3.2.93 \(\int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^3} \, dx\) [193]

Optimal. Leaf size=237 \[ \frac {a^2 x}{c^3}-\frac {\left (3 b^2 c^4 d-2 a b c^3 \left (2 c^2+d^2\right )+a^2 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 (c-d)^{5/2} (c+d)^{5/2} f}-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (3 a d \left (2 c^2-d^2\right )-b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))} \]

[Out]

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

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Rubi [A]
time = 0.56, antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4026, 3067, 3100, 2814, 2738, 214} \begin {gather*} -\frac {\left (a^2 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )-2 a b c^3 \left (2 c^2+d^2\right )+3 b^2 c^4 d\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 f (c-d)^{5/2} (c+d)^{5/2}}+\frac {a^2 x}{c^3}-\frac {(b c-a d) \left (3 a d \left (2 c^2-d^2\right )-b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{2 c^2 f \left (c^2-d^2\right )^2 (c \cos (e+f x)+d)}-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 f \left (c^2-d^2\right ) (c \cos (e+f x)+d)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 214

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

Rule 2738

Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[2*(e/d), Subst[Int[1/(a + b + (a - b)*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 2814

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

Rule 3067

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(B*c - A*d)*(b*c - a*d)^2*Cos[e + f*x]*((c + d*Sin[e + f*x])^(n + 1)/(
f*d^2*(n + 1)*(c^2 - d^2))), x] - Dist[1/(d^2*(n + 1)*(c^2 - d^2)), Int[(c + d*Sin[e + f*x])^(n + 1)*Simp[d*(n
+ 1)*(B*(b*c - a*d)^2 - A*d*(a^2*c + b^2*c - 2*a*b*d)) - ((B*c - A*d)*(a^2*d^2*(n + 2) + b^2*(c^2 + d^2*(n +
1))) + 2*a*b*d*(A*c*d*(n + 2) - B*(c^2 + d^2*(n + 1))))*Sin[e + f*x] - b^2*B*d*(n + 1)*(c^2 - d^2)*Sin[e + f*x
]^2, 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] && LtQ[n, -1]

Rule 3100

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*sin[(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)/(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*(a*A - b*B +
a*C)*(m + 1) - (A*b^2 - a*b*B + a^2*C + b*(A*b - a*B + b*C)*(m + 1))*Sin[e + f*x], x], x], x] /; FreeQ[{a, b,
e, f, A, B, C}, x] && LtQ[m, -1] && NeQ[a^2 - b^2, 0]

