\(\int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx\) [252]

Optimal result

Integrand size = 31, antiderivative size = 247 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\frac {d^3 (4 b c-a d) \text {arctanh}(\sin (e+f x))}{2 b^2 f}+\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) \text {arctanh}(\sin (e+f x))}{b^4 f}+\frac {2 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} f}+\frac {d^4 \tan (e+f x)}{b f}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \tan (e+f x)}{b^3 f}+\frac {d^3 (4 b c-a d) \sec (e+f x) \tan (e+f x)}{2 b^2 f}+\frac {d^4 \tan ^3(e+f x)}{3 b f} \]

[Out]

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

Rubi [A] (verified)

Time = 0.63 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {4073, 3031, 2738, 214, 3855, 3852, 8, 3853} \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\frac {d (2 b c-a d) \left (a^2 d^2-2 a b c d+2 b^2 c^2\right ) \text {arctanh}(\sin (e+f x))}{b^4 f}+\frac {d^2 \left (a^2 d^2-4 a b c d+6 b^2 c^2\right ) \tan (e+f x)}{b^3 f}+\frac {2 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{b^4 f \sqrt {a-b} \sqrt {a+b}}+\frac {d^3 (4 b c-a d) \text {arctanh}(\sin (e+f x))}{2 b^2 f}+\frac {d^3 (4 b c-a d) \tan (e+f x) \sec (e+f x)}{2 b^2 f}+\frac {d^4 \tan ^3(e+f x)}{3 b f}+\frac {d^4 \tan (e+f x)}{b f} \]

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, 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 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*} \text {integral}& = \int \frac {(d+c \cos (e+f x))^4 \sec ^4(e+f x)}{b+a \cos (e+f x)} \, dx \\ & = \int \left (\frac {(b c-a d)^4}{b^4 (b+a \cos (e+f x))}+\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) \sec (e+f x)}{b^4}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \sec ^2(e+f x)}{b^3}+\frac {d^3 (4 b c-a d) \sec ^3(e+f x)}{b^2}+\frac {d^4 \sec ^4(e+f x)}{b}\right ) \, dx \\ & = \frac {d^4 \int \sec ^4(e+f x) \, dx}{b}+\frac {(b c-a d)^4 \int \frac {1}{b+a \cos (e+f x)} \, dx}{b^4}+\frac {\left (d^3 (4 b c-a d)\right ) \int \sec ^3(e+f x) \, dx}{b^2}+\frac {\left (d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right )\right ) \int \sec ^2(e+f x) \, dx}{b^3}+\frac {\left (d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right )\right ) \int \sec (e+f x) \, dx}{b^4} \\ & = \frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) \text {arctanh}(\sin (e+f x))}{b^4 f}+\frac {d^3 (4 b c-a d) \sec (e+f x) \tan (e+f x)}{2 b^2 f}+\frac {\left (d^3 (4 b c-a d)\right ) \int \sec (e+f x) \, dx}{2 b^2}-\frac {d^4 \text {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (e+f x)\right )}{b f}+\frac {\left (2 (b c-a d)^4\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 )}{b^4 f}-\frac {\left (d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right )\right ) \text {Subst}(\int 1 \, dx,x,-\tan (e+f x))}{b^3 f} \\ & = \frac {d^3 (4 b c-a d) \text {arctanh}(\sin (e+f x))}{2 b^2 f}+\frac {d (2 b c-a d) \left (2 b^2 c^2-2 a b c d+a^2 d^2\right ) \text {arctanh}(\sin (e+f x))}{b^4 f}+\frac {2 (b c-a d)^4 \text {arctanh}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a+b}}\right )}{\sqrt {a-b} b^4 \sqrt {a+b} f}+\frac {d^4 \tan (e+f x)}{b f}+\frac {d^2 \left (6 b^2 c^2-4 a b c d+a^2 d^2\right ) \tan (e+f x)}{b^3 f}+\frac {d^3 (4 b c-a d) \sec (e+f x) \tan (e+f x)}{2 b^2 f}+\frac {d^4 \tan ^3(e+f x)}{3 b f} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(580\) vs. \(2(247)=494\).

