\(\int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx\) [685]

Optimal result

Integrand size = 25, antiderivative size = 295 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=-\frac {\left (6 b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-9 \left (2 c^3+3 c d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-3 d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-3 d) \left (15 c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (9 d^2 \left (11 c^2+4 d^2\right )-3 b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \]

[Out]

-(2*a*b*d*(4*c^2+d^2)-b^2*c*(c^2+4*d^2)-a^2*(2*c^3+3*c*d^2))*arctan((d+c*tan(1/2*f*x+1/2*e))/(c^2-d^2)^(1/2))/
(c^2-d^2)^(7/2)/f+1/3*(-a*d+b*c)^2*cos(f*x+e)/d/(c^2-d^2)/f/(c+d*sin(f*x+e))^3-1/6*(-a*d+b*c)*(5*a*c*d+b*(c^2-
6*d^2))*cos(f*x+e)/d/(c^2-d^2)^2/f/(c+d*sin(f*x+e))^2+1/6*(a^2*d^2*(11*c^2+4*d^2)-a*b*(4*c^3*d+26*c*d^3)-b^2*(
c^4-10*c^2*d^2-6*d^4))*cos(f*x+e)/d/(c^2-d^2)^3/f/(c+d*sin(f*x+e))

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.03, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2869, 2833, 12, 2739, 632, 210} \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=-\frac {\left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \arctan \left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\left (5 a c d+b \left (c^2-6 d^2\right )\right ) (b c-a d) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2} \]

[In]

Int[(a + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]

[Out]

-(((2*a*b*d*(4*c^2 + d^2) - b^2*c*(c^2 + 4*d^2) - a^2*(2*c^3 + 3*c*d^2))*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[
c^2 - d^2]])/((c^2 - d^2)^(7/2)*f)) + ((b*c - a*d)^2*Cos[e + f*x])/(3*d*(c^2 - d^2)*f*(c + d*Sin[e + f*x])^3)
- ((b*c - a*d)*(5*a*c*d + b*(c^2 - 6*d^2))*Cos[e + f*x])/(6*d*(c^2 - d^2)^2*f*(c + d*Sin[e + f*x])^2) + ((a^2*
d^2*(11*c^2 + 4*d^2) - a*b*(4*c^3*d + 26*c*d^3) - b^2*(c^4 - 10*c^2*d^2 - 6*d^4))*Cos[e + f*x])/(6*d*(c^2 - d^
2)^3*f*(c + d*Sin[e + f*x]))

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, 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 2833

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

Rule 2869

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

Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {\int \frac {3 d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\left (4 a b c d-2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d \left (c^2-d^2\right )} \\ & = \frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {\int \frac {2 d \left (10 a b c d-a^2 \left (3 c^2+2 d^2\right )-b^2 \left (2 c^2+3 d^2\right )\right )-(b c-a d) \left (b c^2+5 a c d-6 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 d \left (c^2-d^2\right )^2} \\ & = \frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac {\int -\frac {3 d \left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right )}{c+d \sin (e+f x)} \, dx}{6 d \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c 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 )}{\left (c^2-d^2\right )^3 f} \\ & = \frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac {\left (2 \left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\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 )}{\left (c^2-d^2\right )^3 f} \\ & = -\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.34 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.08 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {\frac {12 \left (\left (18+b^2\right ) c^3-24 b c^2 d+\left (27+4 b^2\right ) c d^2-6 b d^3\right ) \arctan \left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac {\cos (e+f x) \left (-72 b c^5+324 c^4 d+25 b^2 c^4 d-132 b c^3 d^2+9 c^2 d^3+14 b^2 c^2 d^3-66 b c d^4+72 d^5+6 b^2 d^5+d \left (-9 d^2 \left (11 c^2+4 d^2\right )+6 b \left (2 c^3 d+13 c d^3\right )+b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (2 (e+f x))-6 \left (-9 c d^2 \left (9 c^2+d^2\right )+b^2 \left (c^5-9 c^3 d^2-2 c d^4\right )+6 b \left (2 c^4 d+9 c^2 d^3-d^5\right )\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^3 (c+d \sin (e+f x))^3}}{12 f} \]