Rule 4026

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

Rubi steps

\begin {align*} \int \frac {(a+b \sec (e+f x))^2}{(c+d \sec (e+f x))^3} \, dx &=\int \frac {\cos (e+f x) (b+a \cos (e+f x))^2}{(d+c \cos (e+f x))^3} \, dx\\ &=-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}-\frac {\int \frac {-2 c (b c-a d)^2+\left (b^2 c^2 d-2 a b c \left (2 c^2-d^2\right )+a^2 \left (2 c^2 d-d^3\right )\right ) \cos (e+f x)-2 a^2 c \left (c^2-d^2\right ) \cos ^2(e+f x)}{(d+c \cos (e+f x))^2} \, dx}{2 c^2 \left (c^2-d^2\right )}\\ &=-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (3 a d \left (2 c^2-d^2\right )-b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}-\frac {\int \frac {-c^2 (b c-a d) \left (4 a c^2-3 b c d-a d^2\right )-2 a^2 c \left (c^2-d^2\right )^2 \cos (e+f x)}{d+c \cos (e+f x)} \, dx}{2 c^3 \left (c^2-d^2\right )^2}\\ &=\frac {a^2 x}{c^3}-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (3 a d \left (2 c^2-d^2\right )-b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}-\frac {\left (3 b^2 c^4 d-2 a b c^3 \left (2 c^2+d^2\right )+a^2 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )\right ) \int \frac {1}{d+c \cos (e+f x)} \, dx}{2 c^3 \left (c^2-d^2\right )^2}\\ &=\frac {a^2 x}{c^3}-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (3 a d \left (2 c^2-d^2\right )-b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}-\frac {\left (3 b^2 c^4 d-2 a b c^3 \left (2 c^2+d^2\right )+a^2 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )\right ) \text {Subst}\left (\int \frac {1}{c+d+(-c+d) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{c^3 \left (c^2-d^2\right )^2 f}\\ &=\frac {a^2 x}{c^3}+\frac {\left (4 a b c^5-6 a^2 c^4 d-3 b^2 c^4 d+2 a b c^3 d^2+5 a^2 c^2 d^3-2 a^2 d^5\right ) \tanh ^{-1}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{c^3 (c-d)^{5/2} (c+d)^{5/2} f}-\frac {d (b c-a d)^2 \sin (e+f x)}{2 c^2 \left (c^2-d^2\right ) f (d+c \cos (e+f x))^2}-\frac {(b c-a d) \left (3 a d \left (2 c^2-d^2\right )-b c \left (2 c^2+d^2\right )\right ) \sin (e+f x)}{2 c^2 \left (c^2-d^2\right )^2 f (d+c \cos (e+f x))}\\ \end {align*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(493\) vs. \(2(237)=474\).
time = 2.05, size = 493, normalized size = 2.08 \begin {gather*} \frac {(d+c \cos (e+f x)) \sec (e+f x) (a+b \sec (e+f x))^2 \left (\frac {4 \left (3 b^2 c^4 d-2 a b c^3 \left (2 c^2+d^2\right )+a^2 \left (6 c^4 d-5 c^2 d^3+2 d^5\right )\right ) \tanh ^{-1}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) (d+c \cos (e+f x))^2}{\left (c^2-d^2\right )^{5/2}}+\frac {2 a^2 c^6 e-6 a^2 c^2 d^4 e+4 a^2 d^6 e+2 a^2 c^6 f x-6 a^2 c^2 d^4 f x+4 a^2 d^6 f x+8 a^2 c d \left (c^2-d^2\right )^2 (e+f x) \cos (e+f x)+2 a^2 c^2 \left (c^2-d^2\right )^2 (e+f x) \cos (2 (e+f x))+2 b^2 c^5 d \sin (e+f x)-12 a b c^4 d^2 \sin (e+f x)+10 a^2 c^3 d^3 \sin (e+f x)+4 b^2 c^3 d^3 \sin (e+f x)-4 a^2 c d^5 \sin (e+f x)+2 b^2 c^6 \sin (2 (e+f x))-8 a b c^5 d \sin (2 (e+f x))+6 a^2 c^4 d^2 \sin (2 (e+f x))+b^2 c^4 d^2 \sin (2 (e+f x))+2 a b c^3 d^3 \sin (2 (e+f x))-3 a^2 c^2 d^4 \sin (2 (e+f x))}{\left (c^2-d^2\right )^2}\right )}{4 c^3 f (b+a \cos (e+f x))^2 (c+d \sec (e+f x))^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((d + c*Cos[e + f*x])*Sec[e + f*x]*(a + b*Sec[e + f*x])^2*((4*(3*b^2*c^4*d - 2*a*b*c^3*(2*c^2 + d^2) + a^2*(6*
c^4*d - 5*c^2*d^3 + 2*d^5))*ArcTanh[((-c + d)*Tan[(e + f*x)/2])/Sqrt[c^2 - d^2]]*(d + c*Cos[e + f*x])^2)/(c^2
- d^2)^(5/2) + (2*a^2*c^6*e - 6*a^2*c^2*d^4*e + 4*a^2*d^6*e + 2*a^2*c^6*f*x - 6*a^2*c^2*d^4*f*x + 4*a^2*d^6*f*
x + 8*a^2*c*d*(c^2 - d^2)^2*(e + f*x)*Cos[e + f*x] + 2*a^2*c^2*(c^2 - d^2)^2*(e + f*x)*Cos[2*(e + f*x)] + 2*b^
2*c^5*d*Sin[e + f*x] - 12*a*b*c^4*d^2*Sin[e + f*x] + 10*a^2*c^3*d^3*Sin[e + f*x] + 4*b^2*c^3*d^3*Sin[e + f*x]
- 4*a^2*c*d^5*Sin[e + f*x] + 2*b^2*c^6*Sin[2*(e + f*x)] - 8*a*b*c^5*d*Sin[2*(e + f*x)] + 6*a^2*c^4*d^2*Sin[2*(
e + f*x)] + b^2*c^4*d^2*Sin[2*(e + f*x)] + 2*a*b*c^3*d^3*Sin[2*(e + f*x)] - 3*a^2*c^2*d^4*Sin[2*(e + f*x)])/(c
^2 - d^2)^2))/(4*c^3*f*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^3)

