3.8.21 \(\int \frac {1}{(a+b \sin (e+f x))^3 (c+d \sin (e+f x))^2} \, dx\) [721]

Optimal. Leaf size=454 \[ -\frac {b^2 \left (8 a^3 b c d-2 a b^3 c d-12 a^4 d^2-a^2 b^2 \left (2 c^2-15 d^2\right )-b^4 \left (c^2+6 d^2\right )\right ) \tan ^{-1}\left (\frac {b+a \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{5/2} (b c-a d)^4 f}-\frac {2 d^3 \left (4 b c^2-a c d-3 b d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(b c-a d)^4 \left (c^2-d^2\right )^{3/2} f}-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-2 b^4 d \left (2 c^2-3 d^2\right )-3 a b^3 c \left (c^2-d^2\right )\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^3 \left (c^2-d^2\right ) f (c+d \sin (e+f x))}+\frac {b^2 \cos (e+f x)}{2 \left (a^2-b^2\right ) (b c-a d) f (a+b \sin (e+f x))^2 (c+d \sin (e+f x))}+\frac {3 b^2 \left (a b c-2 a^2 d+b^2 d\right ) \cos (e+f x)}{2 \left (a^2-b^2\right )^2 (b c-a d)^2 f (a+b \sin (e+f x)) (c+d \sin (e+f x))} \]

[Out]

-b^2*(8*a^3*b*c*d-2*a*b^3*c*d-12*a^4*d^2-a^2*b^2*(2*c^2-15*d^2)-b^4*(c^2+6*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)^4/f-2*d^3*(-a*c*d+4*b*c^2-3*b*d^2)*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)^(3/2)/f-1/2*d*(2*a^4*d^3+a^2*b^2*d*(7*c^2-11*d^2)-2*b^4*d*(2*c^2-3
*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))+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))+3/2*b^2*(-2*a^2*d+a*b*c+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))

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Rubi [A]
time = 1.60, antiderivative size = 454, 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 {3 b^2 \left (-2 a^2 d+a b c+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))}+\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))}-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-3 a b^3 c \left (c^2-d^2\right )-2 b^4 d \left (2 c^2-3 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))}-\frac {b^2 \left (-12 a^4 d^2+8 a^3 b c d-a^2 b^2 \left (2 c^2-15 d^2\right )-2 a b^3 c d-b^4 \left (c^2+6 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)^4}-\frac {2 d^3 \left (-a c d+4 b c^2-3 b d^2\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 )^{3/2} (b c-a d)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((b^2*(8*a^3*b*c*d - 2*a*b^3*c*d - 12*a^4*d^2 - a^2*b^2*(2*c^2 - 15*d^2) - b^4*(c^2 + 6*d^2))*ArcTan[(b + a*T
an[(e + f*x)/2])/Sqrt[a^2 - b^2]])/((a^2 - b^2)^(5/2)*(b*c - a*d)^4*f)) - (2*d^3*(4*b*c^2 - a*c*d - 3*b*d^2)*A
rcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(3/2)*f) - (d*(2*a^4*d^3 + a^2*b^2
*d*(7*c^2 - 11*d^2) - 2*b^4*d*(2*c^2 - 3*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])) + (b^2*Cos[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])) + (3*b^2*(a*b*c - 2*a^2*d + 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]))

