\(\int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx\) [121]

Optimal result

Integrand size = 28, antiderivative size = 262 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {3 a \text {arctanh}(\sin (c+d x))}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (a^2+b^2\right )^2 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {\left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\sec ^5(c+d x)}{5 b d}-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \sec (c+d x) \tan (c+d x)}{2 b^4 d}-\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 b^2 d} \]

[Out]

-3/8*a*arctanh(sin(d*x+c))/b^2/d-1/2*a*(a^2+b^2)*arctanh(sin(d*x+c))/b^4/d-a*(a^2+b^2)^2*arctanh(sin(d*x+c))/b
^6/d-(a^2+b^2)^(5/2)*arctanh((b*cos(d*x+c)-a*sin(d*x+c))/(a^2+b^2)^(1/2))/b^6/d+(a^2+b^2)^2*sec(d*x+c)/b^5/d+1
/3*(a^2+b^2)*sec(d*x+c)^3/b^3/d+1/5*sec(d*x+c)^5/b/d-3/8*a*sec(d*x+c)*tan(d*x+c)/b^2/d-1/2*a*(a^2+b^2)*sec(d*x
+c)*tan(d*x+c)/b^4/d-1/4*a*sec(d*x+c)^3*tan(d*x+c)/b^2/d

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {3183, 3853, 3855, 3153, 212} \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {a \left (a^2+b^2\right )^2 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {\left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}-\frac {a \left (a^2+b^2\right ) \tan (c+d x) \sec (c+d x)}{2 b^4 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}-\frac {3 a \text {arctanh}(\sin (c+d x))}{8 b^2 d}-\frac {a \tan (c+d x) \sec ^3(c+d x)}{4 b^2 d}-\frac {3 a \tan (c+d x) \sec (c+d x)}{8 b^2 d}+\frac {\sec ^5(c+d x)}{5 b d} \]

[In]

Int[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 3153

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

Rule 3183

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

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^5(c+d x)}{5 b d}-\frac {a \int \sec ^5(c+d x) \, dx}{b^2}+\frac {\left (a^2+b^2\right ) \int \frac {\sec ^4(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^2} \\ & = \frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\sec ^5(c+d x)}{5 b d}-\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 b^2 d}-\frac {(3 a) \int \sec ^3(c+d x) \, dx}{4 b^2}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \sec ^3(c+d x) \, dx}{b^4}+\frac {\left (a^2+b^2\right )^2 \int \frac {\sec ^2(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^4} \\ & = \frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\sec ^5(c+d x)}{5 b d}-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \sec (c+d x) \tan (c+d x)}{2 b^4 d}-\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 b^2 d}-\frac {(3 a) \int \sec (c+d x) \, dx}{8 b^2}-\frac {\left (a \left (a^2+b^2\right )\right ) \int \sec (c+d x) \, dx}{2 b^4}-\frac {\left (a \left (a^2+b^2\right )^2\right ) \int \sec (c+d x) \, dx}{b^6}+\frac {\left (a^2+b^2\right )^3 \int \frac {1}{a \cos (c+d x)+b \sin (c+d x)} \, dx}{b^6} \\ & = -\frac {3 a \text {arctanh}(\sin (c+d x))}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (a^2+b^2\right )^2 \text {arctanh}(\sin (c+d x))}{b^6 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\sec ^5(c+d x)}{5 b d}-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \sec (c+d x) \tan (c+d x)}{2 b^4 d}-\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 b^2 d}-\frac {\left (a^2+b^2\right )^3 \text {Subst}\left (\int \frac {1}{a^2+b^2-x^2} \, dx,x,b \cos (c+d x)-a \sin (c+d x)\right )}{b^6 d} \\ & = -\frac {3 a \text {arctanh}(\sin (c+d x))}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \text {arctanh}(\sin (c+d x))}{2 b^4 d}-\frac {a \left (a^2+b^2\right )^2 \text {arctanh}(\sin (c+d x))}{b^6 d}-\frac {\left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b \cos (c+d x)-a \sin (c+d x)}{\sqrt {a^2+b^2}}\right )}{b^6 d}+\frac {\left (a^2+b^2\right )^2 \sec (c+d x)}{b^5 d}+\frac {\left (a^2+b^2\right ) \sec ^3(c+d x)}{3 b^3 d}+\frac {\sec ^5(c+d x)}{5 b d}-\frac {3 a \sec (c+d x) \tan (c+d x)}{8 b^2 d}-\frac {a \left (a^2+b^2\right ) \sec (c+d x) \tan (c+d x)}{2 b^4 d}-\frac {a \sec ^3(c+d x) \tan (c+d x)}{4 b^2 d} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(661\) vs. \(2(262)=524\).

