3.8.13 \(\int \frac {1}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^3} \, dx\) [713]

Optimal. Leaf size=458 \[ \frac {2 b^3 \left (a b c-4 a^2 d+3 b^2 d\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{3/2} (b c-a d)^4 f}-\frac {d^2 \left (2 a b c d \left (4 c^2-d^2\right )-a^2 d^2 \left (2 c^2+d^2\right )-3 b^2 \left (4 c^4-5 c^2 d^2+2 d^4\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^4 \left (c^2-d^2\right )^{5/2} f}+\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-3 a b^2 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))} \]

[Out]

2*b^3*(-4*a^2*d+a*b*c+3*b^2*d)*arctan((b+a*tan(1/2*f*x+1/2*e))/(a^2-b^2)^(1/2))/(a^2-b^2)^(3/2)/(-a*d+b*c)^4/f
-d^2*(2*a*b*c*d*(4*c^2-d^2)-a^2*d^2*(2*c^2+d^2)-3*b^2*(4*c^4-5*c^2*d^2+2*d^4))*arctan((d+c*tan(1/2*f*x+1/2*e))
/(c^2-d^2)^(1/2))/(-a*d+b*c)^4/(c^2-d^2)^(5/2)/f+1/2*d*(a^2*d^2+b^2*(2*c^2-3*d^2))*cos(f*x+e)/(a^2-b^2)/(-a*d+
b*c)^2/(c^2-d^2)/f/(c+d*sin(f*x+e))^2+b^2*cos(f*x+e)/(a^2-b^2)/(-a*d+b*c)/f/(a+b*sin(f*x+e))/(c+d*sin(f*x+e))^
2-1/2*(3*a^3*c*d^4-3*a*b^2*c*d^4-a^2*b*d^3*(7*c^2-4*d^2)-b^3*(2*c^4*d-11*c^2*d^3+6*d^5))*cos(f*x+e)/(a^2-b^2)/
(-a*d+b*c)^3/(c^2-d^2)^2/f/(c+d*sin(f*x+e))

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Rubi [A]
time = 1.58, antiderivative size = 458, normalized size of antiderivative = 1.00, number of steps used = 10, 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^2 \left (-a^2 d^2 \left (2 c^2+d^2\right )+2 a b c d \left (4 c^2-d^2\right )-3 b^2 \left (4 c^4-5 c^2 d^2+2 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)^4}+\frac {2 b^3 \left (-4 a^2 d+a b c+3 b^2 d\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 )^{3/2} (b c-a d)^4}+\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right ) \left (c^2-d^2\right ) (b c-a d)^2 (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{f \left (a^2-b^2\right ) (b c-a d) (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-3 a b^2 c d^4-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 f \left (a^2-b^2\right ) \left (c^2-d^2\right )^2 (b c-a d)^3 (c+d \sin (e+f x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]

[Out]