[In]

Integrate[(3 + b*Sin[e + f*x])^2/(c + d*Sin[e + f*x])^4,x]

[Out]

((12*((18 + b^2)*c^3 - 24*b*c^2*d + (27 + 4*b^2)*c*d^2 - 6*b*d^3)*ArcTan[(d + c*Tan[(e + f*x)/2])/Sqrt[c^2 - d
^2]])/(c^2 - d^2)^(7/2) + (Cos[e + f*x]*(-72*b*c^5 + 324*c^4*d + 25*b^2*c^4*d - 132*b*c^3*d^2 + 9*c^2*d^3 + 14
*b^2*c^2*d^3 - 66*b*c*d^4 + 72*d^5 + 6*b^2*d^5 + d*(-9*d^2*(11*c^2 + 4*d^2) + 6*b*(2*c^3*d + 13*c*d^3) + b^2*(
c^4 - 10*c^2*d^2 - 6*d^4))*Cos[2*(e + f*x)] - 6*(-9*c*d^2*(9*c^2 + d^2) + b^2*(c^5 - 9*c^3*d^2 - 2*c*d^4) + 6*
b*(2*c^4*d + 9*c^2*d^3 - d^5))*Sin[e + f*x]))/((c^2 - d^2)^3*(c + d*Sin[e + f*x])^3))/(12*f)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(932\) vs. \(2(294)=588\).

Time = 2.64 (sec) , antiderivative size = 933, normalized size of antiderivative = 3.16

method	result	size
derivativedivides	\(\frac {\frac {\frac {\left (9 a^{2} c^{4} d^{2}-6 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-8 a b \,c^{5} d -2 a b \,c^{3} d^{3}+b^{2} c^{6}+4 b^{2} c^{4} d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (6 a^{2} c^{6} d +27 a^{2} c^{4} d^{3}-12 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-4 a b \,c^{7}-28 a b \,c^{5} d^{2}-22 a b \,c^{3} d^{4}+4 a b c \,d^{6}+5 b^{2} c^{6} d +20 b^{2} c^{4} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) c^{2}}+\frac {2 d \left (54 a^{2} c^{6} d +21 a^{2} c^{4} d^{3}-4 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-36 a b \,c^{7}-84 a b \,c^{5} d^{2}-34 a b \,c^{3} d^{4}+4 a b c \,d^{6}+39 b^{2} c^{6} d +32 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (6 a^{2} c^{6} d +20 a^{2} c^{4} d^{3}-3 a^{2} c^{2} d^{5}+2 a^{2} d^{7}-4 a b \,c^{7}-20 a b \,c^{5} d^{2}-28 a b \,c^{3} d^{4}+2 a b c \,d^{6}+4 b^{2} c^{6} d +17 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (27 a^{2} c^{4} d^{2}-4 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-16 a b \,c^{5} d -38 a b \,c^{3} d^{3}+4 a b c \,d^{5}-b^{2} c^{6}+22 b^{2} c^{4} d^{2}+4 b^{2} c^{2} d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (18 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-12 a b \,c^{5}-20 a b \,c^{3} d^{2}+2 a b c \,d^{4}+13 b^{2} c^{4} d +2 b^{2} c^{2} d^{3}\right )}{6 c^{6}-18 c^{4} d^{2}+18 c^{2} d^{4}-6 d^{6}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{3}}+\frac {\left (2 a^{2} c^{3}+3 a^{2} c \,d^{2}-8 a b \,c^{2} d -2 a b \,d^{3}+b^{2} c^{3}+4 b^{2} c \,d^{2}\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^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(933\)
default	\(\frac {\frac {\frac {\left (9 a^{2} c^{4} d^{2}-6 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-8 a b \,c^{5} d -2 a b \,c^{3} d^{3}+b^{2} c^{6}+4 b^{2} c^{4} d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (6 a^{2} c^{6} d +27 a^{2} c^{4} d^{3}-12 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-4 a b \,c^{7}-28 a b \,c^{5} d^{2}-22 a b \,c^{3} d^{4}+4 a b c \,d^{6}+5 b^{2} c^{6} d +20 b^{2} c^{4} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) c^{2}}+\frac {2 d \left (54 a^{2} c^{6} d +21 a^{2} c^{4} d^{3}-4 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-36 a b \,c^{7}-84 a b \,c^{5} d^{2}-34 a b \,c^{3} d^{4}+4 a b c \,d^{6}+39 b^{2} c^{6} d +32 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (6 a^{2} c^{6} d +20 a^{2} c^{4} d^{3}-3 a^{2} c^{2} d^{5}+2 a^{2} d^{7}-4 a b \,c^{7}-20 a b \,c^{5} d^{2}-28 a b \,c^{3} d^{4}+2 a b c \,d^{6}+4 b^{2} c^{6} d +17 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (27 a^{2} c^{4} d^{2}-4 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-16 a b \,c^{5} d -38 a b \,c^{3} d^{3}+4 a b c \,d^{5}-b^{2} c^{6}+22 b^{2} c^{4} d^{2}+4 b^{2} c^{2} d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (18 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-12 a b \,c^{5}-20 a b \,c^{3} d^{2}+2 a b c \,d^{4}+13 b^{2} c^{4} d +2 b^{2} c^{2} d^{3}\right )}{6 c^{6}-18 c^{4} d^{2}+18 c^{2} d^{4}-6 d^{6}}}{{\left (\left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right ) c +2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )}^{3}}+\frac {\left (2 a^{2} c^{3}+3 a^{2} c \,d^{2}-8 a b \,c^{2} d -2 a b \,d^{3}+b^{2} c^{3}+4 b^{2} c \,d^{2}\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^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {c^{2}-d^{2}}}}{f}\)	\(933\)
risch	\(\text {Expression too large to display}\)	\(1942\)