Time = 5.68 (sec) , antiderivative size = 580, normalized size of antiderivative = 2.35 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\frac {\cos ^3(e+f x) (b+a \cos (e+f x)) (c+d \sec (e+f x))^4 \left (-\frac {24 (b c-a d)^4 \text {arctanh}\left (\frac {(-a+b) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {a^2-b^2}}\right )}{\sqrt {a^2-b^2}}-6 d \left (8 a^2 b c d^2-2 a^3 d^3+4 b^3 c \left (2 c^2+d^2\right )-a b^2 d \left (12 c^2+d^2\right )\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-6 d \left (-8 a^2 b c d^2+2 a^3 d^3-4 b^3 c \left (2 c^2+d^2\right )+a b^2 d \left (12 c^2+d^2\right )\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+\frac {b^2 d^3 (-3 a d+b (12 c+d))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {2 b^3 d^4 \sin \left (\frac {1}{2} (e+f x)\right )}{\left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}+\frac {4 b d^2 \left (-12 a b c d+3 a^2 d^2+2 b^2 \left (9 c^2+d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )}+\frac {2 b^3 d^4 \sin \left (\frac {1}{2} (e+f x)\right )}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3}-\frac {b^2 d^3 (-3 a d+b (12 c+d))}{\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}+\frac {4 b d^2 \left (-12 a b c d+3 a^2 d^2+2 b^2 \left (9 c^2+d^2\right )\right ) \sin \left (\frac {1}{2} (e+f x)\right )}{\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )}\right )}{12 b^4 f (d+c \cos (e+f x))^4 (a+b \sec (e+f x))} \]

[In]

Integrate[(Sec[e + f*x]*(c + d*Sec[e + f*x])^4)/(a + b*Sec[e + f*x]),x]

[Out]

(Cos[e + f*x]^3*(b + a*Cos[e + f*x])*(c + d*Sec[e + f*x])^4*((-24*(b*c - a*d)^4*ArcTanh[((-a + b)*Tan[(e + f*x
)/2])/Sqrt[a^2 - b^2]])/Sqrt[a^2 - b^2] - 6*d*(8*a^2*b*c*d^2 - 2*a^3*d^3 + 4*b^3*c*(2*c^2 + d^2) - a*b^2*d*(12
*c^2 + d^2))*Log[Cos[(e + f*x)/2] - Sin[(e + f*x)/2]] - 6*d*(-8*a^2*b*c*d^2 + 2*a^3*d^3 - 4*b^3*c*(2*c^2 + d^2
) + a*b^2*d*(12*c^2 + d^2))*Log[Cos[(e + f*x)/2] + Sin[(e + f*x)/2]] + (b^2*d^3*(-3*a*d + b*(12*c + d)))/(Cos[
(e + f*x)/2] - Sin[(e + f*x)/2])^2 + (2*b^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])^3 + (4
*b*d^2*(-12*a*b*c*d + 3*a^2*d^2 + 2*b^2*(9*c^2 + d^2))*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] - Sin[(e + f*x)/2])
+ (2*b^3*d^4*Sin[(e + f*x)/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])^3 - (b^2*d^3*(-3*a*d + b*(12*c + d)))/(C
os[(e + f*x)/2] + Sin[(e + f*x)/2])^2 + (4*b*d^2*(-12*a*b*c*d + 3*a^2*d^2 + 2*b^2*(9*c^2 + d^2))*Sin[(e + f*x)
/2])/(Cos[(e + f*x)/2] + Sin[(e + f*x)/2])))/(12*b^4*f*(d + c*Cos[e + f*x])^4*(a + b*Sec[e + f*x]))

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(479\) vs. \(2(232)=464\).

Time = 1.44 (sec) , antiderivative size = 480, normalized size of antiderivative = 1.94

method	result	size
derivativedivides	\(\frac {-\frac {2 \left (-a^{4} d^{4}+4 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {d^{4}}{3 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d \left (2 a^{3} d^{3}-8 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +a \,b^{2} d^{3}-8 b^{3} c^{3}-4 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{4}}-\frac {d^{2} \left (2 a^{2} d^{2}-8 a b c d +b \,d^{2} a +12 b^{2} c^{2}-4 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{3} \left (a d -4 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{4}}{3 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d \left (2 a^{3} d^{3}-8 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +a \,b^{2} d^{3}-8 b^{3} c^{3}-4 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{4}}-\frac {d^{2} \left (2 a^{2} d^{2}-8 a b c d +b \,d^{2} a +12 b^{2} c^{2}-4 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{3} \left (a d -4 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}}{f}\)	\(480\)
default	\(\frac {-\frac {2 \left (-a^{4} d^{4}+4 a^{3} b c \,d^{3}-6 a^{2} b^{2} c^{2} d^{2}+4 a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \operatorname {arctanh}\left (\frac {\left (a -b \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4} \sqrt {\left (a -b \right ) \left (a +b \right )}}-\frac {d^{4}}{3 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}-\frac {d \left (2 a^{3} d^{3}-8 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +a \,b^{2} d^{3}-8 b^{3} c^{3}-4 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}{2 b^{4}}-\frac {d^{2} \left (2 a^{2} d^{2}-8 a b c d +b \,d^{2} a +12 b^{2} c^{2}-4 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}+\frac {d^{3} \left (a d -4 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {d^{4}}{3 b \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{3}}+\frac {d \left (2 a^{3} d^{3}-8 a^{2} b c \,d^{2}+12 a \,b^{2} c^{2} d +a \,b^{2} d^{3}-8 b^{3} c^{3}-4 c \,d^{2} b^{3}\right ) \ln \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}{2 b^{4}}-\frac {d^{2} \left (2 a^{2} d^{2}-8 a b c d +b \,d^{2} a +12 b^{2} c^{2}-4 c d \,b^{2}+2 b^{2} d^{2}\right )}{2 b^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}-\frac {d^{3} \left (a d -4 b c +b d \right )}{2 b^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )^{2}}}{f}\)	\(480\)
risch	\(\text {Expression too large to display}\)	\(1324\)