(2*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c -
a*d)^4*f) - (d^2*(2*a*b*c*d*(4*c^2 - d^2) - a^2*d^2*(2*c^2 + d^2) - 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcTan
[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(5/2)*f) + (d*(a^2*d^2 + b^2*(2*c^2 - 3
*d^2))*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^2*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^2) + (b^2*Cos[e + f*x])/(
(a^2 - b^2)*(b*c - a*d)*f*(a + b*Sin[e + f*x])*(c + d*Sin[e + f*x])^2) - ((3*a^3*c*d^4 - 3*a*b^2*c*d^4 - a^2*b
*d^3*(7*c^2 - 4*d^2) - b^3*(2*c^4*d - 11*c^2*d^3 + 6*d^5))*Cos[e + f*x])/(2*(a^2 - b^2)*(b*c - a*d)^3*(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))^2 (c+d \sin (e+f x))^3} \, dx &=\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\int \frac {-a b c+a^2 d-3 b^2 d-a b d \sin (e+f x)+2 b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^3} \, dx}{\left (a^2-b^2\right ) (b c-a d)}\\ &=\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\int \frac {-2 \left (a^3 c d^2+a b^2 c \left (c^2-2 d^2\right )-2 a^2 b d \left (c^2-d^2\right )+3 b^3 d \left (c^2-d^2\right )\right )-d \left (2 a^2 b c d-2 b^3 c d-a^3 d^2+a b^2 \left (2 c^2-d^2\right )\right ) \sin (e+f x)+b d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \sin ^2(e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))^2} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right )}\\ &=\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-3 a b^2 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\int \frac {-6 b^4 d \left (c^2-d^2\right )^2-3 a^3 b c d^2 \left (2 c^2-d^2\right )+a^2 b^2 d \left (6 c^4-14 c^2 d^2+5 d^4\right )-a b^3 c \left (2 c^4-10 c^2 d^2+5 d^4\right )+a^4 \left (2 c^2 d^3+d^5\right )-b d \left (3 a^2 b c d \left (2 c^2-d^2\right )-3 b^3 c d \left (2 c^2-d^2\right )-a^3 \left (2 c^2 d^2+d^4\right )+a b^2 \left (2 c^4-2 c^2 d^2+3 d^4\right )\right ) \sin (e+f x)}{(a+b \sin (e+f x)) (c+d \sin (e+f x))} \, dx}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2}\\ &=\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-3 a b^2 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\left (b^3 \left (a b c-4 a^2 d+3 b^2 d\right )\right ) \int \frac {1}{a+b \sin (e+f x)} \, dx}{\left (a^2-b^2\right ) (b c-a d)^4}-\frac {\left (d^2 \left (2 a b c d \left (4 c^2-d^2\right )-a^2 d^2 \left (2 c^2+d^2\right )-3 b^2 \left (4 c^4-5 c^2 d^2+2 d^4\right )\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (b c-a d)^4 \left (c^2-d^2\right )^2}\\ &=\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-3 a b^2 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}+\frac {\left (2 b^3 \left (a b c-4 a^2 d+3 b^2 d\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 ) (b c-a d)^4 f}-\frac {\left (d^2 \left (2 a b c d \left (4 c^2-d^2\right )-a^2 d^2 \left (2 c^2+d^2\right )-3 b^2 \left (4 c^4-5 c^2 d^2+2 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)^4 \left (c^2-d^2\right )^2 f}\\ &=\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-3 a b^2 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}-\frac {\left (4 b^3 \left (a b c-4 a^2 d+3 b^2 d\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 ) (b c-a d)^4 f}+\frac {\left (2 d^2 \left (2 a b c d \left (4 c^2-d^2\right )-a^2 d^2 \left (2 c^2+d^2\right )-3 b^2 \left (4 c^4-5 c^2 d^2+2 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)^4 \left (c^2-d^2\right )^2 f}\\ &=\frac {2 b^3 \left (a b c-4 a^2 d+3 b^2 d\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 )^{3/2} (b c-a d)^4 f}-\frac {d^2 \left (2 a b c d \left (4 c^2-d^2\right )-a^2 d^2 \left (2 c^2+d^2\right )-3 b^2 \left (4 c^4-5 c^2 d^2+2 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)^4 \left (c^2-d^2\right )^{5/2} f}+\frac {d \left (a^2 d^2+b^2 \left (2 c^2-3 d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^2 \left (c^2-d^2\right ) f (c+d \sin (e+f x))^2}+\frac {b^2 \cos (e+f x)}{\left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x)) (c+d \sin (e+f x))^2}-\frac {\left (3 a^3 c d^4-3 a b^2 c d^4-a^2 b d^3 \left (7 c^2-4 d^2\right )-b^3 \left (2 c^4 d-11 c^2 d^3+6 d^5\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d)^3 \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))}\\ \end {align*}

