3.3.58 \(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^5}{(a+b \sec (e+f x))^2} \, dx\) [258]

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

[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*f*x+1/2*e)/(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*f*x+1/2*e)/(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

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Rubi [A]
time = 0.48, 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 = {4073, 3031, 2743, 12, 2738, 214, 3855, 3852, 8, 3853} \begin {gather*} \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} \end {gather*}

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 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 2743

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] && Integ
erQ[2*n]

Rule 3031

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 3852

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 3853

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 3855

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

Rule 4073

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 \text {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 ) \text {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 ) \text {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 ) \text {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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(1137\) vs. \(2(379)=758\).
time = 6.53, size = 1137, normalized size = 3.00 \begin {gather*} -\frac {2 (b c-a d)^4 \left (-a b c-4 a^2 d+5 b^2 d\right ) \tanh ^{-1}\left (\frac {(-a+b) \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 (-a^2+b^2\right ) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2}+\frac {\left (-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\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 (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\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 {(b+a \cos (e+f x)) (c+d \sec (e+f x))^5 \left (-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))\right )}{24 b^4 \left (-a^2+b^2\right ) f (d+c \cos (e+f x))^5 (a+b \sec (e+f x))^2} \end {gather*}

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)

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Maple [A]
time = 1.46, size = 689, normalized size = 1.82

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a -b \right )}-\frac {\left (4 a^{6} d^{5}-15 a^{5} b c \,d^{4}+20 a^{4} b^{2} c^{2} d^{3}-5 a^{4} b^{2} d^{5}-10 a^{3} b^{3} c^{3} d^{2}+20 a^{3} b^{3} c \,d^{4}-30 a^{2} b^{4} c^{2} d^{3}+b^{5} c^{5} a +20 a \,b^{5} c^{3} d^{2}-5 b^{6} c^{4} d \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}}{f}\)	\(689\)
default	\(\frac {-\frac {2 \left (\frac {b \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\left (a^{2}-b^{2}\right ) \left (a \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-a -b \right )}-\frac {\left (4 a^{6} d^{5}-15 a^{5} b c \,d^{4}+20 a^{4} b^{2} c^{2} d^{3}-5 a^{4} b^{2} d^{5}-10 a^{3} b^{3} c^{3} d^{2}+20 a^{3} b^{3} c \,d^{4}-30 a^{2} b^{4} c^{2} d^{3}+b^{5} c^{5} a +20 a \,b^{5} c^{3} d^{2}-5 b^{6} c^{4} d \right ) \arctanh \left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{\left (a +b \right ) \left (a -b \right ) \sqrt {\left (a +b \right ) \left (a -b \right )}}\right )}{b^{5}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{5}}{3 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d^{2} \left (8 a^{3} d^{3}-30 a^{2} b c \,d^{2}+40 a \,b^{2} c^{2} d +2 a \,b^{2} d^{3}-20 c^{3} b^{3}-5 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{5}}-\frac {d^{3} \left (6 a^{2} d^{2}-20 a b d c +2 b \,d^{2} a +20 b^{2} c^{2}-5 b^{2} c d +2 b^{2} d^{2}\right )}{2 b^{4} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{4} \left (2 a d -5 b c +d b \right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}}{f}\)	\(689\)
risch	\(\text {Expression too large to display}\)	\(3699\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(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 de