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Maple [A]
time = 0.40, size = 386, normalized size = 1.63

method	result	size
derivativedivides	\(\frac {\frac {\frac {2 \left (-\frac {\left (6 a^{2} c^{2} d^{2}+a^{2} c \,d^{3}-2 a^{2} d^{4}-8 a b \,c^{3} d -2 a b \,c^{2} d^{2}+2 b^{2} c^{4}+b^{2} c^{3} d +2 b^{2} c^{2} d^{2}\right ) c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c -d \right ) \left (c^{2}+2 c d +d^{2}\right )}+\frac {c \left (6 a^{2} c^{2} d^{2}-a^{2} c \,d^{3}-2 a^{2} d^{4}-8 a b \,c^{3} d +2 a b \,c^{2} d^{2}+2 b^{2} c^{4}-b^{2} c^{3} d +2 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c -d \right )^{2}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (6 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-4 a b \,c^{5}-2 a b \,c^{3} d^{2}+3 b^{2} c^{4} d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{3}}+\frac {2 a^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{3}}}{f}\)	\(386\)
default	\(\frac {\frac {\frac {2 \left (-\frac {\left (6 a^{2} c^{2} d^{2}+a^{2} c \,d^{3}-2 a^{2} d^{4}-8 a b \,c^{3} d -2 a b \,c^{2} d^{2}+2 b^{2} c^{4}+b^{2} c^{3} d +2 b^{2} c^{2} d^{2}\right ) c \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 \left (c -d \right ) \left (c^{2}+2 c d +d^{2}\right )}+\frac {c \left (6 a^{2} c^{2} d^{2}-a^{2} c \,d^{3}-2 a^{2} d^{4}-8 a b \,c^{3} d +2 a b \,c^{2} d^{2}+2 b^{2} c^{4}-b^{2} c^{3} d +2 b^{2} c^{2} d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c -d \right )^{2}}\right )}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-d \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-c -d \right )^{2}}-\frac {\left (6 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-4 a b \,c^{5}-2 a b \,c^{3} d^{2}+3 b^{2} c^{4} d \right ) \arctanh \left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{4}-2 c^{2} d^{2}+d^{4}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{c^{3}}+\frac {2 a^{2} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{3}}}{f}\)	\(386\)
risch	\(\text {Expression too large to display}\)	\(1526\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/f*(2/c^3*((-1/2*(6*a^2*c^2*d^2+a^2*c*d^3-2*a^2*d^4-8*a*b*c^3*d-2*a*b*c^2*d^2+2*b^2*c^4+b^2*c^3*d+2*b^2*c^2*d
^2)*c/(c-d)/(c^2+2*c*d+d^2)*tan(1/2*f*x+1/2*e)^3+1/2*c*(6*a^2*c^2*d^2-a^2*c*d^3-2*a^2*d^4-8*a*b*c^3*d+2*a*b*c^
2*d^2+2*b^2*c^4-b^2*c^3*d+2*b^2*c^2*d^2)/(c+d)/(c-d)^2*tan(1/2*f*x+1/2*e))/(c*tan(1/2*f*x+1/2*e)^2-d*tan(1/2*f
*x+1/2*e)^2-c-d)^2-1/2*(6*a^2*c^4*d-5*a^2*c^2*d^3+2*a^2*d^5-4*a*b*c^5-2*a*b*c^3*d^2+3*b^2*c^4*d)/(c^4-2*c^2*d^
2+d^4)/((c+d)*(c-d))^(1/2)*arctanh((c-d)*tan(1/2*f*x+1/2*e)/((c+d)*(c-d))^(1/2)))+2*a^2/c^3*arctan(tan(1/2*f*x
+1/2*e)))

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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((a+b*sec(f*x+e))^2/(c+d*sec(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*c^2-4*d^2>0)', see `assume?`
for more de