Time = 6.27 (sec) , antiderivative size = 661, normalized size of antiderivative = 2.52 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\sec (c+d x) \left (240 a^4 b+520 a^2 b^3+298 b^5+480 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {-b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )+30 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-30 a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {3 b^4 (-5 a+2 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}+\frac {b^2 \left (-60 a^3+20 a^2 b-105 a b^2+29 b^3\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {12 b^5 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {2 b^3 \left (20 a^2+29 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 b \left (120 a^4+260 a^2 b^2+149 b^4\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )}-\frac {12 b^5 \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^5}+\frac {3 b^4 (5 a+2 b)}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^4}-\frac {2 b^3 \left (20 a^2+29 b^2\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {b^2 \left (60 a^3+20 a^2 b+105 a b^2+29 b^3\right )}{\left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}-\frac {2 b \left (120 a^4+260 a^2 b^2+149 b^4\right ) \sin \left (\frac {1}{2} (c+d x)\right )}{\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )}\right ) (a \cos (c+d x)+b \sin (c+d x))}{240 b^6 d (a+b \tan (c+d x))} \]

[In]

Integrate[Sec[c + d*x]^6/(a*Cos[c + d*x] + b*Sin[c + d*x]),x]

[Out]

(Sec[c + d*x]*(240*a^4*b + 520*a^2*b^3 + 298*b^5 + 480*(a^2 + b^2)^(5/2)*ArcTanh[(-b + a*Tan[(c + d*x)/2])/Sqr
t[a^2 + b^2]] + 30*a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] - 30*a*(8*a^4 + 20
*a^2*b^2 + 15*b^4)*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (3*b^4*(-5*a + 2*b))/(Cos[(c + d*x)/2] - Sin[(c
+ d*x)/2])^4 + (b^2*(-60*a^3 + 20*a^2*b - 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2 + (12*b
^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^5 + (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos
[(c + d*x)/2] - Sin[(c + d*x)/2])^3 + (2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/
2] - Sin[(c + d*x)/2]) - (12*b^5*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^5 + (3*b^4*(5*a + 2*b
))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^4 - (2*b^3*(20*a^2 + 29*b^2)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Si
n[(c + d*x)/2])^3 + (b^2*(60*a^3 + 20*a^2*b + 105*a*b^2 + 29*b^3))/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2 - (
2*b*(120*a^4 + 260*a^2*b^2 + 149*b^4)*Sin[(c + d*x)/2])/(Cos[(c + d*x)/2] + Sin[(c + d*x)/2]))*(a*Cos[c + d*x]
+ b*Sin[c + d*x]))/(240*b^6*d*(a + b*Tan[c + d*x]))

Maple [A] (verified)

Time = 1.46 (sec) , antiderivative size = 479, normalized size of antiderivative = 1.83

method	result	size
derivativedivides	\(\frac {-\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {a +2 b}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 a \,b^{3}+15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{6}}+\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2 b -a}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 a \,b^{3}-15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{6}}-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}}{d}\)	\(479\)
default	\(\frac {-\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{5}}-\frac {a +2 b}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 a \,b^{3}+15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 b^{6}}+\frac {1}{5 b \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{5}}-\frac {2 b -a}{4 b^{2} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 a \,b^{3}-15 b^{4}}{8 b^{5} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{8 b^{6}}-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}}{d}\)	\(479\)
risch	\(\frac {{\mathrm e}^{i \left (d x +c \right )} \left (-330 i a \,b^{3} {\mathrm e}^{2 i \left (d x +c \right )}+330 i a \,b^{3} {\mathrm e}^{6 i \left (d x +c \right )}+120 a^{4} {\mathrm e}^{8 i \left (d x +c \right )}+240 a^{2} b^{2} {\mathrm e}^{8 i \left (d x +c \right )}+120 b^{4} {\mathrm e}^{8 i \left (d x +c \right )}-60 i a^{3} b +60 i a^{3} b \,{\mathrm e}^{8 i \left (d x +c \right )}+480 a^{4} {\mathrm e}^{6 i \left (d x +c \right )}+1120 a^{2} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+640 b^{4} {\mathrm e}^{6 i \left (d x +c \right )}+720 a^{4} {\mathrm e}^{4 i \left (d x +c \right )}+1760 a^{2} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}+1424 b^{4} {\mathrm e}^{4 i \left (d x +c \right )}-105 i a \,b^{3}-120 i a^{3} b \,{\mathrm e}^{2 i \left (d x +c \right )}+480 a^{4} {\mathrm e}^{2 i \left (d x +c \right )}+1120 a^{2} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}+640 b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+120 i a^{3} b \,{\mathrm e}^{6 i \left (d x +c \right )}+105 i a \,b^{3} {\mathrm e}^{8 i \left (d x +c \right )}+120 a^{4}+240 a^{2} b^{2}+120 b^{4}\right )}{60 d \,b^{5} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{5}}+\frac {a^{5} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d \,b^{6}}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 b^{4} d}+\frac {15 a \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 b^{2} d}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \left (i a -b \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\right )}{d \,b^{6}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \left (i a -b \right )}{a^{6}+3 a^{4} b^{2}+3 a^{2} b^{4}+b^{6}}\right )}{d \,b^{6}}-\frac {a^{5} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,b^{6}}-\frac {5 a^{3} \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{2 b^{4} d}-\frac {15 a \ln \left (i+{\mathrm e}^{i \left (d x +c \right )}\right )}{8 b^{2} d}\)	\(660\)