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Mathematica [A]
time = 6.93, size = 346, normalized size = 0.76 \begin {gather*} \frac {\frac {4 b^3 \left (a b c-4 a^2 d+3 b^2 d\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 )^{3/2} (b c-a d)^4}+\frac {2 d^2 \left (2 a b c d \left (-4 c^2+d^2\right )+a^2 d^2 \left (2 c^2+d^2\right )+3 b^2 \left (4 c^4-5 c^2 d^2+2 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)^4 \left (c^2-d^2\right )^{5/2}}-\frac {2 b^4 \cos (e+f x)}{(a-b) (a+b) (-b c+a d)^3 (a+b \sin (e+f x))}+\frac {d^3 \cos (e+f x)}{(c-d) (c+d) (b c-a d)^2 (c+d \sin (e+f x))^2}+\frac {d^3 \left (7 b c^2-3 a c d-4 b d^2\right ) \cos (e+f x)}{(c-d)^2 (c+d)^2 (b c-a d)^3 (c+d \sin (e+f x))}}{2 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sin[e + f*x])^2*(c + d*Sin[e + f*x])^3),x]

[Out]

((4*b^3*(a*b*c - 4*a^2*d + 3*b^2*d)*ArcTan[(b + a*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(3/2)*(b*c
- a*d)^4) + (2*d^2*(2*a*b*c*d*(-4*c^2 + d^2) + a^2*d^2*(2*c^2 + d^2) + 3*b^2*(4*c^4 - 5*c^2*d^2 + 2*d^4))*ArcT
an[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(5/2)) - (2*b^4*Cos[e + f*x])/((a - b
)*(a + b)*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) + (d^3*Cos[e + f*x])/((c - d)*(c + d)*(b*c - a*d)^2*(c + d*Si
n[e + f*x])^2) + (d^3*(7*b*c^2 - 3*a*c*d - 4*b*d^2)*Cos[e + f*x])/((c - d)^2*(c + d)^2*(b*c - a*d)^3*(c + d*Si
n[e + f*x])))/(2*f)

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Maple [A]
time = 9.09, size = 760, normalized size = 1.66

method	result	size
derivativedivides	\(\frac {-\frac {2 b^{3} \left (\frac {\frac {b^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a \left (a^{2}-b^{2}\right )}+\frac {b \left (a d -b c \right )}{a^{2}-b^{2}}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (4 a^{2} d -a b c -3 b^{2} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a d -b c \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (5 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-14 a b \,c^{3} d +8 a b c \,d^{3}+9 b^{2} c^{4}-6 b^{2} c^{2} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{4} d^{2}+7 a^{2} c^{2} d^{4}-2 a^{2} d^{6}-12 a b \,c^{5} d -18 a b \,c^{3} d^{3}+12 a b c \,d^{5}+8 b^{2} c^{6}+11 b^{2} c^{4} d^{2}-10 b^{2} c^{2} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-34 a b \,c^{3} d +16 a b c \,d^{3}+23 b^{2} c^{4}-14 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{2} d^{2}-a^{2} d^{4}-12 a b \,c^{3} d +6 a b c \,d^{3}+8 b^{2} c^{4}-5 b^{2} c^{2} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{2} c^{2} d^{2}+a^{2} d^{4}-8 a b \,c^{3} d +2 a b c \,d^{3}+12 b^{2} c^{4}-15 b^{2} c^{2} d^{2}+6 b^{2} d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right )^{2}}}{f}\)	\(760\)
default	\(\frac {-\frac {2 b^{3} \left (\frac {\frac {b^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a \left (a^{2}-b^{2}\right )}+\frac {b \left (a d -b c \right )}{a^{2}-b^{2}}}{a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 b \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+a}+\frac {\left (4 a^{2} d -a b c -3 b^{2} d \right ) \arctan \left (\frac {2 a \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a^{2}-b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a d -b c \right )^{2} \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {2 d^{2} \left (\frac {\frac {d^{2} \left (5 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-14 a b \,c^{3} d +8 a b c \,d^{3}+9 b^{2} c^{4}-6 b^{2} c^{2} d^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{4} d^{2}+7 a^{2} c^{2} d^{4}-2 a^{2} d^{6}-12 a b \,c^{5} d -18 a b \,c^{3} d^{3}+12 a b c \,d^{5}+8 b^{2} c^{6}+11 b^{2} c^{4} d^{2}-10 b^{2} c^{2} d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c^{2}}+\frac {d^{2} \left (11 a^{2} c^{2} d^{2}-2 a^{2} d^{4}-34 a b \,c^{3} d +16 a b c \,d^{3}+23 b^{2} c^{4}-14 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) c}+\frac {d \left (4 a^{2} c^{2} d^{2}-a^{2} d^{4}-12 a b \,c^{3} d +6 a b c \,d^{3}+8 b^{2} c^{4}-5 b^{2} c^{2} d^{2}\right )}{2 c^{4}-4 c^{2} d^{2}+2 d^{4}}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{2}}+\frac {\left (2 a^{2} c^{2} d^{2}+a^{2} d^{4}-8 a b \,c^{3} d +2 a b c \,d^{3}+12 b^{2} c^{4}-15 b^{2} c^{2} d^{2}+6 b^{2} d^{4}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {c^{2}-d^{2}}}\right )}{\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (a d -b c \right )^{2}}}{f}\)	\(760\)
risch	\(\text {Expression too large to display}\)	\(3317\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x,method=_RETURNVERBOSE)

