[next] [prev] [prev-tail] [tail] [up]

3.258 \(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx\)

Optimal. Leaf size=379 \[ \frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \tan (e+f x)}{b^4 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {d^2 \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^5 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^4 (5 b c-2 a d) \tan (e+f x) \sec (e+f x)}{2 b^3 f}+\frac {2 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^3 f (a-b)^{3/2} (a+b)^{3/2}}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {d^5 \tan (e+f x)}{b^2 f} \]

[Out]

1/2*d^4*(-2*a*d+5*b*c)*arctanh(sin(f*x+e))/b^3/f+d^2*(-4*a^3*d^3+15*a^2*b*c*d^2-20*a*b^2*c^2*d+10*b^3*c^3)*arc
tanh(sin(f*x+e))/b^5/f+2*(-a*d+b*c)^5*arctanh((a-b)^(1/2)*tan(1/2*e+1/2*f*x)/(a+b)^(1/2))/a/(a-b)^(3/2)/b^3/(a
+b)^(3/2)/f-(-a*d+b*c)^5*sin(f*x+e)/b^4/(a^2-b^2)/f/(b+a*cos(f*x+e))+2*(-a*d+b*c)^4*(4*a*d+b*c)*arctanh((a-b)^
(1/2)*tan(1/2*e+1/2*f*x)/(a+b)^(1/2))/a/b^5/f/(a-b)^(1/2)/(a+b)^(1/2)+d^5*tan(f*x+e)/b^2/f+d^3*(3*a^2*d^2-10*a
*b*c*d+10*b^2*c^2)*tan(f*x+e)/b^4/f+1/2*d^4*(-2*a*d+5*b*c)*sec(f*x+e)*tan(f*x+e)/b^3/f+1/3*d^5*tan(f*x+e)^3/b^
2/f

________________________________________________________________________________________

Rubi [A]  time = 0.67, antiderivative size = 379, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 10, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {3988, 2952, 2664, 12, 2659, 208, 3770, 3767, 8, 3768} \[ \frac {d^3 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^2 \left (15 a^2 b c d^2-4 a^3 d^3-20 a b^2 c^2 d+10 b^3 c^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 f \left (a^2-b^2\right ) (a \cos (e+f x)+b)}+\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^4 (5 b c-2 a d) \tan (e+f x) \sec (e+f x)}{2 b^3 f}+\frac {2 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^3 f (a-b)^{3/2} (a+b)^{3/2}}+\frac {2 (b c-a d)^4 (4 a d+b c) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a b^5 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {d^5 \tan (e+f x)}{b^2 f} \]

Antiderivative was successfully verified.

[In]

Int[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^2,x]

[Out]

(d^4*(5*b*c - 2*a*d)*ArcTanh[Sin[e + f*x]])/(2*b^3*f) + (d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4
*a^3*d^3)*ArcTanh[Sin[e + f*x]])/(b^5*f) + (2*(b*c - a*d)^5*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2])/Sqrt[a + b]
])/(a*(a - b)^(3/2)*b^3*(a + b)^(3/2)*f) + (2*(b*c - a*d)^4*(b*c + 4*a*d)*ArcTanh[(Sqrt[a - b]*Tan[(e + f*x)/2
])/Sqrt[a + b]])/(a*Sqrt[a - b]*b^5*Sqrt[a + b]*f) - ((b*c - a*d)^5*Sin[e + f*x])/(b^4*(a^2 - b^2)*f*(b + a*Co
s[e + f*x])) + (d^5*Tan[e + f*x])/(b^2*f) + (d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*Tan[e + f*x])/(b^4*f) +
 (d^4*(5*b*c - 2*a*d)*Sec[e + f*x]*Tan[e + f*x])/(2*b^3*f) + (d^5*Tan[e + f*x]^3)/(3*b^2*f)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 208

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

Rule 2659

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 2664

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

Rule 2952

Int[((g_.)*sin[(e_.) + (f_.)*(x_)])^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*sin[(e_.) +
 (f_.)*(x_)])^(n_), x_Symbol] :> Int[ExpandTrig[(g*sin[e + f*x])^p*(a + b*sin[e + f*x])^m*(c + d*sin[e + f*x])
^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[b*c - a*d, 0] && (IntegersQ[m, n] || IntegersQ[m, p
] || IntegersQ[n, p]) && NeQ[p, 2]

Rule 3767

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
 1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3988

Int[(csc[(e_.) + (f_.)*(x_)]*(g_.))^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]
*(d_.) + (c_))^(n_), x_Symbol] :> Dist[1/g^(m + n), Int[(g*Csc[e + f*x])^(m + n + p)*(b + a*Sin[e + f*x])^m*(d
 + c*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b*c - a*d, 0] && IntegerQ[m] && Inte
gerQ[n]