[In]

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

[Out]

1/f*(-2/b^4*(-a^4*d^4+4*a^3*b*c*d^3-6*a^2*b^2*c^2*d^2+4*a*b^3*c^3*d-b^4*c^4)/((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^4/b/(tan(1/2*f*x+1/2*e)+1)^3-1/2*d*(2*a^3*d^3-8*a^2*b*c*d^2+12*
a*b^2*c^2*d+a*b^2*d^3-8*b^3*c^3-4*b^3*c*d^2)/b^4*ln(tan(1/2*f*x+1/2*e)+1)-1/2*d^2*(2*a^2*d^2-8*a*b*c*d+a*b*d^2
+12*b^2*c^2-4*b^2*c*d+2*b^2*d^2)/b^3/(tan(1/2*f*x+1/2*e)+1)+1/2*d^3*(a*d-4*b*c+b*d)/b^2/(tan(1/2*f*x+1/2*e)+1)
^2-1/3*d^4/b/(tan(1/2*f*x+1/2*e)-1)^3+1/2*d*(2*a^3*d^3-8*a^2*b*c*d^2+12*a*b^2*c^2*d+a*b^2*d^3-8*b^3*c^3-4*b^3*
c*d^2)/b^4*ln(tan(1/2*f*x+1/2*e)-1)-1/2*d^2*(2*a^2*d^2-8*a*b*c*d+a*b*d^2+12*b^2*c^2-4*b^2*c*d+2*b^2*d^2)/b^3/(
tan(1/2*f*x+1/2*e)-1)-1/2*d^3*(a*d-4*b*c+b*d)/b^2/(tan(1/2*f*x+1/2*e)-1)^2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 518 vs. \(2 (232) = 464\).

Time = 102.85 (sec) , antiderivative size = 1093, normalized size of antiderivative = 4.43 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

[1/12*(6*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(a^2 - b^2)*cos(f*x + e)^
3*log((2*a*b*cos(f*x + e) - (a^2 - 2*b^2)*cos(f*x + e)^2 + 2*sqrt(a^2 - b^2)*(b*cos(f*x + e) + a)*sin(f*x + e)
+ 2*a^2 - b^2)/(a^2*cos(f*x + e)^2 + 2*a*b*cos(f*x + e) + b^2)) + 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 -
a*b^4)*c^2*d^2 + 4*(2*a^4*b - a^2*b^3 - b^5)*c*d^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*cos(f*x + e)^3*log(sin(f*x
+ e) + 1) - 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 - a*b^4)*c^2*d^2 + 4*(2*a^4*b - a^2*b^3 - b^5)*c*d^3 - (
2*a^5 - a^3*b^2 - a*b^4)*d^4)*cos(f*x + e)^3*log(-sin(f*x + e) + 1) + 2*(2*(a^2*b^3 - b^5)*d^4 + 2*(18*(a^2*b^
3 - b^5)*c^2*d^2 - 12*(a^3*b^2 - a*b^4)*c*d^3 + (3*a^4*b - a^2*b^3 - 2*b^5)*d^4)*cos(f*x + e)^2 + 3*(4*(a^2*b^
3 - b^5)*c*d^3 - (a^3*b^2 - a*b^4)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^2*b^4 - b^6)*f*cos(f*x + e)^3), 1/12*(
12*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 + a^4*d^4)*sqrt(-a^2 + b^2)*arctan(-sqrt(-a^2
+ b^2)*(b*cos(f*x + e) + a)/((a^2 - b^2)*sin(f*x + e)))*cos(f*x + e)^3 + 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*
b^2 - a*b^4)*c^2*d^2 + 4*(2*a^4*b - a^2*b^3 - b^5)*c*d^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*cos(f*x + e)^3*log(s
in(f*x + e) + 1) - 3*(8*(a^2*b^3 - b^5)*c^3*d - 12*(a^3*b^2 - a*b^4)*c^2*d^2 + 4*(2*a^4*b - a^2*b^3 - b^5)*c*d
^3 - (2*a^5 - a^3*b^2 - a*b^4)*d^4)*cos(f*x + e)^3*log(-sin(f*x + e) + 1) + 2*(2*(a^2*b^3 - b^5)*d^4 + 2*(18*(
a^2*b^3 - b^5)*c^2*d^2 - 12*(a^3*b^2 - a*b^4)*c*d^3 + (3*a^4*b - a^2*b^3 - 2*b^5)*d^4)*cos(f*x + e)^2 + 3*(4*(
a^2*b^3 - b^5)*c*d^3 - (a^3*b^2 - a*b^4)*d^4)*cos(f*x + e))*sin(f*x + e))/((a^2*b^4 - b^6)*f*cos(f*x + e)^3)]