[In]

int((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x,method=_RETURNVERBOSE)

[Out]

1/f*(2*(1/2*(9*a^2*c^4*d^2-6*a^2*c^2*d^4+2*a^2*d^6-8*a*b*c^5*d-2*a*b*c^3*d^3+b^2*c^6+4*b^2*c^4*d^2)/c/(c^6-3*c
^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^5+1/2*(6*a^2*c^6*d+27*a^2*c^4*d^3-12*a^2*c^2*d^5+4*a^2*d^7-4*a*b*c^7-
28*a*b*c^5*d^2-22*a*b*c^3*d^4+4*a*b*c*d^6+5*b^2*c^6*d+20*b^2*c^4*d^3)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)/c^2*tan(1/
2*f*x+1/2*e)^4+1/3/c^3*d*(54*a^2*c^6*d+21*a^2*c^4*d^3-4*a^2*c^2*d^5+4*a^2*d^7-36*a*b*c^7-84*a*b*c^5*d^2-34*a*b
*c^3*d^4+4*a*b*c*d^6+39*b^2*c^6*d+32*b^2*c^4*d^3+4*b^2*c^2*d^5)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*
e)^3+1/c^2*(6*a^2*c^6*d+20*a^2*c^4*d^3-3*a^2*c^2*d^5+2*a^2*d^7-4*a*b*c^7-20*a*b*c^5*d^2-28*a*b*c^3*d^4+2*a*b*c
*d^6+4*b^2*c^6*d+17*b^2*c^4*d^3+4*b^2*c^2*d^5)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)^2+1/2*(27*a^2*
c^4*d^2-4*a^2*c^2*d^4+2*a^2*d^6-16*a*b*c^5*d-38*a*b*c^3*d^3+4*a*b*c*d^5-b^2*c^6+22*b^2*c^4*d^2+4*b^2*c^2*d^4)/
c/(c^6-3*c^4*d^2+3*c^2*d^4-d^6)*tan(1/2*f*x+1/2*e)+1/6*(18*a^2*c^4*d-5*a^2*c^2*d^3+2*a^2*d^5-12*a*b*c^5-20*a*b
*c^3*d^2+2*a*b*c*d^4+13*b^2*c^4*d+2*b^2*c^2*d^3)/(c^6-3*c^4*d^2+3*c^2*d^4-d^6))/(tan(1/2*f*x+1/2*e)^2*c+2*d*ta
n(1/2*f*x+1/2*e)+c)^3+(2*a^2*c^3+3*a^2*c*d^2-8*a*b*c^2*d-2*a*b*d^3+b^2*c^3+4*b^2*c*d^2)/(c^6-3*c^4*d^2+3*c^2*d
^4-d^6)/(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)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 820 vs. \(2 (291) = 582\).