Rubi steps

\begin {align*} \int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx &=\int \frac {(d+c \cos (e+f x))^5 \sec ^4(e+f x)}{(b+a \cos (e+f x))^2} \, dx\\ &=\int \left (\frac {(-b c+a d)^5}{a b^4 (b+a \cos (e+f x))^2}+\frac {(-b c+a d)^4 (b c+4 a d)}{a b^5 (b+a \cos (e+f x))}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \sec (e+f x)}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \sec ^2(e+f x)}{b^4}+\frac {d^4 (5 b c-2 a d) \sec ^3(e+f x)}{b^3}+\frac {d^5 \sec ^4(e+f x)}{b^2}\right ) \, dx\\ &=\frac {d^5 \int \sec ^4(e+f x) \, dx}{b^2}+\frac {\left (d^4 (5 b c-2 a d)\right ) \int \sec ^3(e+f x) \, dx}{b^3}-\frac {(b c-a d)^5 \int \frac {1}{(b+a \cos (e+f x))^2} \, dx}{a b^4}+\frac {\left ((b c-a d)^4 (b c+4 a d)\right ) \int \frac {1}{b+a \cos (e+f x)} \, dx}{a b^5}+\frac {\left (d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \int \sec ^2(e+f x) \, dx}{b^4}+\frac {\left (d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right )\right ) \int \sec (e+f x) \, dx}{b^5}\\ &=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {\left (d^4 (5 b c-2 a d)\right ) \int \sec (e+f x) \, dx}{2 b^3}+\frac {(b c-a d)^5 \int \frac {b}{b+a \cos (e+f x)} \, dx}{a b^4 \left (a^2-b^2\right )}-\frac {d^5 \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{b^2 f}+\frac {\left (2 (b c-a d)^4 (b c+4 a d)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a b^5 f}-\frac {\left (d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right )\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{b^4 f}\\ &=\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {(b c-a d)^5 \int \frac {1}{b+a \cos (e+f x)} \, dx}{a b^3 \left (a^2-b^2\right )}\\ &=\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}+\frac {\left (2 (b c-a d)^5\right ) \operatorname {Subst}\left (\int \frac {1}{a+b+(-a+b) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a b^3 \left (a^2-b^2\right ) f}\\ &=\frac {d^4 (5 b c-2 a d) \tanh ^{-1}(\sin (e+f x))}{2 b^3 f}+\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) \tanh ^{-1}(\sin (e+f x))}{b^5 f}+\frac {2 (b c-a d)^5 \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a (a-b)^{3/2} b^3 (a+b)^{3/2} f}+\frac {2 (b c-a d)^4 (b c+4 a d) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{a \sqrt {a-b} b^5 \sqrt {a+b} f}-\frac {(b c-a d)^5 \sin (e+f x)}{b^4 \left (a^2-b^2\right ) f (b+a \cos (e+f x))}+\frac {d^5 \tan (e+f x)}{b^2 f}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) \tan (e+f x)}{b^4 f}+\frac {d^4 (5 b c-2 a d) \sec (e+f x) \tan (e+f x)}{2 b^3 f}+\frac {d^5 \tan ^3(e+f x)}{3 b^2 f}\\ \end {align*}

________________________________________________________________________________________

Mathematica [B]  time = 6.55, size = 1137, normalized size = 3.00 \[ \frac {(b+a \cos (e+f x)) \left (12 d^5 \sin (e+f x) b^5+60 c^2 d^3 \sin (e+f x) b^5+6 c^5 \sin (2 (e+f x)) b^5+30 c d^4 \sin (2 (e+f x)) b^5+4 d^5 \sin (3 (e+f x)) b^5+60 c^2 d^3 \sin (3 (e+f x)) b^5+3 c^5 \sin (4 (e+f x)) b^5-45 a c d^4 \sin (e+f x) b^4-4 a d^5 \sin (2 (e+f x)) b^4+60 a c^2 d^3 \sin (2 (e+f x)) b^4-30 a c^4 d \sin (2 (e+f x)) b^4-45 a c d^4 \sin (3 (e+f x)) b^4+2 a d^5 \sin (4 (e+f x)) b^4+30 a c^2 d^3 \sin (4 (e+f x)) b^4-15 a c^4 d \sin (4 (e+f x)) b^4-60 a^2 c^2 d^3 \sin (e+f x) b^3-90 a^2 c d^4 \sin (2 (e+f x)) b^3+60 a^2 c^3 d^2 \sin (2 (e+f x)) b^3+8 a^2 d^5 \sin (3 (e+f x)) b^3-60 a^2 c^2 d^3 \sin (3 (e+f x)) b^3-30 a^2 c d^4 \sin (4 (e+f x)) b^3+30 a^2 c^3 d^2 \sin (4 (e+f x)) b^3+45 a^3 c d^4 \sin (e+f x) b^2+22 a^3 d^5 \sin (2 (e+f x)) b^2-120 a^3 c^2 d^3 \sin (2 (e+f x)) b^2+45 a^3 c d^4 \sin (3 (e+f x)) b^2+7 a^3 d^5 \sin (4 (e+f x)) b^2-60 a^3 c^2 d^3 \sin (4 (e+f x)) b^2-12 a^4 d^5 \sin (e+f x) b+90 a^4 c d^4 \sin (2 (e+f x)) b-12 a^4 d^5 \sin (3 (e+f x)) b+45 a^4 c d^4 \sin (4 (e+f x)) b-24 a^5 d^5 \sin (2 (e+f x))-12 a^5 d^5 \sin (4 (e+f x))\right ) (c+d \sec (e+f x))^5}{24 b^4 \left (b^2-a^2\right ) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac {\left (8 a^3 d^5+2 a b^2 d^5-5 b^3 c d^4-30 a^2 b c d^4+40 a b^2 c^2 d^3-20 b^3 c^3 d^2\right ) \cos ^3(e+f x) (b+a \cos (e+f x))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sec (e+f x))^5}{2 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac {\left (-8 a^3 d^5-2 a b^2 d^5+5 b^3 c d^4+30 a^2 b c d^4-40 a b^2 c^2 d^3+20 b^3 c^3 d^2\right ) \cos ^3(e+f x) (b+a \cos (e+f x))^2 \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) (c+d \sec (e+f x))^5}{2 b^5 f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}-\frac {2 (b c-a d)^4 \left (-4 d a^2-b c a+5 b^2 d\right ) \tanh ^{-1}\left (\frac {(b-a) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right ) \cos ^3(e+f x) (b+a \cos (e+f x))^2 (c+d \sec (e+f x))^5}{b^5 \sqrt {a^2-b^2} \left (b^2-a^2\right ) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^5)/(a + b*Sec[e + f*x])^2,x]

[Out]