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))^2} \, 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))}-\frac {\int \frac {-2 a b c+2 a^2 d-3 b^2 d+b (b c-2 a d) \sin (e+f x)+2 b^2 d \sin ^2(e+f x)}{(a+b \sin (e+f x))^2 (c+d \sin (e+f x))^2} \, 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))}+\frac {3 b^2 \left (a b c-2 a^2 d+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))}+\frac {\int \frac {-4 a^3 b c d+4 a b^3 c d+2 a^4 d^2+a^2 b^2 \left (2 c^2-11 d^2\right )+b^4 \left (c^2+6 d^2\right )+b d \left (a^2 b c+2 b^3 c-4 a^3 d+a b^2 d\right ) \sin (e+f x)-3 b^2 d \left (a b c-2 a^2 d+b^2 d\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 )^2 (b c-a d)^2}\\ &=-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-2 b^4 d \left (2 c^2-3 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))}+\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))}+\frac {3 b^2 \left (a b c-2 a^2 d+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))}+\frac {\int \frac {-2 a^5 c d^3-2 a^3 b^2 c d \left (3 c^2-5 d^2\right )+a b^4 c d \left (3 c^2-5 d^2\right )+6 a^4 b d^2 \left (c^2-d^2\right )+b^5 \left (c^4+5 c^2 d^2-6 d^4\right )+2 a^2 b^3 \left (c^4-7 c^2 d^2+6 d^4\right )-b d \left (2 a^4 c d^2-b^4 c \left (c^2-3 d^2\right )+6 a^3 b d \left (c^2-d^2\right )-3 a b^3 d \left (c^2-d^2\right )-2 a^2 b^2 c \left (c^2+d^2\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 )^2 (b c-a d)^3 \left (c^2-d^2\right )}\\ &=-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-2 b^4 d \left (2 c^2-3 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))}+\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))}+\frac {3 b^2 \left (a b c-2 a^2 d+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))}-\frac {\left (d^3 \left (4 b c^2-a c d-3 b d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{(b c-a d)^4 \left (c^2-d^2\right )}-\frac {\left (b^2 \left (8 a^3 b c d-2 a b^3 c d-12 a^4 d^2-a^2 b^2 \left (2 c^2-15 d^2\right )-b^4 \left (c^2+6 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)^4}\\ &=-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-2 b^4 d \left (2 c^2-3 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))}+\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))}+\frac {3 b^2 \left (a b c-2 a^2 d+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))}-\frac {\left (2 d^3 \left (4 b c^2-a c d-3 b d^2\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 ) f}-\frac {\left (b^2 \left (8 a^3 b c d-2 a b^3 c d-12 a^4 d^2-a^2 b^2 \left (2 c^2-15 d^2\right )-b^4 \left (c^2+6 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)^4 f}\\ &=-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-2 b^4 d \left (2 c^2-3 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))}+\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))}+\frac {3 b^2 \left (a b c-2 a^2 d+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))}+\frac {\left (4 d^3 \left (4 b c^2-a c d-3 b d^2\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 ) f}+\frac {\left (2 b^2 \left (8 a^3 b c d-2 a b^3 c d-12 a^4 d^2-a^2 b^2 \left (2 c^2-15 d^2\right )-b^4 \left (c^2+6 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)^4 f}\\ &=-\frac {b^2 \left (8 a^3 b c d-2 a b^3 c d-12 a^4 d^2-a^2 b^2 \left (2 c^2-15 d^2\right )-b^4 \left (c^2+6 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)^4 f}-\frac {2 d^3 \left (4 b c^2-a c d-3 b d^2\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 )^{3/2} f}-\frac {d \left (2 a^4 d^3+a^2 b^2 d \left (7 c^2-11 d^2\right )-2 b^4 d \left (2 c^2-3 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))}+\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))}+\frac {3 b^2 \left (a b c-2 a^2 d+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))}\\ \end {align*}

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Mathematica [A]
time = 5.90, size = 346, normalized size = 0.76 \begin {gather*} \frac {\frac {2 b^2 \left (-8 a^3 b c d+2 a b^3 c d+12 a^4 d^2+a^2 b^2 \left (2 c^2-15 d^2\right )+b^4 \left (c^2+6 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)^4}+\frac {4 d^3 \left (-4 b c^2+a c d+3 b d^2\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 )^{3/2}}+\frac {b^3 \cos (e+f x)}{(a-b) (a+b) (b c-a d)^2 (a+b \sin (e+f x))^2}+\frac {b^3 \left (-3 a b c+7 a^2 d-4 b^2 d\right ) \cos (e+f x)}{(a-b)^2 (a+b)^2 (-b c+a d)^3 (a+b \sin (e+f x))}-\frac {2 d^4 \cos (e+f x)}{(c-d) (c+d) (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])^3*(c + d*Sin[e + f*x])^2),x]

[Out]