[Out]

1/f*(-2*b^3/(a*d-b*c)^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)*((b^2*(a*d-b*c)/a/(a^2-b^2)*tan(1/2*f*x+1/2*e)+b*(a*d-b*c)
/(a^2-b^2))/(a*tan(1/2*f*x+1/2*e)^2+2*b*tan(1/2*f*x+1/2*e)+a)+(4*a^2*d-a*b*c-3*b^2*d)/(a^2-b^2)^(3/2)*arctan(1
/2*(2*a*tan(1/2*f*x+1/2*e)+2*b)/(a^2-b^2)^(1/2)))+2*d^2/(a^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)^2*((1/2*d^2*(5*a
^2*c^2*d^2-2*a^2*d^4-14*a*b*c^3*d+8*a*b*c*d^3+9*b^2*c^4-6*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^6-12*a*b*c^5*d-18*a*b*c^3*d^3+12*a*b*c*d^5+8*b^2*c^6+11*b^2*c^4
*d^2-10*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^4-34*a*b*c^3
*d+16*a*b*c*d^3+23*b^2*c^4-14*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
^4-12*a*b*c^3*d+6*a*b*c*d^3+8*b^2*c^4-5*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^4-8*a*b*c^3*d+2*a*b*c*d^3+12*b^2*c^4-15*b^2*c^2*d^2+6*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))^2/(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*b^2-4*a^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))^2/(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))**2/(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. 1109 vs. \(2 (453) = 906\).
time = 0.57, size = 1109, normalized size = 2.42 \begin {gather*} \frac {\frac {2 \, {\left (a b^{4} c - 4 \, a^{2} b^{3} d + 3 \, b^{5} d\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (a\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + b}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{2} b^{4} c^{4} - b^{6} c^{4} - 4 \, a^{3} b^{3} c^{3} d + 4 \, a b^{5} c^{3} d + 6 \, a^{4} b^{2} c^{2} d^{2} - 6 \, a^{2} b^{4} c^{2} d^{2} - 4 \, a^{5} b c d^{3} + 4 \, a^{3} b^{3} c d^{3} + a^{6} d^{4} - a^{4} b^{2} d^{4}\right )} \sqrt {a^{2} - b^{2}}} + \frac {{\left (12 \, b^{2} c^{4} d^{2} - 8 \, a b c^{3} d^{3} + 2 \, a^{2} c^{2} d^{4} - 15 \, b^{2} c^{2} d^{4} + 2 \, a b c d^{5} + a^{2} d^{6} + 6 \, b^{2} d^{6}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )}}{{\left (b^{4} c^{8} - 4 \, a b^{3} c^{7} d + 6 \, a^{2} b^{2} c^{6} d^{2} - 2 \, b^{4} c^{6} d^{2} - 4 \, a^{3} b c^{5} d^{3} + 8 \, a b^{3} c^{5} d^{3} + a^{4} c^{4} d^{4} - 12 \, a^{2} b^{2} c^{4} d^{4} + b^{4} c^{4} d^{4} + 8 \, a^{3} b c^{3} d^{5} - 4 \, a b^{3} c^{3} d^{5} - 2 \, a^{4} c^{2} d^{6} + 6 \, a^{2} b^{2} c^{2} d^{6} - 4 \, a^{3} b c d^{7} + a^{4} d^{8}\right )} \sqrt {c^{2} - d^{2}}} + \frac {2 \, {\left (b^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a b^{4}\right )}}{{\left (a^{3} b^{3} c^{3} - a b^{5} c^{3} - 3 \, a^{4} b^{2} c^{2} d + 3 \, a^{2} b^{4} c^{2} d + 3 \, a^{5} b c d^{2} - 3 \, a^{3} b^{3} c d^{2} - a^{6} d^{3} + a^{4} b^{2} d^{3}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + a\right )}} + \frac {9 \, b c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, b c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 4 \, a c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, b c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 7 \, a c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 10 \, b c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, a d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 23 \, b c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 11 \, a c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 14 \, b c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, b c^{5} d^{3} - 4 \, a c^{4} d^{4} - 5 \, b c^{3} d^{5} + a c^{2} d^{6}}{{\left (b^{3} c^{9} - 3 \, a b^{2} c^{8} d + 3 \, a^{2} b c^{7} d^{2} - 2 \, b^{3} c^{7} d^{2} - a^{3} c^{6} d^{3} + 6 \, a b^{2} c^{6} d^{3} - 6 \, a^{2} b c^{5} d^{4} + b^{3} c^{5} d^{4} + 2 \, a^{3} c^{4} d^{5} - 3 \, a b^{2} c^{4} d^{5} + 3 \, a^{2} b c^{3} d^{6} - a^{3} c^{2} d^{7}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^3,x, algorithm="giac")