Sympy [F]

\[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\int \frac {\left (c + d \sec {\left (e + f x \right )}\right )^{4} \sec {\left (e + f x \right )}}{a + b \sec {\left (e + f x \right )}}\, dx \]

[In]

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

[Out]

Integral((c + d*sec(e + f*x))**4*sec(e + f*x)/(a + b*sec(e + f*x)), x)

Maxima [F(-2)]

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

[In]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 606 vs. \(2 (232) = 464\).

Time = 0.40 (sec) , antiderivative size = 606, normalized size of antiderivative = 2.45 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\frac {\frac {3 \, {\left (8 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 8 \, a^{2} b c d^{3} + 4 \, b^{3} c d^{3} - 2 \, a^{3} d^{4} - a b^{2} d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1 \right |}\right )}{b^{4}} - \frac {3 \, {\left (8 \, b^{3} c^{3} d - 12 \, a b^{2} c^{2} d^{2} + 8 \, a^{2} b c d^{3} + 4 \, b^{3} c d^{3} - 2 \, a^{3} d^{4} - a b^{2} d^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1 \right |}\right )}{b^{4}} - \frac {12 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\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 )}}{\sqrt {-a^{2} + b^{2}} b^{4}} - \frac {2 \, {\left (36 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 72 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 48 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 12 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, b^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, b^{2} c^{2} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 24 \, a b c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b^{2} c d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a b d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b^{2} d^{4} \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^{3}}}{6 \, f} \]

[In]

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

[Out]

1/6*(3*(8*b^3*c^3*d - 12*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + 4*b^3*c*d^3 - 2*a^3*d^4 - a*b^2*d^4)*log(abs(tan(1/2*
f*x + 1/2*e) + 1))/b^4 - 3*(8*b^3*c^3*d - 12*a*b^2*c^2*d^2 + 8*a^2*b*c*d^3 + 4*b^3*c*d^3 - 2*a^3*d^4 - a*b^2*d
^4)*log(abs(tan(1/2*f*x + 1/2*e) - 1))/b^4 - 12*(b^4*c^4 - 4*a*b^3*c^3*d + 6*a^2*b^2*c^2*d^2 - 4*a^3*b*c*d^3 +
a^4*d^4)*(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)))/(sqrt(-a^2 + b^2)*b^4) - 2*(36*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^5 - 24*a*b*c*d^3*ta
n(1/2*f*x + 1/2*e)^5 - 12*b^2*c*d^3*tan(1/2*f*x + 1/2*e)^5 + 6*a^2*d^4*tan(1/2*f*x + 1/2*e)^5 + 3*a*b*d^4*tan(
1/2*f*x + 1/2*e)^5 + 6*b^2*d^4*tan(1/2*f*x + 1/2*e)^5 - 72*b^2*c^2*d^2*tan(1/2*f*x + 1/2*e)^3 + 48*a*b*c*d^3*t
an(1/2*f*x + 1/2*e)^3 - 12*a^2*d^4*tan(1/2*f*x + 1/2*e)^3 - 4*b^2*d^4*tan(1/2*f*x + 1/2*e)^3 + 36*b^2*c^2*d^2*
tan(1/2*f*x + 1/2*e) - 24*a*b*c*d^3*tan(1/2*f*x + 1/2*e) + 12*b^2*c*d^3*tan(1/2*f*x + 1/2*e) + 6*a^2*d^4*tan(1
/2*f*x + 1/2*e) - 3*a*b*d^4*tan(1/2*f*x + 1/2*e) + 6*b^2*d^4*tan(1/2*f*x + 1/2*e))/((tan(1/2*f*x + 1/2*e)^2 -
1)^3*b^3))/f

Mupad [B] (verification not implemented)