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 857 vs. \(2 (355) = 710\).
time = 0.62, size = 857, normalized size = 2.26 \begin {gather*} -\frac {\frac {12 \, {\left (a b^{5} c^{5} - 5 \, b^{6} c^{4} d - 10 \, a^{3} b^{3} c^{3} d^{2} + 20 \, a b^{5} c^{3} d^{2} + 20 \, a^{4} b^{2} c^{2} d^{3} - 30 \, a^{2} b^{4} c^{2} d^{3} - 15 \, a^{5} b c d^{4} + 20 \, a^{3} b^{3} c d^{4} + 4 \, a^{6} d^{5} - 5 \, a^{4} b^{2} d^{5}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, a - 2 \, b\right ) + \arctan \left (\frac {a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-a^{2} + b^{2}}}\right )\right )}}{{\left (a^{2} b^{5} - b^{7}\right )} \sqrt {-a^{2} + b^{2}}} - \frac {12 \, {\left (b^{5} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 5 \, a b^{4} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 10 \, a^{2} b^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 10 \, a^{3} b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, a^{4} b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - a^{5} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - b \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a - b\right )}} - \frac {3 \, {\left (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}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{5}} + \frac {3 \, {\left (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}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{5}} + \frac {2 \, {\left (60 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 60 \, a b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, b^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 18 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a b d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, b^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 120 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 120 \, a b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 36 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 60 \, b^{2} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 60 \, a b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, b^{2} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 18 \, a^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a b d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} b^{4}}}{6 \, f} \end {gather*}

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]

-1/6*(12*(a*b^5*c^5 - 5*b^6*c^4*d - 10*a^3*b^3*c^3*d^2 + 20*a*b^5*c^3*d^2 + 20*a^4*b^2*c^2*d^3 - 30*a^2*b^4*c^
2*d^3 - 15*a^5*b*c*d^4 + 20*a^3*b^3*c*d^4 + 4*a^6*d^5 - 5*a^4*b^2*d^5)*(pi*floor(1/2*(f*x + e)/pi + 1/2)*sgn(2
*a - 2*b) + arctan((a*tan(1/2*f*x + 1/2*e) - b*tan(1/2*f*x + 1/2*e))/sqrt(-a^2 + b^2)))/((a^2*b^5 - b^7)*sqrt(
-a^2 + b^2)) - 12*(b^5*c^5*tan(1/2*f*x + 1/2*e) - 5*a*b^4*c^4*d*tan(1/2*f*x + 1/2*e) + 10*a^2*b^3*c^3*d^2*tan(
1/2*f*x + 1/2*e) - 10*a^3*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e) + 5*a^4*b*c*d^4*tan(1/2*f*x + 1/2*e) - a^5*d^5*tan(
1/2*f*x + 1/2*e))/((a^2*b^4 - b^6)*(a*tan(1/2*f*x + 1/2*e)^2 - b*tan(1/2*f*x + 1/2*e)^2 - a - b)) - 3*(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)*log(abs(tan(1/2*f*x + 1/2
*e) + 1))/b^5 + 3*(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)
*log(abs(tan(1/2*f*x + 1/2*e) - 1))/b^5 + 2*(60*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^5 - 60*a*b*c*d^4*tan(1/2*f*x
+ 1/2*e)^5 - 15*b^2*c*d^4*tan(1/2*f*x + 1/2*e)^5 + 18*a^2*d^5*tan(1/2*f*x + 1/2*e)^5 + 6*a*b*d^5*tan(1/2*f*x +
1/2*e)^5 + 6*b^2*d^5*tan(1/2*f*x + 1/2*e)^5 - 120*b^2*c^2*d^3*tan(1/2*f*x + 1/2*e)^3 + 120*a*b*c*d^4*tan(1/2*
f*x + 1/2*e)^3 - 36*a^2*d^5*tan(1/2*f*x + 1/2*e)^3 - 4*b^2*d^5*tan(1/2*f*x + 1/2*e)^3 + 60*b^2*c^2*d^3*tan(1/2
*f*x + 1/2*e) - 60*a*b*c*d^4*tan(1/2*f*x + 1/2*e) + 15*b^2*c*d^4*tan(1/2*f*x + 1/2*e) + 18*a^2*d^5*tan(1/2*f*x
+ 1/2*e) - 6*a*b*d^5*tan(1/2*f*x + 1/2*e) + 6*b^2*d^5*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 - 1)^3*b
^4))/f

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Mupad [B]
time = 16.95, size = 2500, normalized size = 6.60 \begin {gather*} \text {Too large to display} \end {gather*}

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 +...

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