[Out]

(2*(a*b^4*c - 4*a^2*b^3*d + 3*b^5*d)*(pi*floor(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^2*b^4*c^4 - b^6*c^4 - 4*a^3*b^3*c^3*d + 4*a*b^5*c^3*d + 6*a^4*b^2*c^2*d^2 - 6*a^2*
b^4*c^2*d^2 - 4*a^5*b*c*d^3 + 4*a^3*b^3*c*d^3 + a^6*d^4 - a^4*b^2*d^4)*sqrt(a^2 - b^2)) + (12*b^2*c^4*d^2 - 8*
a*b*c^3*d^3 + 2*a^2*c^2*d^4 - 15*b^2*c^2*d^4 + 2*a*b*c*d^5 + a^2*d^6 + 6*b^2*d^6)*(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^4*c^8 - 4*a*b^3*c^7*d + 6*a^2*b^2*c^6
*d^2 - 2*b^4*c^6*d^2 - 4*a^3*b*c^5*d^3 + 8*a*b^3*c^5*d^3 + a^4*c^4*d^4 - 12*a^2*b^2*c^4*d^4 + b^4*c^4*d^4 + 8*
a^3*b*c^3*d^5 - 4*a*b^3*c^3*d^5 - 2*a^4*c^2*d^6 + 6*a^2*b^2*c^2*d^6 - 4*a^3*b*c*d^7 + a^4*d^8)*sqrt(c^2 - d^2)
) + 2*(b^5*tan(1/2*f*x + 1/2*e) + a*b^4)/((a^3*b^3*c^3 - a*b^5*c^3 - 3*a^4*b^2*c^2*d + 3*a^2*b^4*c^2*d + 3*a^5
*b*c*d^2 - 3*a^3*b^3*c*d^2 - a^6*d^3 + a^4*b^2*d^3)*(a*tan(1/2*f*x + 1/2*e)^2 + 2*b*tan(1/2*f*x + 1/2*e) + a))
+ (9*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 5*a*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 6*b*c^2*d^6*tan(1/2*f*x + 1/2*e)
^3 + 2*a*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^2 - 4*a*c^4*d^4*tan(1/2*f*x + 1/2*e)^
2 + 11*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 - 7*a*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 - 10*b*c*d^7*tan(1/2*f*x + 1/2*e)
^2 + 2*a*d^8*tan(1/2*f*x + 1/2*e)^2 + 23*b*c^4*d^4*tan(1/2*f*x + 1/2*e) - 11*a*c^3*d^5*tan(1/2*f*x + 1/2*e) -
14*b*c^2*d^6*tan(1/2*f*x + 1/2*e) + 2*a*c*d^7*tan(1/2*f*x + 1/2*e) + 8*b*c^5*d^3 - 4*a*c^4*d^4 - 5*b*c^3*d^5 +
a*c^2*d^6)/((b^3*c^9 - 3*a*b^2*c^8*d + 3*a^2*b*c^7*d^2 - 2*b^3*c^7*d^2 - a^3*c^6*d^3 + 6*a*b^2*c^6*d^3 - 6*a^
2*b*c^5*d^4 + b^3*c^5*d^4 + 2*a^3*c^4*d^5 - 3*a*b^2*c^4*d^5 + 3*a^2*b*c^3*d^6 - a^3*c^2*d^7)*(c*tan(1/2*f*x +
1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2*e) + c)^2))/f