[In]

int(sec(d*x+c)^6/(cos(d*x+c)*a+b*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(-1/5/b/(tan(1/2*d*x+1/2*c)-1)^5-1/4*(a+2*b)/b^2/(tan(1/2*d*x+1/2*c)-1)^4-1/12*(4*a^2+6*a*b+13*b^2)/b^3/(t
an(1/2*d*x+1/2*c)-1)^3-1/8*(4*a^3+4*a^2*b+11*a*b^2+9*b^3)/b^4/(tan(1/2*d*x+1/2*c)-1)^2-1/8*(8*a^4+4*a^3*b+20*a
^2*b^2+9*a*b^3+15*b^4)/b^5/(tan(1/2*d*x+1/2*c)-1)+1/8*a*(8*a^4+20*a^2*b^2+15*b^4)/b^6*ln(tan(1/2*d*x+1/2*c)-1)
+1/5/b/(tan(1/2*d*x+1/2*c)+1)^5-1/4*(2*b-a)/b^2/(tan(1/2*d*x+1/2*c)+1)^4-1/12*(-4*a^2+6*a*b-13*b^2)/b^3/(tan(1
/2*d*x+1/2*c)+1)^3-1/8*(-4*a^3+4*a^2*b-11*a*b^2+9*b^3)/b^4/(tan(1/2*d*x+1/2*c)+1)^2-1/8*(-8*a^4+4*a^3*b-20*a^2
*b^2+9*a*b^3-15*b^4)/b^5/(tan(1/2*d*x+1/2*c)+1)-1/8*a*(8*a^4+20*a^2*b^2+15*b^4)/b^6*ln(tan(1/2*d*x+1/2*c)+1)-2
/b^6*(-a^6-3*a^4*b^2-3*a^2*b^4-b^6)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tan(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.47 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.32 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {120 \, {\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt {a^{2} + b^{2}} \cos \left (d x + c\right )^{5} \log \left (-\frac {2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, a^{2} - b^{2} + 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cos \left (d x + c\right ) - a \sin \left (d x + c\right )\right )}}{2 \, a b \cos \left (d x + c\right ) \sin \left (d x + c\right ) + {\left (a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} + b^{2}}\right ) - 15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \cos \left (d x + c\right )^{5} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 48 \, b^{5} + 240 \, {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{4} + 80 \, {\left (a^{2} b^{3} + b^{5}\right )} \cos \left (d x + c\right )^{2} - 30 \, {\left (2 \, a b^{4} \cos \left (d x + c\right ) + {\left (4 \, a^{3} b^{2} + 7 \, a b^{4}\right )} \cos \left (d x + c\right )^{3}\right )} \sin \left (d x + c\right )}{240 \, b^{6} d \cos \left (d x + c\right )^{5}} \]

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/240*(120*(a^4 + 2*a^2*b^2 + b^4)*sqrt(a^2 + b^2)*cos(d*x + c)^5*log(-(2*a*b*cos(d*x + c)*sin(d*x + c) + (a^2
- b^2)*cos(d*x + c)^2 - 2*a^2 - b^2 + 2*sqrt(a^2 + b^2)*(b*cos(d*x + c) - a*sin(d*x + c)))/(2*a*b*cos(d*x + c
)*sin(d*x + c) + (a^2 - b^2)*cos(d*x + c)^2 + b^2)) - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5*log(si
n(d*x + c) + 1) + 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*cos(d*x + c)^5*log(-sin(d*x + c) + 1) + 48*b^5 + 240*(a^4
*b + 2*a^2*b^3 + b^5)*cos(d*x + c)^4 + 80*(a^2*b^3 + b^5)*cos(d*x + c)^2 - 30*(2*a*b^4*cos(d*x + c) + (4*a^3*b
^2 + 7*a*b^4)*cos(d*x + c)^3)*sin(d*x + c))/(b^6*d*cos(d*x + c)^5)

Sympy [F]

\[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\int \frac {\sec ^{6}{\left (c + d x \right )}}{a \cos {\left (c + d x \right )} + b \sin {\left (c + d x \right )}}\, dx \]

[In]

integrate(sec(d*x+c)**6/(a*cos(d*x+c)+b*sin(d*x+c)),x)

[Out]

Integral(sec(c + d*x)**6/(a*cos(c + d*x) + b*sin(c + d*x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 625 vs. \(2 (244) = 488\).