Time = 0.37 (sec) , antiderivative size = 1724, normalized size of antiderivative = 5.84 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x, algorithm="fricas")

[Out]

[-1/12*(2*(b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11*(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2
)*c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos(f*x + e)^3 - 6*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5
+ 2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2*b^2)*c*d^6)*cos(f*x + e)*sin(f*x
+ e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2
+ 4*b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3*a^2 + 4*b^2)*c^2*d^4)*cos(f*x +
e)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*
a^2 + 4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5)*cos(f*x + e)^
2)*sin(f*x + e))*sqrt(-c^2 + d^2)*log(((2*c^2 - d^2)*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d^2 + 2*(c*co
s(f*x + e)*sin(f*x + e) + d*cos(f*x + e))*sqrt(-c^2 + d^2))/(d^2*cos(f*x + e)^2 - 2*c*d*sin(f*x + e) - c^2 - d
^2)) - 12*(2*a*b*c^7 + 2*a*b*c^5*d^2 + 2*a^2*c^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^
2 + b^2)*d^7)*cos(f*x + e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11
- c^9*d^2 - 6*c^7*d^4 + 14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9
+ d^11)*f*cos(f*x + e)^2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e))
, -1/6*((b^2*c^6*d + 4*a*b*c^5*d^2 + 22*a*b*c^3*d^4 - 26*a*b*c*d^6 - 11*(a^2 + b^2)*c^4*d^3 + (7*a^2 + 4*b^2)*
c^2*d^5 + 2*(2*a^2 + 3*b^2)*d^7)*cos(f*x + e)^3 - 3*(b^2*c^7 + 4*a*b*c^6*d + 14*a*b*c^4*d^3 - 20*a*b*c^2*d^5 +
2*a*b*d^7 - (9*a^2 + 10*b^2)*c^5*d^2 + (8*a^2 + 7*b^2)*c^3*d^4 + (a^2 + 2*b^2)*c*d^6)*cos(f*x + e)*sin(f*x +
e) + 3*(8*a*b*c^5*d + 26*a*b*c^3*d^3 + 6*a*b*c*d^5 - (2*a^2 + b^2)*c^6 - (9*a^2 + 7*b^2)*c^4*d^2 - 3*(3*a^2 +
4*b^2)*c^2*d^4 - 3*(8*a*b*c^3*d^3 + 2*a*b*c*d^5 - (2*a^2 + b^2)*c^4*d^2 - (3*a^2 + 4*b^2)*c^2*d^4)*cos(f*x + e
)^2 + (24*a*b*c^4*d^2 + 14*a*b*c^2*d^4 + 2*a*b*d^6 - 3*(2*a^2 + b^2)*c^5*d - (11*a^2 + 13*b^2)*c^3*d^3 - (3*a^
2 + 4*b^2)*c*d^5 - (8*a*b*c^2*d^4 + 2*a*b*d^6 - (2*a^2 + b^2)*c^3*d^3 - (3*a^2 + 4*b^2)*c*d^5)*cos(f*x + e)^2)
*sin(f*x + e))*sqrt(c^2 - d^2)*arctan(-(c*sin(f*x + e) + d)/(sqrt(c^2 - d^2)*cos(f*x + e))) - 6*(2*a*b*c^7 + 2
*a*b*c^5*d^2 + 2*a^2*c^4*d^3 + b^2*c^2*d^5 - 4*a*b*c*d^6 - (3*a^2 + 2*b^2)*c^6*d + (a^2 + b^2)*d^7)*cos(f*x +
e))/(3*(c^9*d^2 - 4*c^7*d^4 + 6*c^5*d^6 - 4*c^3*d^8 + c*d^10)*f*cos(f*x + e)^2 - (c^11 - c^9*d^2 - 6*c^7*d^4 +
14*c^5*d^6 - 11*c^3*d^8 + 3*c*d^10)*f + ((c^8*d^3 - 4*c^6*d^5 + 6*c^4*d^7 - 4*c^2*d^9 + d^11)*f*cos(f*x + e)^
2 - (3*c^10*d - 11*c^8*d^3 + 14*c^6*d^5 - 6*c^4*d^7 - c^2*d^9 + d^11)*f)*sin(f*x + e))]