Time = 23.13 (sec) , antiderivative size = 9987, normalized size of antiderivative = 40.43 \[ \int \frac {\sec (e+f x) (c+d \sec (e+f x))^4}{a+b \sec (e+f x)} \, dx=\text {Too large to display} \]

[In]

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

[Out]

(atan(((((((8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^
2*b^11*d^4 - 2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^
11*c^3*d - 24*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3 + 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 -
32*a*b^12*c^3*d))/b^9 - (8*tan(e/2 + (f*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*(b^2*((a*d^4)/2 + 6*a*c^2*d
^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^10)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3
+ 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^4 + (8*tan(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8 + 4*a*b^8*c^8 - 16*a
^8*b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8 + 16*a^7*b^2*d^8 - 16
*b^9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 192*a*b^8*c^4*d^4 + 19
2*a*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*d^7 - 104*a^4*b^5*c*d
^7 + 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 336*a^2*b^7*c^3*d^5 - 4
64*a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 560*a^3*b^6*c^3*d^5 + 8
80*a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 784*a^4*b^5*c^3*d^5 - 1
096*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 552*a^5*b^4*c^4*d^4 -
448*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^8*c^7*d - 64*a^8*b*c*
d^7))/b^6)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3)*1i)/b^4 - (((((
8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^2*b^11*d^4 -
2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^11*c^3*d - 2
4*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3 + 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 - 32*a*b^12*c
^3*d))/b^9 + (8*tan(e/2 + (f*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2
*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^10)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) +
a^3*d^4 - 4*a^2*b*c*d^3))/b^4 - (8*tan(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8 + 4*a*b^8*c^8 - 16*a^8*b*d^8 - a
^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8 + 16*a^7*b^2*d^8 - 16*b^9*c^2*d^6
- 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 192*a*b^8*c^4*d^4 + 192*a*b^8*c^5*
d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*d^7 - 104*a^4*b^5*c*d^7 + 128*a^5
*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 336*a^2*b^7*c^3*d^5 - 464*a^2*b^7*c
^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 560*a^3*b^6*c^3*d^5 + 880*a^3*b^6*c
^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 784*a^4*b^5*c^3*d^5 - 1096*a^4*b^5*
c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 552*a^5*b^4*c^4*d^4 - 448*a^6*b^3*
c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^8*c^7*d - 64*a^8*b*c*d^7))/b^6)*(
b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3)*1i)/b^4)/((16*(4*a^11*d^12
- 6*a^10*b*d^12 + 16*b^11*c^11*d - a^6*b^5*d^12 + 2*a^7*b^4*d^12 - 5*a^8*b^3*d^12 + 6*a^9*b^2*d^12 - 16*b^11*c
^6*d^6 - 64*b^11*c^8*d^4 + 8*b^11*c^9*d^3 - 64*b^11*c^10*d^2 + 72*a*b^10*c^5*d^7 + 32*a*b^10*c^6*d^6 + 368*a*b
^10*c^7*d^5 + 62*a*b^10*c^8*d^4 + 440*a*b^10*c^9*d^3 - 24*a*b^10*c^10*d^2 + 12*a^5*b^6*c*d^11 - 24*a^6*b^5*c*d
^11 + 60*a^7*b^4*c*d^11 - 72*a^8*b^3*c*d^11 + 72*a^9*b^2*c*d^11 - 129*a^2*b^9*c^4*d^8 - 144*a^2*b^9*c^5*d^7 -
936*a^2*b^9*c^6*d^6 - 496*a^2*b^9*c^7*d^5 - 1422*a^2*b^9*c^8*d^4 - 240*a^2*b^9*c^9*d^3 + 88*a^2*b^9*c^10*d^2 +
116*a^3*b^8*c^3*d^9 + 258*a^3*b^8*c^4*d^8 + 1384*a^3*b^8*c^5*d^7 + 1336*a^3*b^8*c^6*d^6 + 2848*a^3*b^8*c^7*d^