(-2*(b*c - a*d)^4*(-(a*b*c) - 4*a^2*d + 5*b^2*d)*ArcTanh[((-a + b)*Tan[(e + f*x)/2])/Sqrt[a^2 - b^2]]*Cos[e +
f*x]^3*(b + a*Cos[e + f*x])^2*(c + d*Sec[e + f*x])^5)/(b^5*Sqrt[a^2 - b^2]*(-a^2 + b^2)*f*(d + c*Cos[e + f*x])
^5*(a + b*Sec[e + f*x])^2) + ((-20*b^3*c^3*d^2 + 40*a*b^2*c^2*d^3 - 30*a^2*b*c*d^4 - 5*b^3*c*d^4 + 8*a^3*d^5 +
 2*a*b^2*d^5)*Cos[e + f*x]^3*(b + a*Cos[e + f*x])^2*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*(c + d*Sec[e + f*
x])^5)/(2*b^5*f*(d + c*Cos[e + f*x])^5*(a + b*Sec[e + f*x])^2) + ((20*b^3*c^3*d^2 - 40*a*b^2*c^2*d^3 + 30*a^2*
b*c*d^4 + 5*b^3*c*d^4 - 8*a^3*d^5 - 2*a*b^2*d^5)*Cos[e + f*x]^3*(b + a*Cos[e + f*x])^2*Log[Cos[(e + f*x)/2] +
Sin[(e + f*x)/2]]*(c + d*Sec[e + f*x])^5)/(2*b^5*f*(d + c*Cos[e + f*x])^5*(a + b*Sec[e + f*x])^2) + ((b + a*Co
s[e + f*x])*(c + d*Sec[e + f*x])^5*(-60*a^2*b^3*c^2*d^3*Sin[e + f*x] + 60*b^5*c^2*d^3*Sin[e + f*x] + 45*a^3*b^
2*c*d^4*Sin[e + f*x] - 45*a*b^4*c*d^4*Sin[e + f*x] - 12*a^4*b*d^5*Sin[e + f*x] + 12*b^5*d^5*Sin[e + f*x] + 6*b
^5*c^5*Sin[2*(e + f*x)] - 30*a*b^4*c^4*d*Sin[2*(e + f*x)] + 60*a^2*b^3*c^3*d^2*Sin[2*(e + f*x)] - 120*a^3*b^2*
c^2*d^3*Sin[2*(e + f*x)] + 60*a*b^4*c^2*d^3*Sin[2*(e + f*x)] + 90*a^4*b*c*d^4*Sin[2*(e + f*x)] - 90*a^2*b^3*c*
d^4*Sin[2*(e + f*x)] + 30*b^5*c*d^4*Sin[2*(e + f*x)] - 24*a^5*d^5*Sin[2*(e + f*x)] + 22*a^3*b^2*d^5*Sin[2*(e +
 f*x)] - 4*a*b^4*d^5*Sin[2*(e + f*x)] - 60*a^2*b^3*c^2*d^3*Sin[3*(e + f*x)] + 60*b^5*c^2*d^3*Sin[3*(e + f*x)]
+ 45*a^3*b^2*c*d^4*Sin[3*(e + f*x)] - 45*a*b^4*c*d^4*Sin[3*(e + f*x)] - 12*a^4*b*d^5*Sin[3*(e + f*x)] + 8*a^2*
b^3*d^5*Sin[3*(e + f*x)] + 4*b^5*d^5*Sin[3*(e + f*x)] + 3*b^5*c^5*Sin[4*(e + f*x)] - 15*a*b^4*c^4*d*Sin[4*(e +
 f*x)] + 30*a^2*b^3*c^3*d^2*Sin[4*(e + f*x)] - 60*a^3*b^2*c^2*d^3*Sin[4*(e + f*x)] + 30*a*b^4*c^2*d^3*Sin[4*(e
 + f*x)] + 45*a^4*b*c*d^4*Sin[4*(e + f*x)] - 30*a^2*b^3*c*d^4*Sin[4*(e + f*x)] - 12*a^5*d^5*Sin[4*(e + f*x)] +
 7*a^3*b^2*d^5*Sin[4*(e + f*x)] + 2*a*b^4*d^5*Sin[4*(e + f*x)]))/(24*b^4*(-a^2 + b^2)*f*(d + c*Cos[e + f*x])^5
*(a + b*Sec[e + f*x])^2)

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError >> Unable to parse Giac output: Unable to check sign: (4*pi/x/2)>(-4*pi/
x/2)Unable to check sign: (4*pi/x/2)>(-4*pi/x/2)2/f*((-tan((f*x+exp(1))/2)*c^5*b^5+5*tan((f*x+exp(1))/2)*c^4*a
*b^4*d-10*tan((f*x+exp(1))/2)*c^3*a^2*b^3*d^2+10*tan((f*x+exp(1))/2)*c^2*a^3*b^2*d^3-5*tan((f*x+exp(1))/2)*c*a
^4*b*d^4+tan((f*x+exp(1))/2)*a^5*d^5)/(-a^2*b^4+b^6)/(tan((f*x+exp(1))/2)^2*a-tan((f*x+exp(1))/2)^2*b-a-b)+(-6
0*tan((f*x+exp(1))/2)^5*c^2*b^2*d^3+60*tan((f*x+exp(1))/2)^5*c*a*b*d^4+15*tan((f*x+exp(1))/2)^5*c*b^2*d^4-18*t
an((f*x+exp(1))/2)^5*a^2*d^5-6*tan((f*x+exp(1))/2)^5*a*b*d^5-6*tan((f*x+exp(1))/2)^5*b^2*d^5+120*tan((f*x+exp(
1))/2)^3*c^2*b^2*d^3-120*tan((f*x+exp(1))/2)^3*c*a*b*d^4+36*tan((f*x+exp(1))/2)^3*a^2*d^5+4*tan((f*x+exp(1))/2
)^3*b^2*d^5-60*tan((f*x+exp(1))/2)*c^2*b^2*d^3+60*tan((f*x+exp(1))/2)*c*a*b*d^4-15*tan((f*x+exp(1))/2)*c*b^2*d
^4-18*tan((f*x+exp(1))/2)*a^2*d^5+6*tan((f*x+exp(1))/2)*a*b*d^5-6*tan((f*x+exp(1))/2)*b^2*d^5)*1/6/b^4/(tan((f
*x+exp(1))/2)^2-1)^3+(2*c^5*a*b^5-10*c^4*b^6*d-20*c^3*a^3*b^3*d^2+40*c^3*a*b^5*d^2+40*c^2*a^4*b^2*d^3-60*c^2*a
^2*b^4*d^3-30*c*a^5*b*d^4+40*c*a^3*b^3*d^4+8*a^6*d^5-10*a^4*b^2*d^5)*1/2/(-a^2*b^5+b^7)/sqrt(-a^2+b^2)*(atan((
a*tan((f*x+exp(1))/2)-b*tan((f*x+exp(1))/2))/sqrt(-a^2+b^2))+pi*sign(2*a-2*b)*floor((f*x+exp(1))/2/pi+1/2))-(2
0*c^3*b^3*d^2-40*c^2*a*b^2*d^3+30*c*a^2*b*d^4+5*c*b^3*d^4-8*a^3*d^5-2*a*b^2*d^5)*1/4/b^5*ln(abs(tan((f*x+exp(1
))/2)-1))+(20*c^3*b^3*d^2-40*c^2*a*b^2*d^3+30*c*a^2*b*d^4+5*c*b^3*d^4-8*a^3*d^5-2*a*b^2*d^5)*1/4/b^5*ln(abs(ta
n((f*x+exp(1))/2)+1)))