Sympy [F(-1)]

Timed out. \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Timed out} \]

[In]

integrate((a+b*sin(f*x+e))**2/(c+d*sin(f*x+e))**4,x)

[Out]

Timed out

Maxima [F(-2)]

Exception generated. \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1264 vs. \(2 (291) = 582\).

Time = 0.36 (sec) , antiderivative size = 1264, normalized size of antiderivative = 4.28 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\text {Too large to display} \]

[In]

integrate((a+b*sin(f*x+e))^2/(c+d*sin(f*x+e))^4,x, algorithm="giac")

[Out]

1/3*(3*(2*a^2*c^3 + b^2*c^3 - 8*a*b*c^2*d + 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2*a*b*d^3)*(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)))/((c^6 - 3*c^4*d^2 + 3*c^2*d^4 - d^6)*sqr
t(c^2 - d^2)) + (3*b^2*c^8*tan(1/2*f*x + 1/2*e)^5 - 24*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^5 + 27*a^2*c^6*d^2*tan(1
/2*f*x + 1/2*e)^5 + 12*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^5 - 6*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^5 - 18*a^2*c^4*
d^4*tan(1/2*f*x + 1/2*e)^5 + 6*a^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^5 - 12*a*b*c^8*tan(1/2*f*x + 1/2*e)^4 + 18*a^2
*c^7*d*tan(1/2*f*x + 1/2*e)^4 + 15*b^2*c^7*d*tan(1/2*f*x + 1/2*e)^4 - 84*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^4 +
81*a^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 + 60*b^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^4 - 66*a*b*c^4*d^4*tan(1/2*f*x + 1
/2*e)^4 - 36*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^4 + 12*a*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^4 + 12*a^2*c*d^7*tan(1/2
*f*x + 1/2*e)^4 - 72*a*b*c^7*d*tan(1/2*f*x + 1/2*e)^3 + 108*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e)^3 + 78*b^2*c^6*d^
2*tan(1/2*f*x + 1/2*e)^3 - 168*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e)^3 + 42*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 + 64
*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e)^3 - 68*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e)^3 - 8*a^2*c^2*d^6*tan(1/2*f*x + 1/2*
e)^3 + 8*b^2*c^2*d^6*tan(1/2*f*x + 1/2*e)^3 + 8*a*b*c*d^7*tan(1/2*f*x + 1/2*e)^3 + 8*a^2*d^8*tan(1/2*f*x + 1/2
*e)^3 - 24*a*b*c^8*tan(1/2*f*x + 1/2*e)^2 + 36*a^2*c^7*d*tan(1/2*f*x + 1/2*e)^2 + 24*b^2*c^7*d*tan(1/2*f*x + 1
/2*e)^2 - 120*a*b*c^6*d^2*tan(1/2*f*x + 1/2*e)^2 + 120*a^2*c^5*d^3*tan(1/2*f*x + 1/2*e)^2 + 102*b^2*c^5*d^3*ta
n(1/2*f*x + 1/2*e)^2 - 168*a*b*c^4*d^4*tan(1/2*f*x + 1/2*e)^2 - 18*a^2*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 24*b^2
*c^3*d^5*tan(1/2*f*x + 1/2*e)^2 + 12*a*b*c^2*d^6*tan(1/2*f*x + 1/2*e)^2 + 12*a^2*c*d^7*tan(1/2*f*x + 1/2*e)^2
- 3*b^2*c^8*tan(1/2*f*x + 1/2*e) - 48*a*b*c^7*d*tan(1/2*f*x + 1/2*e) + 81*a^2*c^6*d^2*tan(1/2*f*x + 1/2*e) + 6
6*b^2*c^6*d^2*tan(1/2*f*x + 1/2*e) - 114*a*b*c^5*d^3*tan(1/2*f*x + 1/2*e) - 12*a^2*c^4*d^4*tan(1/2*f*x + 1/2*e
) + 12*b^2*c^4*d^4*tan(1/2*f*x + 1/2*e) + 12*a*b*c^3*d^5*tan(1/2*f*x + 1/2*e) + 6*a^2*c^2*d^6*tan(1/2*f*x + 1/
2*e) - 12*a*b*c^8 + 18*a^2*c^7*d + 13*b^2*c^7*d - 20*a*b*c^6*d^2 - 5*a^2*c^5*d^3 + 2*b^2*c^5*d^3 + 2*a*b*c^4*d
^4 + 2*a^2*c^3*d^5)/((c^9 - 3*c^7*d^2 + 3*c^5*d^4 - c^3*d^6)*(c*tan(1/2*f*x + 1/2*e)^2 + 2*d*tan(1/2*f*x + 1/2
*e) + c)^3))/f