5 + 1148*a^3*b^8*c^8*d^4 - 208*a^3*b^8*c^9*d^3 - 54*a^4*b^7*c^2*d^10 - 232*a^4*b^7*c^3*d^9 - 1301*a^4*b^7*c^4*
d^8 - 1952*a^4*b^7*c^5*d^7 - 3888*a^4*b^7*c^6*d^6 - 2496*a^4*b^7*c^7*d^5 + 276*a^4*b^7*c^8*d^4 + 108*a^5*b^6*c
^2*d^10 + 788*a^5*b^6*c^3*d^9 + 1756*a^5*b^6*c^4*d^8 + 3744*a^5*b^6*c^5*d^7 + 3360*a^5*b^6*c^6*d^6 - 224*a^5*b
^6*c^7*d^5 - 294*a^6*b^5*c^2*d^10 - 1008*a^6*b^5*c^3*d^9 - 2556*a^6*b^5*c^4*d^8 - 3072*a^6*b^5*c^5*d^7 + 112*a
^6*b^5*c^6*d^6 + 360*a^7*b^4*c^2*d^10 + 1216*a^7*b^4*c^3*d^9 + 1968*a^7*b^4*c^4*d^8 - 32*a^7*b^4*c^5*d^7 - 384
*a^8*b^3*c^2*d^10 - 880*a^8*b^3*c^3*d^9 + 4*a^8*b^3*c^4*d^8 + 264*a^9*b^2*c^2*d^10 - 16*a*b^10*c^11*d - 48*a^1
0*b*c*d^11))/b^9 + (((((8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^1
1*c^4 + 2*a^2*b^11*d^4 - 2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3
+ 16*a^2*b^11*c^3*d - 24*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3 + 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*
b^12*c*d^3 - 32*a*b^12*c^3*d))/b^9 - (8*tan(e/2 + (f*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*(b^2*((a*d^4)/2
+ 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^10)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b
^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^4 + (8*tan(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8 + 4*a*b^
8*c^8 - 16*a^8*b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8 + 16*a^7*
b^2*d^8 - 16*b^9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 192*a*b^8*
c^4*d^4 + 192*a*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*d^7 - 104
*a^4*b^5*c*d^7 + 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 336*a^2*b^7
*c^3*d^5 - 464*a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 560*a^3*b^6
*c^3*d^5 + 880*a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 784*a^4*b^5
*c^3*d^5 - 1096*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 552*a^5*b^
4*c^4*d^4 - 448*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^8*c^7*d -
64*a^8*b*c*d^7))/b^6)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^
4 + (((((8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^2*b
^11*d^4 - 2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^11*
c^3*d - 24*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3 + 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 - 32
*a*b^12*c^3*d))/b^9 + (8*tan(e/2 + (f*x)/2)*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2)
- b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^10)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4
*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^4 - (8*tan(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8 + 4*a*b^8*c^8 - 16*a^8*
b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8 + 16*a^7*b^2*d^8 - 16*b^
9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 192*a*b^8*c^4*d^4 + 192*a
*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*d^7 - 104*a^4*b^5*c*d^7
+ 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 336*a^2*b^7*c^3*d^5 - 464*
a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 560*a^3*b^6*c^3*d^5 + 880*
a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 784*a^4*b^5*c^3*d^5 - 1096
*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 552*a^5*b^4*c^4*d^4 - 448
*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^8*c^7*d - 64*a^8*b*c*d^7
))/b^6)*(b^2*((a*d^4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3))/b^4))*(b^2*((a*d^
4)/2 + 6*a*c^2*d^2) - b^3*(2*c*d^3 + 4*c^3*d) + a^3*d^4 - 4*a^2*b*c*d^3)*2i)/(b^4*f) - ((tan(e/2 + (f*x)/2)*(2
*a^2*d^4 + 2*b^2*d^4 + 4*b^2*c*d^3 + 12*b^2*c^2*d^2 - a*b*d^4 - 8*a*b*c*d^3))/b^3 - (4*tan(e/2 + (f*x)/2)^3*(3
*a^2*d^4 + b^2*d^4 + 18*b^2*c^2*d^2 - 12*a*b*c*d^3))/(3*b^3) + (tan(e/2 + (f*x)/2)^5*(2*a^2*d^4 + 2*b^2*d^4 -