((2*b^2*(-8*a^3*b*c*d + 2*a*b^3*c*d + 12*a^4*d^2 + a^2*b^2*(2*c^2 - 15*d^2) + b^4*(c^2 + 6*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)^4) + (4*d^3*(-4*b*c^2 + a*c*d + 3*b*d^2)*A
rcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]])/((b*c - a*d)^4*(c^2 - d^2)^(3/2)) + (b^3*Cos[e + f*x])/((a -
b)*(a + b)*(b*c - a*d)^2*(a + b*Sin[e + f*x])^2) + (b^3*(-3*a*b*c + 7*a^2*d - 4*b^2*d)*Cos[e + f*x])/((a - b)^
2*(a + b)^2*(-(b*c) + a*d)^3*(a + b*Sin[e + f*x])) - (2*d^4*Cos[e + f*x])/((c - d)*(c + d)*(b*c - a*d)^3*(c +
d*Sin[e + f*x])))/(2*f)

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Maple [A]
time = 9.08, size = 736, normalized size = 1.62

method	result	size
derivativedivides	\(\frac {\frac {2 b^{2} \left (\frac {\frac {b^{2} \left (9 a^{4} d^{2}-14 a^{3} b c d +5 a^{2} b^{2} c^{2}-6 a^{2} b^{2} d^{2}+8 a \,b^{3} c d -2 b^{4} c^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}+\frac {b \left (8 a^{6} d^{2}-12 a^{5} b c d +4 a^{4} b^{2} c^{2}+11 a^{4} b^{2} d^{2}-18 a^{3} b^{3} c d +7 a^{2} b^{4} c^{2}-10 a^{2} b^{4} d^{2}+12 a \,b^{5} c d -2 b^{6} c^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b^{2} \left (23 a^{4} d^{2}-34 a^{3} b c d +11 a^{2} b^{2} c^{2}-14 a^{2} b^{2} d^{2}+16 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (8 a^{4} d^{2}-12 a^{3} b c d +4 a^{2} b^{2} c^{2}-5 a^{2} b^{2} d^{2}+6 a \,b^{3} c d -b^{4} c^{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}}{\left (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 \right )^{2}}+\frac {\left (12 a^{4} d^{2}-8 a^{3} b c d +2 a^{2} b^{2} c^{2}-15 a^{2} b^{2} d^{2}+2 a \,b^{3} c d +b^{4} c^{2}+6 b^{4} d^{2}\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 )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{3} \left (\frac {\frac {d^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{2}-d^{2}\right )}+\frac {d \left (a d -b c \right )}{c^{2}-d^{2}}}{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}+\frac {\left (a c d -4 b \,c^{2}+3 d^{2} b \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 )}{\left (c^{2}-d^{2}\right )^{\frac {3}{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}\)	\(736\)
default	\(\frac {\frac {2 b^{2} \left (\frac {\frac {b^{2} \left (9 a^{4} d^{2}-14 a^{3} b c d +5 a^{2} b^{2} c^{2}-6 a^{2} b^{2} d^{2}+8 a \,b^{3} c d -2 b^{4} c^{2}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a}+\frac {b \left (8 a^{6} d^{2}-12 a^{5} b c d +4 a^{4} b^{2} c^{2}+11 a^{4} b^{2} d^{2}-18 a^{3} b^{3} c d +7 a^{2} b^{4} c^{2}-10 a^{2} b^{4} d^{2}+12 a \,b^{5} c d -2 b^{6} c^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) a^{2}}+\frac {b^{2} \left (23 a^{4} d^{2}-34 a^{3} b c d +11 a^{2} b^{2} c^{2}-14 a^{2} b^{2} d^{2}+16 a \,b^{3} c d -2 b^{4} c^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a \left (a^{4}-2 a^{2} b^{2}+b^{4}\right )}+\frac {b \left (8 a^{4} d^{2}-12 a^{3} b c d +4 a^{2} b^{2} c^{2}-5 a^{2} b^{2} d^{2}+6 a \,b^{3} c d -b^{4} c^{2}\right )}{2 a^{4}-4 a^{2} b^{2}+2 b^{4}}}{\left (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 \right )^{2}}+\frac {\left (12 a^{4} d^{2}-8 a^{3} b c d +2 a^{2} b^{2} c^{2}-15 a^{2} b^{2} d^{2}+2 a \,b^{3} c d +b^{4} c^{2}+6 b^{4} d^{2}\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 )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {a^{2}-b^{2}}}\right )}{\left (a d -b c \right )^{4}}+\frac {2 d^{3} \left (\frac {\frac {d^{2} \left (a d -b c \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{2}-d^{2}\right )}+\frac {d \left (a d -b c \right )}{c^{2}-d^{2}}}{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}+\frac {\left (a c d -4 b \,c^{2}+3 d^{2} b \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 )}{\left (c^{2}-d^{2}\right )^{\frac {3}{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}\)	\(736\)
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))^3/(c+d*sin(f*x+e))^2,x,method=_RETURNVERBOSE)