Mupad [B] (verification not implemented)

Time = 11.92 (sec) , antiderivative size = 1220, normalized size of antiderivative = 4.14 \[ \int \frac {(3+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx=\frac {\frac {18\,a^2\,c^4\,d-5\,a^2\,c^2\,d^3+2\,a^2\,d^5-12\,a\,b\,c^5-20\,a\,b\,c^3\,d^2+2\,a\,b\,c\,d^4+13\,b^2\,c^4\,d+2\,b^2\,c^2\,d^3}{3\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,a^2\,c^6\,d+27\,a^2\,c^4\,d^3-12\,a^2\,c^2\,d^5+4\,a^2\,d^7-4\,a\,b\,c^7-28\,a\,b\,c^5\,d^2-22\,a\,b\,c^3\,d^4+4\,a\,b\,c\,d^6+5\,b^2\,c^6\,d+20\,b^2\,c^4\,d^3\right )}{c^2\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (27\,a^2\,c^4\,d^2-4\,a^2\,c^2\,d^4+2\,a^2\,d^6-16\,a\,b\,c^5\,d-38\,a\,b\,c^3\,d^3+4\,a\,b\,c\,d^5-b^2\,c^6+22\,b^2\,c^4\,d^2+4\,b^2\,c^2\,d^4\right )}{c\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (6\,a^2\,c^6\,d+20\,a^2\,c^4\,d^3-3\,a^2\,c^2\,d^5+2\,a^2\,d^7-4\,a\,b\,c^7-20\,a\,b\,c^5\,d^2-28\,a\,b\,c^3\,d^4+2\,a\,b\,c\,d^6+4\,b^2\,c^6\,d+17\,b^2\,c^4\,d^3+4\,b^2\,c^2\,d^5\right )}{c^2\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (9\,a^2\,c^4\,d^2-6\,a^2\,c^2\,d^4+2\,a^2\,d^6-8\,a\,b\,c^5\,d-2\,a\,b\,c^3\,d^3+b^2\,c^6+4\,b^2\,c^4\,d^2\right )}{c\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (18\,a^2\,c^4\,d-5\,a^2\,c^2\,d^3+2\,a^2\,d^5-12\,a\,b\,c^5-20\,a\,b\,c^3\,d^2+2\,a\,b\,c\,d^4+13\,b^2\,c^4\,d+2\,b^2\,c^2\,d^3\right )}{3\,c^3\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}}{f\,\left (c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (12\,c^2\,d+8\,d^3\right )+c^3+6\,c^2\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,c^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )}+\frac {\mathrm {atan}\left (\frac {\left (\frac {c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2\right )}{{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}}+\frac {\left (2\,c^6\,d-6\,c^4\,d^3+6\,c^2\,d^5-2\,d^7\right )\,\left (2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2\right )}{2\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}\right )\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}{2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2}\right )\,\left (2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}} \]