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Mupad [B]
time = 45.34, size = 2500, normalized size = 5.46 \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))^2*(c + d*sin(e + f*x))^3),x)

[Out]

(d^2*atan(((d^2*(-(c + d)^5*(c - d)^5)^(1/2)*((8*tan(e/2 + (f*x)/2)*(4*a^3*b^11*c^16 - a^14*c*d^15 - 4*a^14*c^
3*d^13 - 4*a^14*c^5*d^11 - 144*a*b^13*c^4*d^12 + 684*a*b^13*c^6*d^10 - 1314*a*b^13*c^8*d^8 + 1224*a*b^13*c^10*
d^6 - 504*a*b^13*c^12*d^4 + 36*a*b^13*c^14*d^2 + 24*a^2*b^12*c^15*d + 144*a^4*b^10*c*d^15 - 44*a^4*b^10*c^15*d
- 348*a^6*b^8*c*d^15 + 214*a^8*b^6*c*d^15 + 7*a^10*b^4*c*d^15 - 8*a^12*b^2*c*d^15 - a^13*b*c^2*d^14 + 20*a^13
*b*c^4*d^12 + 44*a^13*b*c^6*d^10 + 432*a^2*b^12*c^3*d^13 - 2148*a^2*b^12*c^5*d^11 + 4470*a^2*b^12*c^7*d^9 - 46
32*a^2*b^12*c^9*d^7 + 2232*a^2*b^12*c^11*d^5 - 252*a^2*b^12*c^13*d^3 - 432*a^3*b^11*c^2*d^14 + 2688*a^3*b^11*c
^4*d^12 - 7294*a^3*b^11*c^6*d^10 + 10105*a^3*b^11*c^8*d^8 - 7104*a^3*b^11*c^10*d^6 + 1892*a^3*b^11*c^12*d^4 -
192*a^3*b^11*c^14*d^2 - 2016*a^4*b^10*c^3*d^13 + 8378*a^4*b^10*c^5*d^11 - 15815*a^4*b^10*c^7*d^9 + 14976*a^4*b
^10*c^9*d^7 - 5932*a^4*b^10*c^11*d^5 + 624*a^4*b^10*c^13*d^3 + 1140*a^5*b^9*c^2*d^14 - 6574*a^5*b^9*c^4*d^12 +
16053*a^5*b^9*c^6*d^10 - 19912*a^5*b^9*c^8*d^8 + 11320*a^5*b^9*c^10*d^6 - 1920*a^5*b^9*c^12*d^4 + 172*a^5*b^9
*c^14*d^2 + 2938*a^6*b^8*c^3*d^13 - 10619*a^6*b^8*c^5*d^11 + 18608*a^6*b^8*c^7*d^9 - 15576*a^6*b^8*c^9*d^7 + 4
344*a^6*b^8*c^11*d^5 - 292*a^6*b^8*c^13*d^3 - 818*a^7*b^7*c^2*d^14 + 5107*a^7*b^7*c^4*d^12 - 12464*a^7*b^7*c^6
*d^10 + 14693*a^7*b^7*c^8*d^8 - 6184*a^7*b^7*c^10*d^6 + 368*a^7*b^7*c^12*d^4 - 1485*a^8*b^6*c^3*d^13 + 5064*a^
8*b^6*c^5*d^11 - 8939*a^8*b^6*c^7*d^9 + 6104*a^8*b^6*c^9*d^7 - 688*a^8*b^6*c^11*d^5 + 55*a^9*b^5*c^2*d^14 - 10
