 10875200*a^13*b^11*c^5*d^19 - 28461040*a^13*b^11*c^7*d^17 + 39987520*a^13*b^11*c^9*d^15 - 29358696*a^13
*b^11*c^11*d^13 + 9722048*a^13*b^11*c^13*d^11 - 1104320*a^13*b^11*c^15*d^9 + 547088*a^14*b^10*c^2*d^22 - 41479
52*a^14*b^10*c^4*d^20 + 13887520*a^14*b^10*c^6*d^18 - 24199280*a^14*b^10*c^8*d^16 + 21989928*a^14*b^10*c^10*d^
14 - 8958208*a^14*b^10*c^12*d^12 + 1124032*a^14*b^10*c^14*d^10 + 1210560*a^15*b^9*c^3*d^21 - 5365072*a^15*b^9*
c^5*d^19 + 11779808*a^15*b^9*c^7*d^17 - 13462088*a^15*b^9*c^9*d^15 + 6995840*a^15*b^9*c^11*d^13 - 1104320*a^15
*b^9*c^13*d^11 - 263320*a^16*b^8*c^2*d^22 + 1586920*a^16*b^8*c^4*d^20 - 4506428*a^16*b^8*c^6*d^18 + 6661036*a^
16*b^8*c^8*d^16 - 4591360*a^16*b^8*c^10*d^14 + 999040*a^16*b^8*c^12*d^12 - 335040*a^17*b^7*c^3*d^21 + 1310168*
a^17*b^7*c^5*d^19 - 2621008*a^17*b^7*c^7*d^17 + 2478528*a^17*b^7*c^9*d^15 - 762560*a^17*b^7*c^11*d^13 + 44120*
a^18*b^6*c^2*d^22 - 276020*a^18*b^6*c^4*d^20 + 793756*a^18*b^6*c^6*d^18 - 1058448*a^18*b^6*c^8*d^16 + 460480*a
^18*b^6*c^10*d^14 + 37680*a^19*b^5*c^3*d^21 - 170968*a^19*b^5*c^5*d^19 + 336688*a^19*b^5*c^7*d^17 - 212032*a^1
9*b^5*c^9*d^15 - 1564*a^20*b^4*c^2*d^22 + 21124*a^20*b^4*c^4*d^20 - 72240*a^20*b^4*c^6*d^18 + 72560*a^20*b^4*c
^8*d^16 - 288*a^21*b^3*c^3*d^21 + 8160*a^21*b^3*c^5*d^19 - 17920*a^21*b^3*c^7*d^17 - 196*a^22*b^2*c^2*d^22 + 1
76*a^22*b^2*c^4*d^20 + 3040*a^22*b^2*c^6*d^18 -...

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