________________________________________________________________________________________

maple [B]  time = 0.76, size = 1870, normalized size = 4.93 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(f*x+e))^2,x)

[Out]

-1/f*d^5/b^2/(tan(1/2*e+1/2*f*x)-1)-1/2/f*d^5/b^2/(tan(1/2*e+1/2*f*x)-1)^2-1/3/f*d^5/b^2/(tan(1/2*e+1/2*f*x)+1
)^3-1/f*d^5/b^2/(tan(1/2*e+1/2*f*x)+1)+1/2/f*d^5/b^2/(tan(1/2*e+1/2*f*x)+1)^2-1/3/f*d^5/b^2/(tan(1/2*e+1/2*f*x
)-1)^3-20/f/b^2/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*a^3*c^3*
d^2-10/f/b^3/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*a^4*d^5+40/
f/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*a*c^3*d^2+8/f/b^5/(a-b
)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*a^6*d^5-10/f*b/(a-b)/(a+b)/(
(a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*c^4*d-20/f/b^2/(a^2-b^2)*tan(1/2*e+1/
2*f*x)/(a*tan(1/2*e+1/2*f*x)^2-tan(1/2*e+1/2*f*x)^2*b-a-b)*a^3*c^2*d^3+20/f/b/(a^2-b^2)*tan(1/2*e+1/2*f*x)/(a*
tan(1/2*e+1/2*f*x)^2-tan(1/2*e+1/2*f*x)^2*b-a-b)*a^2*c^3*d^2+10/f/b^3/(a^2-b^2)*tan(1/2*e+1/2*f*x)/(a*tan(1/2*
e+1/2*f*x)^2-tan(1/2*e+1/2*f*x)^2*b-a-b)*a^4*c*d^4-60/f/b/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/
2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*a^2*c^2*d^3-30/f/b^4/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e
+1/2*f*x)/((a-b)*(a+b))^(1/2))*a^5*c*d^4+40/f/b^3/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*
f*x)/((a-b)*(a+b))^(1/2))*a^4*c^2*d^3+40/f/b^2/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x
)/((a-b)*(a+b))^(1/2))*a^3*c*d^4-10/f/(a^2-b^2)*tan(1/2*e+1/2*f*x)/(a*tan(1/2*e+1/2*f*x)^2-tan(1/2*e+1/2*f*x)^
2*b-a-b)*a*c^4*d+2/f/(a-b)/(a+b)/((a-b)*(a+b))^(1/2)*arctanh((a-b)*tan(1/2*e+1/2*f*x)/((a-b)*(a+b))^(1/2))*c^5
*a-2/f/b^4/(a^2-b^2)*tan(1/2*e+1/2*f*x)/(a*tan(1/2*e+1/2*f*x)^2-tan(1/2*e+1/2*f*x)^2*b-a-b)*a^5*d^5-15/f*d^4/b
^4*ln(tan(1/2*e+1/2*f*x)-1)*a^2*c+20/f*d^3/b^3*ln(tan(1/2*e+1/2*f*x)-1)*a*c^2+10/f*d^4/b^3/(tan(1/2*e+1/2*f*x)
-1)*a*c+15/f*d^4/b^4*ln(tan(1/2*e+1/2*f*x)+1)*a^2*c-20/f*d^3/b^3*ln(tan(1/2*e+1/2*f*x)+1)*a*c^2+10/f*d^4/b^3/(
tan(1/2*e+1/2*f*x)+1)*a*c+2/f*b/(a^2-b^2)*tan(1/2*e+1/2*f*x)/(a*tan(1/2*e+1/2*f*x)^2-tan(1/2*e+1/2*f*x)^2*b-a-
b)*c^5-1/f*d^5/b^3/(tan(1/2*e+1/2*f*x)-1)^2*a+5/2/f*d^4/b^2/(tan(1/2*e+1/2*f*x)-1)^2*c+1/f*d^5/b^3/(tan(1/2*e+
1/2*f*x)+1)^2*a-5/2/f*d^4/b^2/(tan(1/2*e+1/2*f*x)+1)^2*c-3/f*d^5/b^4/(tan(1/2*e+1/2*f*x)+1)*a^2-1/f*d^5/b^3/(t
an(1/2*e+1/2*f*x)+1)*a-10/f*d^3/b^2/(tan(1/2*e+1/2*f*x)+1)*c^2+5/2/f*d^4/b^2/(tan(1/2*e+1/2*f*x)+1)*c-1/f*d^5/
b^3*ln(tan(1/2*e+1/2*f*x)+1)*a+4/f*d^5/b^5*ln(tan(1/2*e+1/2*f*x)-1)*a^3+1/f*d^5/b^3*ln(tan(1/2*e+1/2*f*x)-1)*a
-10/f*d^2/b^2*ln(tan(1/2*e+1/2*f*x)-1)*c^3-5/2/f*d^4/b^2*ln(tan(1/2*e+1/2*f*x)-1)*c-3/f*d^5/b^4/(tan(1/2*e+1/2
*f*x)-1)*a^2-1/f*d^5/b^3/(tan(1/2*e+1/2*f*x)-1)*a-10/f*d^3/b^2/(tan(1/2*e+1/2*f*x)-1)*c^2+5/2/f*d^4/b^2/(tan(1
/2*e+1/2*f*x)-1)*c+10/f*d^2/b^2*ln(tan(1/2*e+1/2*f*x)+1)*c^3+5/2/f*d^4/b^2*ln(tan(1/2*e+1/2*f*x)+1)*c-4/f*d^5/
b^5*ln(tan(1/2*e+1/2*f*x)+1)*a^3

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))^5/(a+b*sec(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*a^2-4*b^2>0)', see `assume?`
 for more details)Is 4*a^2-4*b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 16.95, size = 17256, normalized size = 45.53 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d/cos(e + f*x))^5/(cos(e + f*x)*(a + b/cos(e + f*x))^2),x)