[Out]

1/f*(2*b^2/(a*d-b*c)^4*((1/2*b^2*(9*a^4*d^2-14*a^3*b*c*d+5*a^2*b^2*c^2-6*a^2*b^2*d^2+8*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*(8*a^6*d^2-12*a^5*b*c*d+4*a^4*b^2*c^2+11*a^4*b^2*d^2-18*a^3*b^3
*c*d+7*a^2*b^4*c^2-10*a^2*b^4*d^2+12*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
*(23*a^4*d^2-34*a^3*b*c*d+11*a^2*b^2*c^2-14*a^2*b^2*d^2+16*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*(8*a^4*d^2-12*a^3*b*c*d+4*a^2*b^2*c^2-5*a^2*b^2*d^2+6*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*(12*a^4*d^2-8*a^3*b*c*d+2*a^2*b^2*c^2-15*a^2*b^2*d^2+
2*a*b^3*c*d+b^4*c^2+6*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^2*d^2-2*a*b*c*d+b^2*c^2)/(a*d-b*c)^2*((d^2*(a*d-b*c)/c/(c^2-d^2)*tan(1/2*f*x+1/2*e)+d*
(a*d-b*c)/(c^2-d^2))/(c*tan(1/2*f*x+1/2*e)^2+2*d*tan(1/2*f*x+1/2*e)+c)+(a*c*d-4*b*c^2+3*b*d^2)/(c^2-d^2)^(3/2)
*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))^2,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))^2,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))**2,x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1111 vs. \(2 (448) = 896\).
time = 0.59, size = 1111, normalized size = 2.45 \begin {gather*} \frac {\frac {{\left (2 \, a^{2} b^{4} c^{2} + b^{6} c^{2} - 8 \, a^{3} b^{3} c d + 2 \, a b^{5} c d + 12 \, a^{4} b^{2} d^{2} - 15 \, a^{2} b^{4} d^{2} + 6 \, b^{6} d^{2}\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^{4} b^{4} c^{4} - 2 \, a^{2} b^{6} c^{4} + b^{8} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 8 \, a^{3} b^{5} c^{3} d - 4 \, a b^{7} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 12 \, a^{4} b^{4} c^{2} d^{2} + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + 8 \, a^{5} b^{3} c d^{3} - 4 \, a^{3} b^{5} c d^{3} + a^{8} d^{4} - 2 \, a^{6} b^{2} d^{4} + a^{4} b^{4} d^{4}\right )} \sqrt {a^{2} - b^{2}}} - \frac {2 \, {\left (4 \, b c^{2} d^{3} - a c d^{4} - 3 \, b d^{5}\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^{6} - 4 \, a b^{3} c^{5} d + 6 \, a^{2} b^{2} c^{4} d^{2} - b^{4} c^{4} d^{2} - 4 \, a^{3} b c^{3} d^{3} + 4 \, a b^{3} c^{3} d^{3} + a^{4} c^{2} d^{4} - 6 \, a^{2} b^{2} c^{2} d^{4} + 4 \, a^{3} b c d^{5} - a^{4} d^{6}\right )} \sqrt {c^{2} - d^{2}}} - \frac {2 \, {\left (d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c d^{4}\right )}}{{\left (b^{3} c^{6} - 3 \, a b^{2} c^{5} d + 3 \, a^{2} b c^{4} d^{2} - b^{3} c^{4} d^{2} - a^{3} c^{3} d^{3} + 3 \, a b^{2} c^{3} d^{3} - 3 \, a^{2} b c^{2} d^{4} + a^{3} c d^{5}\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 )}} + \frac {5 \, a^{3} b^{5} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, a b^{7} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 9 \, a^{4} b^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{2} b^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, a^{4} b^{4} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 7 \, a^{2} b^{6} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, b^{8} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 8 \, a^{5} b^{3} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 11 \, a^{3} b^{5} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 10 \, a b^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 11 \, a^{3} b^{5} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b^{7} c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 23 \, a^{4} b^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 14 \, a^{2} b^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 4 \, a^{4} b^{4} c - a^{2} b^{6} c - 8 \, a^{5} b^{3} d + 5 \, a^{3} b^{5} d}{{\left (a^{6} b^{3} c^{3} - 2 \, a^{4} b^{5} c^{3} + a^{2} b^{7} c^{3} - 3 \, a^{7} b^{2} c^{2} d + 6 \, a^{5} b^{4} c^{2} d - 3 \, a^{3} b^{6} c^{2} d + 3 \, a^{8} b c d^{2} - 6 \, a^{6} b^{3} c d^{2} + 3 \, a^{4} b^{5} c d^{2} - a^{9} d^{3} + 2 \, a^{7} b^{2} d^{3} - a^{5} b^{4} 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 )}^{2}}}{f} \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))^2,x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 43.67, size = 2500, normalized size = 5.51 \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))^2),x)