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 685 vs. \(2 (228) = 456\).
time = 3.08, size = 1433, normalized size = 6.05 \begin {gather*} \left [\frac {4 \, {\left (a^{2} c^{8} - 3 \, a^{2} c^{6} d^{2} + 3 \, a^{2} c^{4} d^{4} - a^{2} c^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} + 8 \, {\left (a^{2} c^{7} d - 3 \, a^{2} c^{5} d^{3} + 3 \, a^{2} c^{3} d^{5} - a^{2} c d^{7}\right )} f x \cos \left (f x + e\right ) + 4 \, {\left (a^{2} c^{6} d^{2} - 3 \, a^{2} c^{4} d^{4} + 3 \, a^{2} c^{2} d^{6} - a^{2} d^{8}\right )} f x - {\left (4 \, a b c^{5} d^{2} + 2 \, a b c^{3} d^{4} + 5 \, a^{2} c^{2} d^{5} - 2 \, a^{2} d^{7} - 3 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{3} + {\left (4 \, a b c^{7} + 2 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3} - 2 \, a^{2} c^{2} d^{5} - 3 \, {\left (2 \, a^{2} + b^{2}\right )} c^{6} d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (4 \, a b c^{6} d + 2 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4} - 2 \, a^{2} c d^{6} - 3 \, {\left (2 \, a^{2} + b^{2}\right )} c^{5} d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \log \left (\frac {2 \, c d \cos \left (f x + e\right ) - {\left (c^{2} - 2 \, d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, \sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )} \sin \left (f x + e\right ) + 2 \, c^{2} - d^{2}}{c^{2} \cos \left (f x + e\right )^{2} + 2 \, c d \cos \left (f x + e\right ) + d^{2}}\right ) + 2 \, {\left (b^{2} c^{7} d - 6 \, a b c^{6} d^{2} + 6 \, a b c^{4} d^{4} + 2 \, a^{2} c d^{7} + {\left (5 \, a^{2} + b^{2}\right )} c^{5} d^{3} - {\left (7 \, a^{2} + 2 \, b^{2}\right )} c^{3} d^{5} + {\left (2 \, b^{2} c^{8} - 8 \, a b c^{7} d + 10 \, a b c^{5} d^{3} - 2 \, a b c^{3} d^{5} + 3 \, a^{2} c^{2} d^{6} + {\left (6 \, a^{2} - b^{2}\right )} c^{6} d^{2} - {\left (9 \, a^{2} + b^{2}\right )} c^{4} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{4 \, {\left ({\left (c^{11} - 3 \, c^{9} d^{2} + 3 \, c^{7} d^{4} - c^{5} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{10} d - 3 \, c^{8} d^{3} + 3 \, c^{6} d^{5} - c^{4} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{9} d^{2} - 3 \, c^{7} d^{4} + 3 \, c^{5} d^{6} - c^{3} d^{8}\right )} f\right )}}, \frac {2 \, {\left (a^{2} c^{8} - 3 \, a^{2} c^{6} d^{2} + 3 \, a^{2} c^{4} d^{4} - a^{2} c^{2} d^{6}\right )} f x \cos \left (f x + e\right )^{2} + 4 \, {\left (a^{2} c^{7} d - 3 \, a^{2} c^{5} d^{3} + 3 \, a^{2} c^{3} d^{5} - a^{2} c d^{7}\right )} f x \cos \left (f x + e\right ) + 2 \, {\left (a^{2} c^{6} d^{2} - 3 \, a^{2} c^{4} d^{4} + 3 \, a^{2} c^{2} d^{6} - a^{2} d^{8}\right )} f x + {\left (4 \, a b c^{5} d^{2} + 2 \, a b c^{3} d^{4} + 5 \, a^{2} c^{2} d^{5} - 2 \, a^{2} d^{7} - 3 \, {\left (2 \, a^{2} + b^{2}\right )} c^{4} d^{3} + {\left (4 \, a b c^{7} + 2 \, a b c^{5} d^{2} + 5 \, a^{2} c^{4} d^{3} - 2 \, a^{2} c^{2} d^{5} - 3 \, {\left (2 \, a^{2} + b^{2}\right )} c^{6} d\right )} \cos \left (f x + e\right )^{2} + 2 \, {\left (4 \, a b c^{6} d + 2 \, a b c^{4} d^{3} + 5 \, a^{2} c^{3} d^{4} - 2 \, a^{2} c d^{6} - 3 \, {\left (2 \, a^{2} + b^{2}\right )} c^{5} d^{2}\right )} \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \arctan \left (-\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + c\right )}}{{\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}\right ) + {\left (b^{2} c^{7} d - 6 \, a b c^{6} d^{2} + 6 \, a b c^{4} d^{4} + 2 \, a^{2} c d^{7} + {\left (5 \, a^{2} + b^{2}\right )} c^{5} d^{3} - {\left (7 \, a^{2} + 2 \, b^{2}\right )} c^{3} d^{5} + {\left (2 \, b^{2} c^{8} - 8 \, a b c^{7} d + 10 \, a b c^{5} d^{3} - 2 \, a b c^{3} d^{5} + 3 \, a^{2} c^{2} d^{6} + {\left (6 \, a^{2} - b^{2}\right )} c^{6} d^{2} - {\left (9 \, a^{2} + b^{2}\right )} c^{4} d^{4}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (c^{11} - 3 \, c^{9} d^{2} + 3 \, c^{7} d^{4} - c^{5} d^{6}\right )} f \cos \left (f x + e\right )^{2} + 2 \, {\left (c^{10} d - 3 \, c^{8} d^{3} + 3 \, c^{6} d^{5} - c^{4} d^{7}\right )} f \cos \left (f x + e\right ) + {\left (c^{9} d^{2} - 3 \, c^{7} d^{4} + 3 \, c^{5} d^{6} - c^{3} d^{8}\right )} f\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*(4*(a^2*c^8 - 3*a^2*c^6*d^2 + 3*a^2*c^4*d^4 - a^2*c^2*d^6)*f*x*cos(f*x + e)^2 + 8*(a^2*c^7*d - 3*a^2*c^5*
d^3 + 3*a^2*c^3*d^5 - a^2*c*d^7)*f*x*cos(f*x + e) + 4*(a^2*c^6*d^2 - 3*a^2*c^4*d^4 + 3*a^2*c^2*d^6 - a^2*d^8)*
f*x - (4*a*b*c^5*d^2 + 2*a*b*c^3*d^4 + 5*a^2*c^2*d^5 - 2*a^2*d^7 - 3*(2*a^2 + b^2)*c^4*d^3 + (4*a*b*c^7 + 2*a*
b*c^5*d^2 + 5*a^2*c^4*d^3 - 2*a^2*c^2*d^5 - 3*(2*a^2 + b^2)*c^6*d)*cos(f*x + e)^2 + 2*(4*a*b*c^6*d + 2*a*b*c^4
*d^3 + 5*a^2*c^3*d^4 - 2*a^2*c*d^6 - 3*(2*a^2 + b^2)*c^5*d^2)*cos(f*x + e))*sqrt(c^2 - d^2)*log((2*c*d*cos(f*x
+ e) - (c^2 - 2*d^2)*cos(f*x + e)^2 - 2*sqrt(c^2 - d^2)*(d*cos(f*x + e) + c)*sin(f*x + e) + 2*c^2 - d^2)/(c^2
*cos(f*x + e)^2 + 2*c*d*cos(f*x + e) + d^2)) + 2*(b^2*c^7*d - 6*a*b*c^6*d^2 + 6*a*b*c^4*d^4 + 2*a^2*c*d^7 + (5
*a^2 + b^2)*c^5*d^3 - (7*a^2 + 2*b^2)*c^3*d^5 + (2*b^2*c^8 - 8*a*b*c^7*d + 10*a*b*c^5*d^3 - 2*a*b*c^3*d^5 + 3*
a^2*c^2*d^6 + (6*a^2 - b^2)*c^6*d^2 - (9*a^2 + b^2)*c^4*d^4)*cos(f*x + e))*sin(f*x + e))/((c^11 - 3*c^9*d^2 +
3*c^7*d^4 - c^5*d^6)*f*cos(f*x + e)^2 + 2*(c^10*d - 3*c^8*d^3 + 3*c^6*d^5 - c^4*d^7)*f*cos(f*x + e) + (c^9*d^2
- 3*c^7*d^4 + 3*c^5*d^6 - c^3*d^8)*f), 1/2*(2*(a^2*c^8 - 3*a^2*c^6*d^2 + 3*a^2*c^4*d^4 - a^2*c^2*d^6)*f*x*cos
(f*x + e)^2 + 4*(a^2*c^7*d - 3*a^2*c^5*d^3 + 3*a^2*c^3*d^5 - a^2*c*d^7)*f*x*cos(f*x + e) + 2*(a^2*c^6*d^2 - 3*
a^2*c^4*d^4 + 3*a^2*c^2*d^6 - a^2*d^8)*f*x + (4*a*b*c^5*d^2 + 2*a*b*c^3*d^4 + 5*a^2*c^2*d^5 - 2*a^2*d^7 - 3*(2
*a^2 + b^2)*c^4*d^3 + (4*a*b*c^7 + 2*a*b*c^5*d^2 + 5*a^2*c^4*d^3 - 2*a^2*c^2*d^5 - 3*(2*a^2 + b^2)*c^6*d)*cos(
f*x + e)^2 + 2*(4*a*b*c^6*d + 2*a*b*c^4*d^3 + 5*a^2*c^3*d^4 - 2*a^2*c*d^6 - 3*(2*a^2 + b^2)*c^5*d^2)*cos(f*x +
e))*sqrt(-c^2 + d^2)*arctan(-sqrt(-c^2 + d^2)*(d*cos(f*x + e) + c)/((c^2 - d^2)*sin(f*x + e))) + (b^2*c^7*d -
6*a*b*c^6*d^2 + 6*a*b*c^4*d^4 + 2*a^2*c*d^7 + (5*a^2 + b^2)*c^5*d^3 - (7*a^2 + 2*b^2)*c^3*d^5 + (2*b^2*c^8 -
8*a*b*c^7*d + 10*a*b*c^5*d^3 - 2*a*b*c^3*d^5 + 3*a^2*c^2*d^6 + (6*a^2 - b^2)*c^6*d^2 - (9*a^2 + b^2)*c^4*d^4)*
cos(f*x + e))*sin(f*x + e))/((c^11 - 3*c^9*d^2 + 3*c^7*d^4 - c^5*d^6)*f*cos(f*x + e)^2 + 2*(c^10*d - 3*c^8*d^3
+ 3*c^6*d^5 - c^4*d^7)*f*cos(f*x + e) + (c^9*d^2 - 3*c^7*d^4 + 3*c^5*d^6 - c^3*d^8)*f)]