[Out]

(atan(((((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^
9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d
^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11
*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7
*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140
*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2
*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 -
 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d
^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c
^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b
^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6
*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 100
80*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7
 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^
9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (((8*(4*a*b^17*c^5 + 4*a*b^17*d
^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^
4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^1
7*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d
+ 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a
^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 4
0*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(e/
2 + (f*x)/2)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b
^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b
^8)))*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a
*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*1i)/b^5 + (((8*tan(e/2 + (
f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^1
0 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10
 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c
^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^
9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a
^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^
2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 -
2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^
3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c
^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b
^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^
6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 56
00*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6
 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 9
60*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20
*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^1
3*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c
^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 1
40*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a
^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80
*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(e/2 + (f*x)/2)*(b^2*(a*d^
5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b^15 - 8*a^2*b^14 - 16*a
^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(b^2*(a*d^5 + 20*
a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a*d^5 + 20*a*c^2*d^3) +
4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*1i)/b^5)/((16*(256*a^14*d^15 - 128*a^13*b*d^15 +
20*a^6*b^8*d^15 - 20*a^7*b^7*d^15 + 124*a^8*b^6*d^15 - 24*a^9*b^5*d^15 + 48*a^10*b^4*d^15 + 192*a^11*b^3*d^15
- 448*a^12*b^2*d^15 + 125*b^14*c^6*d^9 + 1000*b^14*c^8*d^7 - 250*b^14*c^9*d^6 + 2000*b^14*c^10*d^5 - 1000*b^14
*c^11*d^4 - 600*a*b^13*c^5*d^10 - 125*a*b^13*c^6*d^9 - 6425*a*b^13*c^7*d^8 + 1100*a*b^13*c^8*d^7 - 16200*a*b^1
3*c^9*d^6 + 8100*a*b^13*c^10*d^5 - 400*a*b^13*c^11*d^4 + 400*a*b^13*c^12*d^3 - 180*a^5*b^9*c*d^14 + 180*a^6*b^
8*c*d^14 - 1320*a^7*b^7*c*d^14 + 270*a^8*b^6*c*d^14 - 900*a^9*b^5*c*d^14 - 2160*a^10*b^4*c*d^14 + 5280*a^11*b^
3*c*d^14 + 1440*a^12*b^2*c*d^14 + 1170*a^2*b^12*c^4*d^11 + 600*a^2*b^12*c^5*d^10 + 17795*a^2*b^12*c^6*d^9 - 13
75*a^2*b^12*c^7*d^8 + 57480*a^2*b^12*c^8*d^7 - 29740*a^2*b^12*c^9*d^6 - 400*a^2*b^12*c^10*d^5 - 2010*a^2*b^12*
c^11*d^4 - 40*a^2*b^12*c^13*d^2 - 1180*a^3*b^11*c^3*d^12 - 1170*a^3*b^11*c^4*d^11 - 27754*a^3*b^11*c^5*d^10 -
995*a^3*b^11*c^6*d^9 - 117635*a^3*b^11*c^7*d^8 + 66680*a^3*b^11*c^8*d^7 + 17400*a^3*b^11*c^9*d^6 + 2604*a^3*b^
11*c^10*d^5 + 400*a^3*b^11*c^11*d^4 + 80*a^3*b^11*c^12*d^3 + 645*a^4*b^10*c^2*d^13 + 1180*a^4*b^10*c^3*d^12 +
26690*a^4*b^10*c^4*d^11 + 4654*a^4*b^10*c^5*d^10 + 153580*a^4*b^10*c^6*d^9 - 103805*a^4*b^10*c^7*d^8 - 79760*a
^4*b^10*c^8*d^7 + 5840*a^4*b^10*c^9*d^6 - 1600*a^4*b^10*c^10*d^5 + 340*a^4*b^10*c^11*d^4 - 645*a^5*b^9*c^2*d^1
3 - 16245*a^5*b^9*c^3*d^12 - 5690*a^5*b^9*c^4*d^11 - 133278*a^5*b^9*c^5*d^10 + 119980*a^5*b^9*c^6*d^9 + 188520
*a^5*b^9*c^7*d^8 - 28880*a^5*b^9*c^8*d^7 - 1200*a^5*b^9*c^9*d^6 - 1584*a^5*b^9*c^10*d^5 + 6135*a^6*b^8*c^2*d^1
3 + 3645*a^6*b^8*c^3*d^12 + 77460*a^6*b^8*c^4*d^11 - 105562*a^6*b^8*c^5*d^10 - 279820*a^6*b^8*c^6*d^9 + 57980*
a^6*b^8*c^7*d^8 + 21280*a^6*b^8*c^8*d^7 + 2800*a^6*b^8*c^9*d^6 - 1335*a^7*b^7*c^2*d^13 - 29515*a^7*b^7*c^3*d^1
2 + 69980*a^7*b^7*c^4*d^11 + 279768*a^7*b^7*c^5*d^10 - 74940*a^7*b^7*c^6*d^9 - 64460*a^7*b^7*c^7*d^8 - 2720*a^
7*b^7*c^8*d^7 + 6960*a^8*b^6*c^2*d^13 - 33645*a^8*b^6*c^3*d^12 - 192920*a^8*b^6*c^4*d^11 + 69104*a^8*b^6*c^5*d
^10 + 108320*a^8*b^6*c^6*d^9 + 1540*a^8*b^6*c^7*d^8 + 10980*a^9*b^5*c^2*d^13 + 91160*a^9*b^5*c^3*d^12 - 46520*
a^9*b^5*c^4*d^11 - 118136*a^9*b^5*c^5*d^10 - 480*a^9*b^5*c^6*d^9 - 28380*a^10*b^4*c^2*d^13 + 22430*a^10*b^4*c^