[Out]

((2*a^6*d^4 + b^6*c^4 - 4*a^2*b^4*c^4 + 2*a^2*b^4*d^4 - 4*a^4*b^2*d^4 - b^6*c^2*d^2 - 8*a^3*b^3*c*d^3 + 8*a^3*
b^3*c^3*d + 4*a^2*b^4*c^2*d^2 + 5*a*b^5*c*d^3 - 5*a*b^5*c^3*d)/((a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c
*d^2)*(a^4*c^2 - a^4*d^2 + b^4*c^2 - b^4*d^2 - 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2)) + (tan(e/2 + (f*x)/2)*(2*a^7*d^
5 + 2*b^7*c^5 - 11*a^2*b^5*c^5 + 2*a^3*b^4*d^5 - 4*a^5*b^2*d^5 - 2*b^7*c^3*d^2 + 12*a*b^6*c^2*d^3 + 18*a^2*b^5
*c*d^4 + 15*a^3*b^4*c^4*d - 32*a^4*b^3*c*d^4 + a^2*b^5*c^3*d^2 - 15*a^3*b^4*c^2*d^3 + 16*a^4*b^3*c^3*d^2 - 12*
a*b^6*c^4*d + 8*a^6*b*c*d^4))/(a*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 - a^4*d^2 + b^
4*c^2 - b^4*d^2 - 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2)) + (tan(e/2 + (f*x)/2)^5*(2*a^7*d^5 + 2*b^7*c^5 - 5*a^2*b^5*c
^5 + 2*a^3*b^4*d^5 - 4*a^5*b^2*d^5 - 2*b^7*c^3*d^2 + 6*a*b^6*c^2*d^3 + 9*a^3*b^4*c^4*d + 5*a^2*b^5*c^3*d^2 - 9
*a^3*b^4*c^2*d^3 - 6*a*b^6*c^4*d))/(a*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 - a^4*d^2
+ b^4*c^2 - b^4*d^2 - 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2)) + (2*tan(e/2 + (f*x)/2)^2*(b^8*c^5 + 4*a^7*b*d^5 + 2*a^
8*c*d^4 - 3*a^2*b^6*c^5 - 4*a^4*b^4*c^5 + 4*a^3*b^5*d^5 - 8*a^5*b^3*d^5 - b^8*c^3*d^2 + 3*a*b^7*c^2*d^3 + 18*a
^2*b^6*c*d^4 - 8*a^3*b^5*c^4*d - 29*a^4*b^4*c*d^4 + 8*a^5*b^3*c^4*d - 11*a^2*b^6*c^3*d^2 + 8*a^3*b^5*c^2*d^3 +
27*a^4*b^4*c^3*d^2 - 8*a^5*b^3*c^2*d^3 - 3*a*b^7*c^4*d))/(a^2*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*
c*d^2)*(a^4*c^2 - a^4*d^2 + b^4*c^2 - b^4*d^2 - 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2)) - (tan(e/2 + (f*x)/2)^4*(7*a^2
*b^6*c^5 - 8*a^7*b*d^5 - 2*a^8*c*d^4 - 2*b^8*c^5 + 4*a^4*b^4*c^5 - 8*a^3*b^5*d^5 + 16*a^5*b^3*d^5 + 2*b^8*c^3*