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \sec {\left (e + f x \right )}\right )^{2}}{\left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sec(f*x+e))**2/(c+d*sec(f*x+e))**3,x)

[Out]

Integral((a + b*sec(e + f*x))**2/(c + d*sec(e + f*x))**3, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 658 vs. \(2 (223) = 446\).
time = 0.61, size = 658, normalized size = 2.78 \begin {gather*} \frac {\frac {{\left (4 \, a b c^{5} - 6 \, a^{2} c^{4} d - 3 \, b^{2} c^{4} d + 2 \, a b c^{3} d^{2} + 5 \, a^{2} c^{2} d^{3} - 2 \, a^{2} d^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, c + 2 \, d\right ) + \arctan \left (-\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{7} - 2 \, c^{5} d^{2} + c^{3} d^{4}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {{\left (f x + e\right )} a^{2}}{c^{3}} - \frac {2 \, b^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, a b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - b^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, a b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + b^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 5 \, a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 2 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 2 \, b^{2} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 8 \, a b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b^{2} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b^{2} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, a b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 2 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{6} - 2 \, c^{4} d^{2} + c^{2} d^{4}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{2}}}{f} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((4*a*b*c^5 - 6*a^2*c^4*d - 3*b^2*c^4*d + 2*a*b*c^3*d^2 + 5*a^2*c^2*d^3 - 2*a^2*d^5)*(pi*floor(1/2*(f*x + e)/p
i + 1/2)*sgn(-2*c + 2*d) + arctan(-(c*tan(1/2*f*x + 1/2*e) - d*tan(1/2*f*x + 1/2*e))/sqrt(-c^2 + d^2)))/((c^7
- 2*c^5*d^2 + c^3*d^4)*sqrt(-c^2 + d^2)) + (f*x + e)*a^2/c^3 - (2*b^2*c^5*tan(1/2*f*x + 1/2*e)^3 - 8*a*b*c^4*d
*tan(1/2*f*x + 1/2*e)^3 - b^2*c^4*d*tan(1/2*f*x + 1/2*e)^3 + 6*a^2*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 + 6*a*b*c^3*
d^2*tan(1/2*f*x + 1/2*e)^3 + b^2*c^3*d^2*tan(1/2*f*x + 1/2*e)^3 - 5*a^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 + 2*a*b
*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 2*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 - 3*a^2*c*d^4*tan(1/2*f*x + 1/2*e)^3 +
2*a^2*d^5*tan(1/2*f*x + 1/2*e)^3 - 2*b^2*c^5*tan(1/2*f*x + 1/2*e) + 8*a*b*c^4*d*tan(1/2*f*x + 1/2*e) - b^2*c^4
*d*tan(1/2*f*x + 1/2*e) - 6*a^2*c^3*d^2*tan(1/2*f*x + 1/2*e) + 6*a*b*c^3*d^2*tan(1/2*f*x + 1/2*e) - b^2*c^3*d^
2*tan(1/2*f*x + 1/2*e) - 5*a^2*c^2*d^3*tan(1/2*f*x + 1/2*e) - 2*a*b*c^2*d^3*tan(1/2*f*x + 1/2*e) - 2*b^2*c^2*d
^3*tan(1/2*f*x + 1/2*e) + 3*a^2*c*d^4*tan(1/2*f*x + 1/2*e) + 2*a^2*d^5*tan(1/2*f*x + 1/2*e))/((c^6 - 2*c^4*d^2
+ c^2*d^4)*(c*tan(1/2*f*x + 1/2*e)^2 - d*tan(1/2*f*x + 1/2*e)^2 - c - d)^2))/f

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Mupad [B]
time = 12.14, size = 2500, normalized size = 10.55 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/cos(e + f*x))^2/(c + d/cos(e + f*x))^3,x)

[Out]