3*d^12 + 87600*a^10*b^4*c^4*d^11 + 64*a^10*b^4*c^5*d^10 - 7320*a^11*b^3*c^2*d^13 - 44220*a^11*b^3*c^3*d^12 + 1
4640*a^12*b^2*c^2*d^13 - 2880*a^13*b*c*d^14))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (((8*tan(e/2 + (f*x)/2)*
(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a
^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^
12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 -
 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*
a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c
*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c
^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3
*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055
*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 -
 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d
^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^
4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b
^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*
a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*
b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^
4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 +
36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 -
 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b
^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*
c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^1
2*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(e/2 + (f*x)/2)*(b^2*(a*d^5 + 20*a
*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13
+ 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(b^2*(a*d^5 + 20*a*c^2*d^
3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^
5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5 - (((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*
b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d
^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*
d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600
*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^
8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9
+ 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5
*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^
9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^
4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290
*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 +
 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d
^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*
c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9
*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 -
 a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a
^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^
11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^
2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*
d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3
*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*
d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(e/2 + (f*x)/2)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^
3*((5*c*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11
 - 8*a^6*b^10))/(b^5*(a*b^10 + b^11 - a^2*b^9 - a^3*b^8)))*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c
*d^4)/2 + 10*c^3*d^2) - 15*a^2*b*c*d^4))/b^5)*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*
c^3*d^2) - 15*a^2*b*c*d^4))/b^5))*(b^2*(a*d^5 + 20*a*c^2*d^3) + 4*a^3*d^5 - b^3*((5*c*d^4)/2 + 10*c^3*d^2) - 1
5*a^2*b*c*d^4)*2i)/(b^5*f) - ((tan(e/2 + (f*x)/2)^5*(18*b^5*c^5 - 72*a^5*d^5 + 2*b^5*d^5 + 16*a*b^4*d^5 + 12*a
^4*b*d^5 - 15*b^5*c*d^4 - 14*a^2*b^3*d^5 + 38*a^3*b^2*d^5 - 60*b^5*c^2*d^3 + 180*a*b^4*c^2*d^3 - 165*a^2*b^3*c
*d^4 - 45*a^3*b^2*c*d^4 + 60*a^2*b^3*c^2*d^3 + 180*a^2*b^3*c^3*d^2 - 360*a^3*b^2*c^2*d^3 + 45*a*b^4*c*d^4 - 90
*a*b^4*c^4*d + 270*a^4*b*c*d^4))/(3*(a*b^4 - b^5)*(a + b)) - (tan(e/2 + (f*x)/2)^7*(2*b^5*c^5 - 8*a^5*d^5 - 2*
b^5*d^5 + 4*a^4*b*d^5 + 5*b^5*c*d^4 - 2*a^2*b^3*d^5 + 6*a^3*b^2*d^5 - 20*b^5*c^2*d^3 + 20*a*b^4*c^2*d^3 - 25*a
^2*b^3*c*d^4 - 15*a^3*b^2*c*d^4 + 20*a^2*b^3*c^2*d^3 + 20*a^2*b^3*c^3*d^2 - 40*a^3*b^2*c^2*d^3 + 15*a*b^4*c*d^
4 - 10*a*b^4*c^4*d + 30*a^4*b*c*d^4))/((a*b^4 - b^5)*(a + b)) + (tan(e/2 + (f*x)/2)*(2*b^5*c^5 - 8*a^5*d^5 + 2
*b^5*d^5 - 4*a^4*b*d^5 + 5*b^5*c*d^4 + 2*a^2*b^3*d^5 + 6*a^3*b^2*d^5 + 20*b^5*c^2*d^3 + 20*a*b^4*c^2*d^3 - 25*
a^2*b^3*c*d^4 + 15*a^3*b^2*c*d^4 - 20*a^2*b^3*c^2*d^3 + 20*a^2*b^3*c^3*d^2 - 40*a^3*b^2*c^2*d^3 - 15*a*b^4*c*d
^4 - 10*a*b^4*c^4*d + 30*a^4*b*c*d^4))/((a*b^4 - b^5)*(a + b)) + (tan(e/2 + (f*x)/2)^3*(72*a^5*d^5 - 18*b^5*c^
5 + 2*b^5*d^5 - 16*a*b^4*d^5 + 12*a^4*b*d^5 + 15*b^5*c*d^4 - 14*a^2*b^3*d^5 - 38*a^3*b^2*d^5 - 60*b^5*c^2*d^3
- 180*a*b^4*c^2*d^3 + 165*a^2*b^3*c*d^4 - 45*a^3*b^2*c*d^4 + 60*a^2*b^3*c^2*d^3 - 180*a^2*b^3*c^3*d^2 + 360*a^
3*b^2*c^2*d^3 + 45*a*b^4*c*d^4 + 90*a*b^4*c^4*d - 270*a^4*b*c*d^4))/(3*b^4*(a + b)*(a - b)))/(f*(a + b + tan(e
/2 + (f*x)/2)^8*(a - b) - tan(e/2 + (f*x)/2)^2*(4*a + 2*b) - tan(e/2 + (f*x)/2)^6*(4*a - 2*b) + 6*a*tan(e/2 +
(f*x)/2)^4)) + (atan((((a*d - b*c)^4*((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10
 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8
*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 +
 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*
b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*