d^2 - 6*a*b^7*c^2*d^3 - 12*a^2*b^6*c*d^4 - a^3*b^5*c^4*d + 16*a^4*b^4*c*d^4 - 8*a^5*b^3*c^4*d + 4*a^6*b^2*c*d^
4 + 5*a^2*b^6*c^3*d^2 + a^3*b^5*c^2*d^3 - 22*a^4*b^4*c^3*d^2 + 8*a^5*b^3*c^2*d^3 + 6*a*b^7*c^4*d))/(a^2*c*(a^3
*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 - a^4*d^2 + b^4*c^2 - b^4*d^2 - 2*a^2*b^2*c^2 + 2*a^2
*b^2*d^2)) + (2*tan(e/2 + (f*x)/2)^3*(a^2*d + 2*b^2*d + 2*a*b*c)*(2*a^6*d^4 + b^6*c^4 - 4*a^2*b^4*c^4 + 2*a^2*
b^4*d^4 - 4*a^4*b^2*d^4 - b^6*c^2*d^2 - 8*a^3*b^3*c*d^3 + 8*a^3*b^3*c^3*d + 4*a^2*b^4*c^2*d^2 + 5*a*b^5*c*d^3
- 5*a*b^5*c^3*d))/(a^2*c*(a^3*d^3 - b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)*(a^4*c^2 - a^4*d^2 + b^4*c^2 - b^
4*d^2 - 2*a^2*b^2*c^2 + 2*a^2*b^2*d^2)))/(f*(tan(e/2 + (f*x)/2)^2*(3*a^2*c + 4*b^2*c + 8*a*b*d) + tan(e/2 + (f
*x)/2)^4*(3*a^2*c + 4*b^2*c + 8*a*b*d) + tan(e/2 + (f*x)/2)^3*(4*a^2*d + 8*b^2*d + 8*a*b*c) + a^2*c + tan(e/2
+ (f*x)/2)*(2*a^2*d + 4*a*b*c) + tan(e/2 + (f*x)/2)^5*(2*a^2*d + 4*a*b*c) + a^2*c*tan(e/2 + (f*x)/2)^6)) - (d^
3*atan(((d^3*(-(c + d)^3*(c - d)^3)^(1/2)*((8*(60*a*b^15*c^7*d^7 - 36*a*b^15*c^5*d^9 - 13*a*b^15*c^9*d^5 - 10*
a*b^15*c^11*d^3 - 4*a^3*b^13*c^13*d + 36*a^5*b^11*c*d^13 - 4*a^5*b^11*c^13*d - 144*a^7*b^9*c*d^13 + 216*a^9*b^
7*c*d^13 - 144*a^11*b^5*c*d^13 + 36*a^13*b^3*c*d^13 + 4*a^15*b*c^3*d^11 + 72*a^2*b^14*c^4*d^10 - 108*a^2*b^14*
c^6*d^8 + 19*a^2*b^14*c^8*d^6 + 14*a^2*b^14*c^10*d^4 - a^2*b^14*c^12*d^2 + 120*a^3*b^13*c^5*d^9 - 305*a^3*b^13
*c^7*d^7 + 190*a^3*b^13*c^9*d^5 + 19*a^3*b^13*c^11*d^3 - 72*a^4*b^12*c^2*d^12 - 168*a^4*b^12*c^4*d^10 + 699*a^
4*b^12*c^6*d^8 - 602*a^4*b^12*c^8*d^6 + 99*a^4*b^12*c^10*d^4 + 20*a^4*b^12*c^12*d^2 - 36*a^5*b^11*c^3*d^11 - 5