4*b^2*c*d^3 + 12*b^2*c^2*d^2 + a*b*d^4 - 8*a*b*c*d^3))/b^3)/(f*(3*tan(e/2 + (f*x)/2)^2 - 3*tan(e/2 + (f*x)/2)^
4 + tan(e/2 + (f*x)/2)^6 - 1)) + (atan(((((a + b)*(a - b))^(1/2)*((8*tan(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8
+ 4*a*b^8*c^8 - 16*a^8*b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8
+ 16*a^7*b^2*d^8 - 16*b^9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 1
92*a*b^8*c^4*d^4 + 192*a*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*
d^7 - 104*a^4*b^5*c*d^7 + 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 33
6*a^2*b^7*c^3*d^5 - 464*a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 56
0*a^3*b^6*c^3*d^5 + 880*a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 78
4*a^4*b^5*c^3*d^5 - 1096*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 5
52*a^5*b^4*c^4*d^4 - 448*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^
8*c^7*d - 64*a^8*b*c*d^7))/b^6 + (((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b
^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^2*b^11*d^4 - 2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*
a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^11*c^3*d - 24*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3
+ 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 - 32*a*b^12*c^3*d))/b^9 - (8*tan(e/2 + (f*x)/2)*(
(a + b)*(a - b))^(1/2)*(a*d - b*c)^4*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/(b^6*(b^6 - a^2*b^4))))/(b^6 - a^2*b
^4))*(a*d - b*c)^4*1i)/(b^6 - a^2*b^4) + (((a + b)*(a - b))^(1/2)*((8*tan(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^
8 + 4*a*b^8*c^8 - 16*a^8*b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8
+ 16*a^7*b^2*d^8 - 16*b^9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 +
192*a*b^8*c^4*d^4 + 192*a*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c
*d^7 - 104*a^4*b^5*c*d^7 + 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 3
36*a^2*b^7*c^3*d^5 - 464*a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 5
60*a^3*b^6*c^3*d^5 + 880*a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 7
84*a^4*b^5*c^3*d^5 - 1096*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 +
552*a^5*b^4*c^4*d^4 - 448*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b
^8*c^7*d - 64*a^8*b*c*d^7))/b^6 - (((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*
b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^2*b^11*d^4 - 2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4
*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^11*c^3*d - 24*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3
+ 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 - 32*a*b^12*c^3*d))/b^9 + (8*tan(e/2 + (f*x)/2)*
((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/(b^6*(b^6 - a^2*b^4))))/(b^6 - a^2*
b^4))*(a*d - b*c)^4*1i)/(b^6 - a^2*b^4))/((16*(4*a^11*d^12 - 6*a^10*b*d^12 + 16*b^11*c^11*d - a^6*b^5*d^12 + 2
*a^7*b^4*d^12 - 5*a^8*b^3*d^12 + 6*a^9*b^2*d^12 - 16*b^11*c^6*d^6 - 64*b^11*c^8*d^4 + 8*b^11*c^9*d^3 - 64*b^11
*c^10*d^2 + 72*a*b^10*c^5*d^7 + 32*a*b^10*c^6*d^6 + 368*a*b^10*c^7*d^5 + 62*a*b^10*c^8*d^4 + 440*a*b^10*c^9*d^
3 - 24*a*b^10*c^10*d^2 + 12*a^5*b^6*c*d^11 - 24*a^6*b^5*c*d^11 + 60*a^7*b^4*c*d^11 - 72*a^8*b^3*c*d^11 + 72*a^
9*b^2*c*d^11 - 129*a^2*b^9*c^4*d^8 - 144*a^2*b^9*c^5*d^7 - 936*a^2*b^9*c^6*d^6 - 496*a^2*b^9*c^7*d^5 - 1422*a^
2*b^9*c^8*d^4 - 240*a^2*b^9*c^9*d^3 + 88*a^2*b^9*c^10*d^2 + 116*a^3*b^8*c^3*d^9 + 258*a^3*b^8*c^4*d^8 + 1384*a
^3*b^8*c^5*d^7 + 1336*a^3*b^8*c^6*d^6 + 2848*a^3*b^8*c^7*d^5 + 1148*a^3*b^8*c^8*d^4 - 208*a^3*b^8*c^9*d^3 - 54
*a^4*b^7*c^2*d^10 - 232*a^4*b^7*c^3*d^9 - 1301*a^4*b^7*c^4*d^8 - 1952*a^4*b^7*c^5*d^7 - 3888*a^4*b^7*c^6*d^6 -
2496*a^4*b^7*c^7*d^5 + 276*a^4*b^7*c^8*d^4 + 108*a^5*b^6*c^2*d^10 + 788*a^5*b^6*c^3*d^9 + 1756*a^5*b^6*c^4*d^