56*a^9*b^5*c^4*d^12 + 3649*a^9*b^5*c^6*d^10 - 4524*a^9*b^5*c^8*d^8 + 1120*a^9*b^5*c^10*d^6 + 152*a^10*b^4*c^3*
d^13 - 975*a^10*b^4*c^5*d^11 + 2300*a^10*b^4*c^7*d^9 - 1088*a^10*b^4*c^9*d^7 + 16*a^11*b^3*c^2*d^14 + 59*a^11*
b^3*c^4*d^12 - 640*a^11*b^3*c^6*d^10 + 628*a^11*b^3*c^8*d^8 + 27*a^12*b^2*c^3*d^13 + 48*a^12*b^2*c^5*d^11 - 22
0*a^12*b^2*c^7*d^9))/(a^13*d^17 - b^13*c^17 + 2*a^2*b^11*c^17 - a^4*b^9*c^17 + a^9*b^4*d^17 - 2*a^11*b^2*d^17
- 4*a^13*c^2*d^15 + 6*a^13*c^4*d^13 - 4*a^13*c^6*d^11 + a^13*c^8*d^9 - b^13*c^9*d^8 + 4*b^13*c^11*d^6 - 6*b^13
*c^13*d^4 + 4*b^13*c^15*d^2 + 9*a*b^12*c^8*d^9 - 36*a*b^12*c^10*d^7 + 54*a*b^12*c^12*d^5 - 36*a*b^12*c^14*d^3
- 18*a^3*b^10*c^16*d + 9*a^5*b^8*c^16*d - 9*a^8*b^5*c*d^16 + 18*a^10*b^3*c*d^16 + 36*a^12*b*c^3*d^14 - 54*a^12
*b*c^5*d^12 + 36*a^12*b*c^7*d^10 - 9*a^12*b*c^9*d^8 - 36*a^2*b^11*c^7*d^10 + 146*a^2*b^11*c^9*d^8 - 224*a^2*b^
11*c^11*d^6 + 156*a^2*b^11*c^13*d^4 - 44*a^2*b^11*c^15*d^2 + 84*a^3*b^10*c^6*d^11 - 354*a^3*b^10*c^8*d^9 + 576
*a^3*b^10*c^10*d^7 - 444*a^3*b^10*c^12*d^5 + 156*a^3*b^10*c^14*d^3 - 126*a^4*b^9*c^5*d^12 + 576*a^4*b^9*c^7*d^
10 - 1045*a^4*b^9*c^9*d^8 + 940*a^4*b^9*c^11*d^6 - 420*a^4*b^9*c^13*d^4 + 76*a^4*b^9*c^15*d^2 + 126*a^5*b^8*c^
4*d^13 - 672*a^5*b^8*c^6*d^11 + 1437*a^5*b^8*c^8*d^9 - 1548*a^5*b^8*c^10*d^7 + 852*a^5*b^8*c^12*d^5 - 204*a^5*
b^8*c^14*d^3 - 84*a^6*b^7*c^3*d^14 + 588*a^6*b^7*c^5*d^12 - 1548*a^6*b^7*c^7*d^10 + 1992*a^6*b^7*c^9*d^8 - 130
8*a^6*b^7*c^11*d^6 + 396*a^6*b^7*c^13*d^4 - 36*a^6*b^7*c^15*d^2 + 36*a^7*b^6*c^2*d^15 - 396*a^7*b^6*c^4*d^13 +
1308*a^7*b^6*c^6*d^11 - 1992*a^7*b^6*c^8*d^9 + 1548*a^7*b^6*c^10*d^7 - 588*a^7*b^6*c^12*d^5 + 84*a^7*b^6*c^14
*d^3 + 204*a^8*b^5*c^3*d^14 - 852*a^8*b^5*c^5*d^12 + 1548*a^8*b^5*c^7*d^10 - 1437*a^8*b^5*c^9*d^8 + 672*a^8*b^