((tan(e/2 + (f*x)/2)^3*(2*b^2*c^4 - 2*a^2*d^4 + a^2*c*d^3 + b^2*c^3*d + 6*a^2*c^2*d^2 + 2*b^2*c^2*d^2 - 8*a*b*
c^3*d - 2*a*b*c^2*d^2))/((c^2*d - c^3)*(c + d)^2) - (tan(e/2 + (f*x)/2)*(2*a^2*d^4 - 2*b^2*c^4 + a^2*c*d^3 + b
^2*c^3*d - 6*a^2*c^2*d^2 - 2*b^2*c^2*d^2 + 8*a*b*c^3*d - 2*a*b*c^2*d^2))/((c + d)*(c^4 - 2*c^3*d + c^2*d^2)))/
(f*(2*c*d - tan(e/2 + (f*x)/2)^2*(2*c^2 - 2*d^2) + tan(e/2 + (f*x)/2)^4*(c^2 - 2*c*d + d^2) + c^2 + d^2)) - (2
*a^2*atan(((a^2*((a^2*((8*(4*a^2*c^15 - 12*a^2*c^14*d - 6*b^2*c^14*d - 4*a^2*c^6*d^9 + 2*a^2*c^7*d^8 + 18*a^2*
c^8*d^7 - 4*a^2*c^9*d^6 - 36*a^2*c^10*d^5 + 6*a^2*c^11*d^4 + 34*a^2*c^12*d^3 - 8*a^2*c^13*d^2 + 6*b^2*c^9*d^6
- 6*b^2*c^10*d^5 - 12*b^2*c^11*d^4 + 12*b^2*c^12*d^3 + 6*b^2*c^13*d^2 + 8*a*b*c^15 - 8*a*b*c^14*d - 4*a*b*c^8*
d^7 + 4*a*b*c^9*d^6 + 12*a*b*c^12*d^3 - 12*a*b*c^13*d^2))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6 + 3*c^8*d^5 + 3*c
^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) - (a^2*tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10 + 8*c^7*d^9 + 32*c^8*d^8 -
32*c^9*d^7 - 48*c^10*d^6 + 48*c^11*d^5 + 32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i)/(c^3*(c^10*d + c^11 - c^4
*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3 + (8*tan(e/2 + (f*x)/2)*(4*a^4*c^10
+ 8*a^4*d^10 - 8*a^4*c*d^9 - 8*a^4*c^9*d + 16*a^2*b^2*c^10 - 32*a^4*c^2*d^8 + 32*a^4*c^3*d^7 + 57*a^4*c^4*d^6
- 48*a^4*c^5*d^5 - 52*a^4*c^6*d^4 + 32*a^4*c^7*d^3 + 24*a^4*c^8*d^2 + 9*b^4*c^8*d^2 - 12*a*b^3*c^7*d^3 - 8*a^3
*b*c^3*d^7 + 4*a^3*b*c^5*d^5 + 16*a^3*b*c^7*d^3 + 12*a^2*b^2*c^4*d^6 - 26*a^2*b^2*c^6*d^4 + 52*a^2*b^2*c^8*d^2
- 24*a*b^3*c^9*d - 48*a^3*b*c^9*d))/(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 -
3*c^9*d^2)))/c^3 - (a^2*((a^2*((8*(4*a^2*c^15 - 12*a^2*c^14*d - 6*b^2*c^14*d - 4*a^2*c^6*d^9 + 2*a^2*c^7*d^8 +
18*a^2*c^8*d^7 - 4*a^2*c^9*d^6 - 36*a^2*c^10*d^5 + 6*a^2*c^11*d^4 + 34*a^2*c^12*d^3 - 8*a^2*c^13*d^2 + 6*b^2*
c^9*d^6 - 6*b^2*c^10*d^5 - 12*b^2*c^11*d^4 + 12*b^2*c^12*d^3 + 6*b^2*c^13*d^2 + 8*a*b*c^15 - 8*a*b*c^14*d - 4*
a*b*c^8*d^7 + 4*a*b*c^9*d^6 + 12*a*b*c^12*d^3 - 12*a*b*c^13*d^2))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6 + 3*c^8*d
^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) + (a^2*tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10 + 8*c^7*d^9 + 32*c^
8*d^8 - 32*c^9*d^7 - 48*c^10*d^6 + 48*c^11*d^5 + 32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i)/(c^3*(c^10*d + c^
11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3 - (8*tan(e/2 + (f*x)/2)*(4*a
^4*c^10 + 8*a^4*d^10 - 8*a^4*c*d^9 - 8*a^4*c^9*d + 16*a^2*b^2*c^10 - 32*a^4*c^2*d^8 + 32*a^4*c^3*d^7 + 57*a^4*