8 + 3744*a^5*b^6*c^5*d^7 + 3360*a^5*b^6*c^6*d^6 - 224*a^5*b^6*c^7*d^5 - 294*a^6*b^5*c^2*d^10 - 1008*a^6*b^5*c^
3*d^9 - 2556*a^6*b^5*c^4*d^8 - 3072*a^6*b^5*c^5*d^7 + 112*a^6*b^5*c^6*d^6 + 360*a^7*b^4*c^2*d^10 + 1216*a^7*b^
4*c^3*d^9 + 1968*a^7*b^4*c^4*d^8 - 32*a^7*b^4*c^5*d^7 - 384*a^8*b^3*c^2*d^10 - 880*a^8*b^3*c^3*d^9 + 4*a^8*b^3
*c^4*d^8 + 264*a^9*b^2*c^2*d^10 - 16*a*b^10*c^11*d - 48*a^10*b*c*d^11))/b^9 + (((a + b)*(a - b))^(1/2)*((8*tan
(e/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8 + 4*a*b^8*c^8 - 16*a^8*b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*
d^8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8 + 16*a^7*b^2*d^8 - 16*b^9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48
*a*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 192*a*b^8*c^4*d^4 + 192*a*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*
d^7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*d^7 - 104*a^4*b^5*c*d^7 + 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^
7*b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 336*a^2*b^7*c^3*d^5 - 464*a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*
b^7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 560*a^3*b^6*c^3*d^5 + 880*a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*
b^6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 784*a^4*b^5*c^3*d^5 - 1096*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5
*b^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 552*a^5*b^4*c^4*d^4 - 448*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7
*b^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^8*c^7*d - 64*a^8*b*c*d^7))/b^6 + (((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*
((8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^2*b^11*d^4
- 2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^11*c^3*d -
24*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3 + 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 - 32*a*b^12
*c^3*d))/b^9 - (8*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8)
)/(b^6*(b^6 - a^2*b^4))))/(b^6 - a^2*b^4))*(a*d - b*c)^4)/(b^6 - a^2*b^4) - (((a + b)*(a - b))^(1/2)*((8*tan(e
/2 + (f*x)/2)*(8*a^9*d^8 - 4*b^9*c^8 + 4*a*b^8*c^8 - 16*a^8*b*d^8 - a^2*b^7*d^8 + 3*a^3*b^6*d^8 - 7*a^4*b^5*d^
8 + 13*a^5*b^4*d^8 - 16*a^6*b^3*d^8 + 16*a^7*b^2*d^8 - 16*b^9*c^2*d^6 - 64*b^9*c^4*d^4 - 64*b^9*c^6*d^2 + 48*a
*b^8*c^2*d^6 + 112*a*b^8*c^3*d^5 + 192*a*b^8*c^4*d^4 + 192*a*b^8*c^5*d^3 + 192*a*b^8*c^6*d^2 - 24*a^2*b^7*c*d^
7 - 32*a^2*b^7*c^7*d + 56*a^3*b^6*c*d^7 - 104*a^4*b^5*c*d^7 + 128*a^5*b^4*c*d^7 - 128*a^6*b^3*c*d^7 + 128*a^7*
b^2*c*d^7 - 136*a^2*b^7*c^2*d^6 - 336*a^2*b^7*c^3*d^5 - 464*a^2*b^7*c^4*d^4 - 576*a^2*b^7*c^5*d^3 - 304*a^2*b^
7*c^6*d^2 + 280*a^3*b^6*c^2*d^6 + 560*a^3*b^6*c^3*d^5 + 880*a^3*b^6*c^4*d^4 + 800*a^3*b^6*c^5*d^3 + 176*a^3*b^
6*c^6*d^2 - 376*a^4*b^5*c^2*d^6 - 784*a^4*b^5*c^3*d^5 - 1096*a^4*b^5*c^4*d^4 - 416*a^4*b^5*c^5*d^3 + 424*a^5*b
^4*c^2*d^6 + 896*a^5*b^4*c^3*d^5 + 552*a^5*b^4*c^4*d^4 - 448*a^6*b^3*c^2*d^6 - 448*a^6*b^3*c^3*d^5 + 224*a^7*b
^2*c^2*d^6 + 8*a*b^8*c*d^7 + 32*a*b^8*c^7*d - 64*a^8*b*c*d^7))/b^6 - (((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*((
8*(4*b^13*c^4 - 8*a*b^12*c^4 - 2*a*b^12*d^4 + 8*b^13*c*d^3 + 16*b^13*c^3*d + 4*a^2*b^11*c^4 + 2*a^2*b^11*d^4 -
2*a^3*b^10*d^4 + 6*a^4*b^9*d^4 - 4*a^5*b^8*d^4 - 24*a*b^12*c^2*d^2 + 8*a^2*b^11*c*d^3 + 16*a^2*b^11*c^3*d - 2
4*a^3*b^10*c*d^3 + 16*a^4*b^9*c*d^3 + 48*a^2*b^11*c^2*d^2 - 24*a^3*b^10*c^2*d^2 - 8*a*b^12*c*d^3 - 32*a*b^12*c
^3*d))/b^9 + (8*tan(e/2 + (f*x)/2)*((a + b)*(a - b))^(1/2)*(a*d - b*c)^4*(8*a*b^10 - 16*a^2*b^9 + 8*a^3*b^8))/
(b^6*(b^6 - a^2*b^4))))/(b^6 - a^2*b^4))*(a*d - b*c)^4)/(b^6 - a^2*b^4)))*((a + b)*(a - b))^(1/2)*(a*d - b*c)^
4*2i)/(f*(b^6 - a^2*b^4))