d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 4
35*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6
*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9
*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4
*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*
a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 8
25*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4
 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c
^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10
*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + ((
(8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14
*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5
 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^
15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4
- 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*
d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2
*b^13 - a^3*b^12) + (8*tan(e/2 + (f*x)/2)*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c
)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a
^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*
d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*
1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) + ((a*d - b*c)^4*((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11
*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*
d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4
*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 160
0*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b
^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9
 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^
5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b
^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a
^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 129
0*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4
+ 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*
d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5
*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^
9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11
- a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*
a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b
^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a
^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c
*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^
3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4
*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*tan(e/2 + (f*x)/2)*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(
4*a^2*d - 5*b^2*d + a*b*c)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*
b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3
)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*
a^2*d - 5*b^2*d + a*b*c)*1i)/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5))/((16*(256*a^14*d^15 - 128*a^13*b*d^15 +
 20*a^6*b^8*d^15 - 20*a^7*b^7*d^15 + 124*a^8*b^6*d^15 - 24*a^9*b^5*d^15 + 48*a^10*b^4*d^15 + 192*a^11*b^3*d^15
 - 448*a^12*b^2*d^15 + 125*b^14*c^6*d^9 + 1000*b^14*c^8*d^7 - 250*b^14*c^9*d^6 + 2000*b^14*c^10*d^5 - 1000*b^1
4*c^11*d^4 - 600*a*b^13*c^5*d^10 - 125*a*b^13*c^6*d^9 - 6425*a*b^13*c^7*d^8 + 1100*a*b^13*c^8*d^7 - 16200*a*b^
13*c^9*d^6 + 8100*a*b^13*c^10*d^5 - 400*a*b^13*c^11*d^4 + 400*a*b^13*c^12*d^3 - 180*a^5*b^9*c*d^14 + 180*a^6*b
^8*c*d^14 - 1320*a^7*b^7*c*d^14 + 270*a^8*b^6*c*d^14 - 900*a^9*b^5*c*d^14 - 2160*a^10*b^4*c*d^14 + 5280*a^11*b
^3*c*d^14 + 1440*a^12*b^2*c*d^14 + 1170*a^2*b^12*c^4*d^11 + 600*a^2*b^12*c^5*d^10 + 17795*a^2*b^12*c^6*d^9 - 1
375*a^2*b^12*c^7*d^8 + 57480*a^2*b^12*c^8*d^7 - 29740*a^2*b^12*c^9*d^6 - 400*a^2*b^12*c^10*d^5 - 2010*a^2*b^12
*c^11*d^4 - 40*a^2*b^12*c^13*d^2 - 1180*a^3*b^11*c^3*d^12 - 1170*a^3*b^11*c^4*d^11 - 27754*a^3*b^11*c^5*d^10 -
 995*a^3*b^11*c^6*d^9 - 117635*a^3*b^11*c^7*d^8 + 66680*a^3*b^11*c^8*d^7 + 17400*a^3*b^11*c^9*d^6 + 2604*a^3*b
^11*c^10*d^5 + 400*a^3*b^11*c^11*d^4 + 80*a^3*b^11*c^12*d^3 + 645*a^4*b^10*c^2*d^13 + 1180*a^4*b^10*c^3*d^12 +
 26690*a^4*b^10*c^4*d^11 + 4654*a^4*b^10*c^5*d^10 + 153580*a^4*b^10*c^6*d^9 - 103805*a^4*b^10*c^7*d^8 - 79760*
a^4*b^10*c^8*d^7 + 5840*a^4*b^10*c^9*d^6 - 1600*a^4*b^10*c^10*d^5 + 340*a^4*b^10*c^11*d^4 - 645*a^5*b^9*c^2*d^
13 - 16245*a^5*b^9*c^3*d^12 - 5690*a^5*b^9*c^4*d^11 - 133278*a^5*b^9*c^5*d^10 + 119980*a^5*b^9*c^6*d^9 + 18852
0*a^5*b^9*c^7*d^8 - 28880*a^5*b^9*c^8*d^7 - 1200*a^5*b^9*c^9*d^6 - 1584*a^5*b^9*c^10*d^5 + 6135*a^6*b^8*c^2*d^
13 + 3645*a^6*b^8*c^3*d^12 + 77460*a^6*b^8*c^4*d^11 - 105562*a^6*b^8*c^5*d^10 - 279820*a^6*b^8*c^6*d^9 + 57980
*a^6*b^8*c^7*d^8 + 21280*a^6*b^8*c^8*d^7 + 2800*a^6*b^8*c^9*d^6 - 1335*a^7*b^7*c^2*d^13 - 29515*a^7*b^7*c^3*d^
12 + 69980*a^7*b^7*c^4*d^11 + 279768*a^7*b^7*c^5*d^10 - 74940*a^7*b^7*c^6*d^9 - 64460*a^7*b^7*c^7*d^8 - 2720*a
^7*b^7*c^8*d^7 + 6960*a^8*b^6*c^2*d^13 - 33645*a^8*b^6*c^3*d^12 - 192920*a^8*b^6*c^4*d^11 + 69104*a^8*b^6*c^5*
d^10 + 108320*a^8*b^6*c^6*d^9 + 1540*a^8*b^6*c^7*d^8 + 10980*a^9*b^5*c^2*d^13 + 91160*a^9*b^5*c^3*d^12 - 46520
*a^9*b^5*c^4*d^11 - 118136*a^9*b^5*c^5*d^10 - 480*a^9*b^5*c^6*d^9 - 28380*a^10*b^4*c^2*d^13 + 22430*a^10*b^4*c
^3*d^12 + 87600*a^10*b^4*c^4*d^11 + 64*a^10*b^4*c^5*d^10 - 7320*a^11*b^3*c^2*d^13 - 44220*a^11*b^3*c^3*d^12 +
14640*a^12*b^2*c^2*d^13 - 2880*a^13*b*c*d^14))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + ((a*d - b*c)^4*((8*tan(