35*a^5*b^11*c^5*d^9 + 1354*a^5*b^11*c^7*d^7 - 895*a^5*b^11*c^9*d^5 + 40*a^5*b^11*c^11*d^3 + 276*a^6*b^10*c^2*d
^12 + 233*a^6*b^10*c^4*d^10 - 2046*a^6*b^10*c^6*d^8 + 2161*a^6*b^10*c^8*d^6 - 552*a^6*b^10*c^10*d^4 + 44*a^6*b
^10*c^12*d^2 + 61*a^7*b^9*c^3*d^11 + 1386*a^7*b^9*c^5*d^9 - 2979*a^7*b^9*c^7*d^7 + 1860*a^7*b^9*c^9*d^5 - 220*
a^7*b^9*c^11*d^3 - 375*a^8*b^8*c^2*d^12 - 270*a^8*b^8*c^4*d^10 + 2885*a^8*b^8*c^6*d^8 - 3012*a^8*b^8*c^8*d^6 +
628*a^8*b^8*c^10*d^4 - 88*a^9*b^7*c^3*d^11 - 1544*a^9*b^7*c^5*d^9 + 2648*a^9*b^7*c^7*d^7 - 1088*a^9*b^7*c^9*d
^5 + 216*a^10*b^6*c^2*d^12 + 100*a^10*b^6*c^4*d^10 - 1336*a^10*b^6*c^6*d^8 + 1056*a^10*b^6*c^8*d^6 + 180*a^11*
b^5*c^3*d^11 + 248*a^11*b^5*c^5*d^9 - 400*a^11*b^5*c^7*d^7 - 60*a^12*b^4*c^2*d^12 + 248*a^12*b^4*c^4*d^10 - 14
8*a^12*b^4*c^6*d^8 - 184*a^13*b^3*c^3*d^11 + 172*a^13*b^3*c^5*d^9 + 24*a^14*b^2*c^2*d^12 - 44*a^14*b^2*c^4*d^1
0 - a*b^15*c^13*d))/(a^17*d^13 - b^17*c^13 + 4*a^2*b^15*c^13 - 6*a^4*b^13*c^13 + 4*a^6*b^11*c^13 - a^8*b^9*c^1
3 + a^9*b^8*d^13 - 4*a^11*b^6*d^13 + 6*a^13*b^4*d^13 - 4*a^15*b^2*d^13 - 2*a^17*c^2*d^11 + a^17*c^4*d^9 - b^17
*c^9*d^4 + 2*b^17*c^11*d^2 + 9*a*b^16*c^8*d^5 - 18*a*b^16*c^10*d^3 - 36*a^3*b^14*c^12*d + 54*a^5*b^12*c^12*d -
36*a^7*b^10*c^12*d - 9*a^8*b^9*c*d^12 + 9*a^9*b^8*c^12*d + 36*a^10*b^7*c*d^12 - 54*a^12*b^5*c*d^12 + 36*a^14*
b^3*c*d^12 + 18*a^16*b*c^3*d^10 - 9*a^16*b*c^5*d^8 - 36*a^2*b^15*c^7*d^6 + 76*a^2*b^15*c^9*d^4 - 44*a^2*b^15*c
^11*d^2 + 84*a^3*b^14*c^6*d^7 - 204*a^3*b^14*c^8*d^5 + 156*a^3*b^14*c^10*d^3 - 126*a^4*b^13*c^5*d^8 + 396*a^4*
b^13*c^7*d^6 - 420*a^4*b^13*c^9*d^4 + 156*a^4*b^13*c^11*d^2 + 126*a^5*b^12*c^4*d^9 - 588*a^5*b^12*c^6*d^7 + 85
2*a^5*b^12*c^8*d^5 - 444*a^5*b^12*c^10*d^3 - 84...

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