c^4*d^6 - 48*a^4*c^5*d^5 - 52*a^4*c^6*d^4 + 32*a^4*c^7*d^3 + 24*a^4*c^8*d^2 + 9*b^4*c^8*d^2 - 12*a*b^3*c^7*d^3
- 8*a^3*b*c^3*d^7 + 4*a^3*b*c^5*d^5 + 16*a^3*b*c^7*d^3 + 12*a^2*b^2*c^4*d^6 - 26*a^2*b^2*c^6*d^4 + 52*a^2*b^2
*c^8*d^2 - 24*a*b^3*c^9*d - 48*a^3*b*c^9*d))/(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^
8*d^3 - 3*c^9*d^2)))/c^3)/((a^2*((a^2*((8*(4*a^2*c^15 - 12*a^2*c^14*d - 6*b^2*c^14*d - 4*a^2*c^6*d^9 + 2*a^2*c
^7*d^8 + 18*a^2*c^8*d^7 - 4*a^2*c^9*d^6 - 36*a^2*c^10*d^5 + 6*a^2*c^11*d^4 + 34*a^2*c^12*d^3 - 8*a^2*c^13*d^2
+ 6*b^2*c^9*d^6 - 6*b^2*c^10*d^5 - 12*b^2*c^11*d^4 + 12*b^2*c^12*d^3 + 6*b^2*c^13*d^2 + 8*a*b*c^15 - 8*a*b*c^1
4*d - 4*a*b*c^8*d^7 + 4*a*b*c^9*d^6 + 12*a*b*c^12*d^3 - 12*a*b*c^13*d^2))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6 +
3*c^8*d^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) - (a^2*tan(e/2 + (f*x)/2)*(8*c^15*d - 8*c^6*d^10 + 8*c^7*d^9
+ 32*c^8*d^8 - 32*c^9*d^7 - 48*c^10*d^6 + 48*c^11*d^5 + 32*c^12*d^4 - 32*c^13*d^3 - 8*c^14*d^2)*8i)/(c^3*(c^1
0*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^4 - 3*c^8*d^3 - 3*c^9*d^2)))*1i)/c^3 + (8*tan(e/2 + (f*x)
/2)*(4*a^4*c^10 + 8*a^4*d^10 - 8*a^4*c*d^9 - 8*a^4*c^9*d + 16*a^2*b^2*c^10 - 32*a^4*c^2*d^8 + 32*a^4*c^3*d^7 +
57*a^4*c^4*d^6 - 48*a^4*c^5*d^5 - 52*a^4*c^6*d^4 + 32*a^4*c^7*d^3 + 24*a^4*c^8*d^2 + 9*b^4*c^8*d^2 - 12*a*b^3
*c^7*d^3 - 8*a^3*b*c^3*d^7 + 4*a^3*b*c^5*d^5 + 16*a^3*b*c^7*d^3 + 12*a^2*b^2*c^4*d^6 - 26*a^2*b^2*c^6*d^4 + 52
*a^2*b^2*c^8*d^2 - 24*a*b^3*c^9*d - 48*a^3*b*c^9*d))/(c^10*d + c^11 - c^4*d^7 - c^5*d^6 + 3*c^6*d^5 + 3*c^7*d^
4 - 3*c^8*d^3 - 3*c^9*d^2))*1i)/c^3 - (16*(4*a^6*d^9 - 8*a^5*b*c^9 - 2*a^6*c*d^8 + 12*a^6*c^8*d + 16*a^4*b^2*c
^9 - 18*a^6*c^2*d^7 + 13*a^6*c^3*d^6 + 36*a^6*c^4*d^5 - 26*a^6*c^5*d^4 - 34*a^6*c^6*d^3 + 24*a^6*c^7*d^2 - 24*
a^3*b^3*c^8*d + 6*a^4*b^2*c^8*d - 4*a^5*b*c^2*d^7 - 4*a^5*b*c^3*d^6 + 4*a^5*b*c^4*d^5 + 4*a^5*b*c^6*d^3 + 12*a
^5*b*c^7*d^2 + 9*a^2*b^4*c^7*d^2 - 12*a^3*b^3*c^6*d^3 + 6*a^4*b^2*c^3*d^6 + 6*a^4*b^2*c^4*d^5 - 14*a^4*b^2*c^5
*d^4 - 12*a^4*b^2*c^6*d^3 + 46*a^4*b^2*c^7*d^2 - 40*a^5*b*c^8*d))/(c^12*d + c^13 - c^6*d^7 - c^7*d^6 + 3*c^8*d
^5 + 3*c^9*d^4 - 3*c^10*d^3 - 3*c^11*d^2) + (a^2*((a^2*((8*(4*a^2*c^15 - 12*a^2*c^14*d - 6*b^2*c^14*d - 4*a^2*
c^6*d^9 + 2*a^2*c^7*d^8 + 18*a^2*c^8*d^7 - 4*a^2*c^9*d^6 - 36*a^2*c^10*d^5 + 6*a^2*c^11*d^4 + 34*a^2*c^12*d^3
- 8*a^2*c^13*d^2 + 6*b^2*c^9*d^6 - 6*b^2*c^10*d^5 - 12*b^2*c^11*d^4 + 12*b^2*c^12*d^3 + 6*b^2*c^13*d^2 + 8*a*b
*c^15 - 8*a*b*c^14*d - 4*a*b*c^8*d^7 + 4*a*b*c^...

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