e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*a^3*b^9*d^10 + 28*a^4*
b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9*b^3*d^10 - 192*a^10*b
^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*a*b^11*c^2*d^8 - 480*a
*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^11*c^7*d^3 + 40*a^2*b^
10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9 - 140*a^7*b^5*c*d^9 -
 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 960*a^2*b^10*c^3*d^7 +
2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8*d^2 - 820*a^3*b^9*c^2
*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9*c^6*d^4 + 160*a^3*b^9
*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4*b^8*c^5*d^5 - 2640*a^
4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800*a^5*b^7*c^4*d^6 + 776
0*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 9920*a^6*b^6*c^3*d^7 - 1
1560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8 + 10080*a^7*b^5*c^3*d
^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c^3*d^7 + 5600*a^8*b^4*
c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^11*c*d^9 - 40*a*b^11*c^
9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) + (((8*(4*a*b^17*c^5 + 4*a*b^17*d^5 - 10*b^18*c*d
^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 - 20*a^4*b^14*d^5 - 16*
a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80*a*b^17*c^2*d^3 + 80*a
*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*c^4*d + 70*a^4*b^14*c*
d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3 + 40*a^2*b^16*c^3*d^2
- 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d^3 + 40*a^5*b^13*c^3*d
^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) + (8*tan(e/2 + (f*x)/2)*(a*
d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*(8*a*b^15 - 8*a^2*b^14 - 16*a^3*b^13 + 16*a
^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b
^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a
^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*a^4*b^7 - a^6*b^5) - (
(a*d - b*c)^4*((8*tan(e/2 + (f*x)/2)*(128*a^12*d^10 - 128*a^11*b*d^10 + 4*a^2*b^10*c^10 + 4*a^2*b^10*d^10 - 8*
a^3*b^9*d^10 + 28*a^4*b^8*d^10 - 48*a^5*b^7*d^10 + 28*a^6*b^6*d^10 - 8*a^7*b^5*d^10 + 8*a^8*b^4*d^10 + 192*a^9
*b^3*d^10 - 192*a^10*b^2*d^10 + 25*b^12*c^2*d^8 + 200*b^12*c^4*d^6 + 400*b^12*c^6*d^4 + 100*b^12*c^8*d^2 - 50*
a*b^11*c^2*d^8 - 480*a*b^11*c^3*d^7 - 400*a*b^11*c^4*d^6 - 1600*a*b^11*c^5*d^5 - 800*a*b^11*c^6*d^4 - 800*a*b^
11*c^7*d^3 + 40*a^2*b^10*c*d^9 - 180*a^3*b^9*c*d^9 + 320*a^4*b^8*c*d^9 - 260*a^5*b^7*c*d^9 + 200*a^6*b^6*c*d^9
 - 140*a^7*b^5*c*d^9 - 1520*a^8*b^4*c*d^9 + 1520*a^9*b^3*c*d^9 + 960*a^10*b^2*c*d^9 + 435*a^2*b^10*c^2*d^8 + 9
60*a^2*b^10*c^3*d^7 + 2600*a^2*b^10*c^4*d^6 + 3200*a^2*b^10*c^5*d^5 + 2400*a^2*b^10*c^6*d^4 + 160*a^2*b^10*c^8
*d^2 - 820*a^3*b^9*c^2*d^8 - 2240*a^3*b^9*c^3*d^7 - 4800*a^3*b^9*c^4*d^6 - 4000*a^3*b^9*c^5*d^5 + 1600*a^3*b^9
*c^6*d^4 + 160*a^3*b^9*c^7*d^3 + 1055*a^4*b^8*c^2*d^8 + 3520*a^4*b^8*c^3*d^7 + 4000*a^4*b^8*c^4*d^6 - 6400*a^4
*b^8*c^5*d^5 - 2640*a^4*b^8*c^6*d^4 - 80*a^4*b^8*c^8*d^2 - 1290*a^5*b^7*c^2*d^8 - 2400*a^5*b^7*c^3*d^7 + 10800
*a^5*b^7*c^4*d^6 + 7760*a^5*b^7*c^5*d^5 - 800*a^5*b^7*c^6*d^4 + 160*a^5*b^7*c^7*d^3 + 825*a^6*b^6*c^2*d^8 - 99
20*a^6*b^6*c^3*d^7 - 11560*a^6*b^6*c^4*d^6 + 3200*a^6*b^6*c^5*d^5 + 680*a^6*b^6*c^6*d^4 + 5240*a^7*b^5*c^2*d^8
 + 10080*a^7*b^5*c^3*d^7 - 5600*a^7*b^5*c^4*d^6 - 3168*a^7*b^5*c^5*d^5 - 5240*a^8*b^4*c^2*d^8 + 5440*a^8*b^4*c
^3*d^7 + 5600*a^8*b^4*c^4*d^6 - 3080*a^9*b^3*c^2*d^8 - 5440*a^9*b^3*c^3*d^7 + 3080*a^10*b^2*c^2*d^8 - 20*a*b^1
1*c*d^9 - 40*a*b^11*c^9*d - 960*a^11*b*c*d^9))/(a*b^10 + b^11 - a^2*b^9 - a^3*b^8) - (((8*(4*a*b^17*c^5 + 4*a*
b^17*d^5 - 10*b^18*c*d^4 - 20*b^18*c^4*d - 4*a^2*b^16*c^5 - 4*a^3*b^15*c^5 + 4*a^4*b^14*c^5 + 4*a^3*b^15*d^5 -
 20*a^4*b^14*d^5 - 16*a^5*b^13*d^5 + 36*a^6*b^12*d^5 + 8*a^7*b^11*d^5 - 16*a^8*b^10*d^5 - 40*b^18*c^3*d^2 + 80
*a*b^17*c^2*d^3 + 80*a*b^17*c^3*d^2 - 30*a^2*b^16*c*d^4 + 20*a^2*b^16*c^4*d + 80*a^3*b^15*c*d^4 - 20*a^3*b^15*
c^4*d + 70*a^4*b^14*c*d^4 - 140*a^5*b^13*c*d^4 - 30*a^6*b^12*c*d^4 + 60*a^7*b^11*c*d^4 - 120*a^2*b^16*c^2*d^3
+ 40*a^2*b^16*c^3*d^2 - 120*a^3*b^15*c^2*d^3 - 120*a^3*b^15*c^3*d^2 + 200*a^4*b^14*c^2*d^3 + 40*a^5*b^13*c^2*d
^3 + 40*a^5*b^13*c^3*d^2 - 80*a^6*b^12*c^2*d^3 + 20*a*b^17*c^4*d))/(a*b^14 + b^15 - a^2*b^13 - a^3*b^12) - (8*
tan(e/2 + (f*x)/2)*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*(8*a*b^15 - 8*a^2*b^1
4 - 16*a^3*b^13 + 16*a^4*b^12 + 8*a^5*b^11 - 8*a^6*b^10))/((a*b^10 + b^11 - a^2*b^9 - a^3*b^8)*(b^11 - 3*a^2*b
^9 + 3*a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a
^2*b^9 + 3*a^4*b^7 - a^6*b^5))*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c))/(b^11 - 3*a^2*b^9 + 3*
a^4*b^7 - a^6*b^5)))*(a*d - b*c)^4*((a + b)^3*(a - b)^3)^(1/2)*(4*a^2*d - 5*b^2*d + a*b*c)*2i)/(f*(b^11 - 3*a^
2*b^9 + 3*a^4*b^7 - a^6*b^5))

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{5} \sec {\left (e + f x \right )}}{\left (a + b \sec {\left (e + f x \right )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)*(c+d*sec(f*x+e))**5/(a+b*sec(f*x+e))**2,x)

[Out]

Integral((c + d*sec(e + f*x))**5*sec(e + f*x)/(a + b*sec(e + f*x))**2, x)

________________________________________________________________________________________

[next] [prev] [prev-tail] [front] [up]