5*c^11*d^6 - 126*a^8*b^5*c^13*d^4 - 76*a^9*b^4*c^2*d^15 + 420*a^9*b^4*c^4*d^13 - 940*a^9*b^4*c^6*d^11 + 1045*a
^9*b^4*c^8*d^9 - 576*a^9*b^4*c^10*d^7 + 126*a^9*b^4*c^12*d^5 - 156*a^10*b^3*c^3*d^14 + 444*a^10*b^3*c^5*d^12 -
576*a^10*b^3*c^7*d^10 + 354*a^10*b^3*c^9*d^8 - 84*a^10*b^3*c^11*d^6 + 44*a^11*b^2*c^2*d^15 - 156*a^11*b^2*c^4
*d^13 + 224*a^11*b^2*c^6*d^11 - 146*a^11*b^2*c^8*d^9 + 36*a^11*b^2*c^10*d^7 + 9*a*b^12*c^16*d - 9*a^12*b*c*d^1
6) - (8*(36*a*b^13*c^5*d^11 - 144*a*b^13*c^7*d^9 + 216*a*b^13*c^9*d^7 - 144*a*b^13*c^11*d^5 + 36*a*b^13*c^13*d
^3 + 4*a^3*b^11*c^15*d - 36*a^5*b^9*c*d^15 + 60*a^7*b^7*c*d^15 - 13*a^9*b^5*c*d^15 - 10*a^11*b^3*c*d^15 - 4*a^
13*b*c^3*d^13 - 4*a^13*b*c^5*d^11 - 72*a^2*b^12*c^4*d^12 + 276*a^2*b^12*c^6*d^10 - 375*a^2*b^12*c^8*d^8 + 216*
a^2*b^12*c^10*d^6 - 60*a^2*b^12*c^12*d^4 + 24*a^2*b^12*c^14*d^2 - 36*a^3*b^11*c^5*d^11 + 61*a^3*b^11*c^7*d^9 -
88*a^3*b^11*c^9*d^7 + 180*a^3*b^11*c^11*d^5 - 184*a^3*b^11*c^13*d^3 + 72*a^4*b^10*c^2*d^14 - 168*a^4*b^10*c^4
*d^12 + 233*a^4*b^10*c^6*d^10 - 270*a^4*b^10*c^8*d^8 + 100*a^4*b^10*c^10*d^6 + 248*a^4*b^10*c^12*d^4 - 44*a^4*
b^10*c^14*d^2 + 120*a^5*b^9*c^3*d^13 - 535*a^5*b^9*c^5*d^11 + 1386*a^5*b^9*c^7*d^9 - 1544*a^5*b^9*c^9*d^7 + 24
8*a^5*b^9*c^11*d^5 + 172*a^5*b^9*c^13*d^3 - 108*a^6*b^8*c^2*d^14 + 699*a^6*b^8*c^4*d^12 - 2046*a^6*b^8*c^6*d^1
0 + 2885*a^6*b^8*c^8*d^8 - 1336*a^6*b^8*c^10*d^6 - 148*a^6*b^8*c^12*d^4 - 305*a^7*b^7*c^3*d^13 + 1354*a^7*b^7*
c^5*d^11 - 2979*a^7*b^7*c^7*d^9 + 2648*a^7*b^7*c^9*d^7 - 400*a^7*b^7*c^11*d^5 + 19*a^8*b^6*c^2*d^14 - 602*a^8*
b^6*c^4*d^12 + 2161*a^8*b^6*c^6*d^10 - 3012*a^8*b^6*c^8*d^8 + 1056*a^8*b^6*c^10*d^6 + 190*a^9*b^5*c^3*d^13 - 8
95*a^9*b^5*c^5*d^11 + 1860*a^9*b^5*c^7*d^9 - 10...

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