[In]

int((a + b*sin(e + f*x))^2/(c + d*sin(e + f*x))^4,x)

[Out]

((2*a^2*d^5 + 18*a^2*c^4*d + 13*b^2*c^4*d - 5*a^2*c^2*d^3 + 2*b^2*c^2*d^3 - 12*a*b*c^5 + 2*a*b*c*d^4 - 20*a*b*
c^3*d^2)/(3*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (tan(e/2 + (f*x)/2)^4*(4*a^2*d^7 + 6*a^2*c^6*d + 5*b^2*c^6*
d - 12*a^2*c^2*d^5 + 27*a^2*c^4*d^3 + 20*b^2*c^4*d^3 - 4*a*b*c^7 + 4*a*b*c*d^6 - 22*a*b*c^3*d^4 - 28*a*b*c^5*d
^2))/(c^2*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (tan(e/2 + (f*x)/2)*(2*a^2*d^6 - b^2*c^6 - 4*a^2*c^2*d^4 + 27
*a^2*c^4*d^2 + 4*b^2*c^2*d^4 + 22*b^2*c^4*d^2 + 4*a*b*c*d^5 - 16*a*b*c^5*d - 38*a*b*c^3*d^3))/(c*(c^6 - d^6 +
3*c^2*d^4 - 3*c^4*d^2)) + (2*tan(e/2 + (f*x)/2)^2*(2*a^2*d^7 + 6*a^2*c^6*d + 4*b^2*c^6*d - 3*a^2*c^2*d^5 + 20*
a^2*c^4*d^3 + 4*b^2*c^2*d^5 + 17*b^2*c^4*d^3 - 4*a*b*c^7 + 2*a*b*c*d^6 - 28*a*b*c^3*d^4 - 20*a*b*c^5*d^2))/(c^
2*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (tan(e/2 + (f*x)/2)^5*(2*a^2*d^6 + b^2*c^6 - 6*a^2*c^2*d^4 + 9*a^2*c^
4*d^2 + 4*b^2*c^4*d^2 - 8*a*b*c^5*d - 2*a*b*c^3*d^3))/(c*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)) + (2*d*tan(e/2 +
(f*x)/2)^3*(3*c^2 + 2*d^2)*(2*a^2*d^5 + 18*a^2*c^4*d + 13*b^2*c^4*d - 5*a^2*c^2*d^3 + 2*b^2*c^2*d^3 - 12*a*b*
c^5 + 2*a*b*c*d^4 - 20*a*b*c^3*d^2))/(3*c^3*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)))/(f*(c^3*tan(e/2 + (f*x)/2)^6
+ tan(e/2 + (f*x)/2)^2*(12*c*d^2 + 3*c^3) + tan(e/2 + (f*x)/2)^4*(12*c*d^2 + 3*c^3) + tan(e/2 + (f*x)/2)^3*(1
2*c^2*d + 8*d^3) + c^3 + 6*c^2*d*tan(e/2 + (f*x)/2) + 6*c^2*d*tan(e/2 + (f*x)/2)^5)) + (atan((((c*tan(e/2 + (f
*x)/2)*(2*a^2*c^3 + b^2*c^3 + 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2*a*b*d^3 - 8*a*b*c^2*d))/((c + d)^(7/2)*(c - d)^(7/
2)) + ((2*c^6*d - 2*d^7 + 6*c^2*d^5 - 6*c^4*d^3)*(2*a^2*c^3 + b^2*c^3 + 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2*a*b*d^3
- 8*a*b*c^2*d))/(2*(c + d)^(7/2)*(c - d)^(7/2)*(c^6 - d^6 + 3*c^2*d^4 - 3*c^4*d^2)))*(c^6 - d^6 + 3*c^2*d^4 -
3*c^4*d^2))/(2*a^2*c^3 + b^2*c^3 + 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2*a*b*d^3 - 8*a*b*c^2*d))*(2*a^2*c^3 + b^2*c^3
+ 3*a^2*c*d^2 + 4*b^2*c*d^2 - 2*a*b*d^3 - 8*a*b*c^2*d))/(f*(c + d)^(7/2)*(c - d)^(7/2))