Time = 0.33 (sec) , antiderivative size = 625, normalized size of antiderivative = 2.39 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\frac {\frac {2 \, {\left (120 \, a^{4} + 280 \, a^{2} b^{2} + 184 \, b^{4} - \frac {15 \, {\left (4 \, a^{3} b + 9 \, a b^{3}\right )} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {80 \, {\left (6 \, a^{4} + 13 \, a^{2} b^{2} + 7 \, b^{4}\right )} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {30 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {80 \, {\left (9 \, a^{4} + 20 \, a^{2} b^{2} + 14 \, b^{4}\right )} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {240 \, {\left (2 \, a^{4} + 5 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} - \frac {30 \, {\left (4 \, a^{3} b + 5 \, a b^{3}\right )} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {120 \, {\left (a^{4} + 3 \, a^{2} b^{2} + 3 \, b^{4}\right )} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {15 \, {\left (4 \, a^{3} b + 9 \, a b^{3}\right )} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}}\right )}}{b^{5} - \frac {5 \, b^{5} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, b^{5} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {10 \, b^{5} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, b^{5} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {b^{5} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}} - \frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}{b^{6}} + \frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - 1\right )}{b^{6}} - \frac {120 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \sqrt {a^{2} + b^{2}}}{b - \frac {a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}}}{120 \, d} \]

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="maxima")

[Out]

1/120*(2*(120*a^4 + 280*a^2*b^2 + 184*b^4 - 15*(4*a^3*b + 9*a*b^3)*sin(d*x + c)/(cos(d*x + c) + 1) - 80*(6*a^4
+ 13*a^2*b^2 + 7*b^4)*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 30*(4*a^3*b + 5*a*b^3)*sin(d*x + c)^3/(cos(d*x +
c) + 1)^3 + 80*(9*a^4 + 20*a^2*b^2 + 14*b^4)*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 240*(2*a^4 + 5*a^2*b^2 + 3*
b^4)*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 - 30*(4*a^3*b + 5*a*b^3)*sin(d*x + c)^7/(cos(d*x + c) + 1)^7 + 120*(a
^4 + 3*a^2*b^2 + 3*b^4)*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 + 15*(4*a^3*b + 9*a*b^3)*sin(d*x + c)^9/(cos(d*x +
c) + 1)^9)/(b^5 - 5*b^5*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 10*b^5*sin(d*x + c)^4/(cos(d*x + c) + 1)^4 - 10
*b^5*sin(d*x + c)^6/(cos(d*x + c) + 1)^6 + 5*b^5*sin(d*x + c)^8/(cos(d*x + c) + 1)^8 - b^5*sin(d*x + c)^10/(co
s(d*x + c) + 1)^10) - 15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) + 1)/b^6 + 15*(8*
a^5 + 20*a^3*b^2 + 15*a*b^4)*log(sin(d*x + c)/(cos(d*x + c) + 1) - 1)/b^6 - 120*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 +
b^6)*log((b - a*sin(d*x + c)/(cos(d*x + c) + 1) + sqrt(a^2 + b^2))/(b - a*sin(d*x + c)/(cos(d*x + c) + 1) - s
qrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6))/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 554 vs. \(2 (244) = 488\).

Time = 0.36 (sec) , antiderivative size = 554, normalized size of antiderivative = 2.11 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=-\frac {\frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right )}{b^{6}} - \frac {15 \, {\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right )}{b^{6}} + \frac {120 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, b + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} + \frac {2 \, {\left (60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} + 120 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 360 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 360 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} - 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} - 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1200 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 720 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} + 720 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1600 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 1120 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 120 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 150 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 480 \, a^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1040 \, a^{2} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 560 \, b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 60 \, a^{3} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 135 \, a b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 120 \, a^{4} + 280 \, a^{2} b^{2} + 184 \, b^{4}\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{5} b^{5}}}{120 \, d} \]

[In]

integrate(sec(d*x+c)^6/(a*cos(d*x+c)+b*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/120*(15*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) + 1))/b^6 - 15*(8*a^5 + 20*a^3*b^2 + 1
5*a*b^4)*log(abs(tan(1/2*d*x + 1/2*c) - 1))/b^6 + 120*(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log(abs(2*a*tan(1/2*
d*x + 1/2*c) - 2*b - 2*sqrt(a^2 + b^2))/abs(2*a*tan(1/2*d*x + 1/2*c) - 2*b + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b
^2)*b^6) + 2*(60*a^3*b*tan(1/2*d*x + 1/2*c)^9 + 135*a*b^3*tan(1/2*d*x + 1/2*c)^9 + 120*a^4*tan(1/2*d*x + 1/2*c
)^8 + 360*a^2*b^2*tan(1/2*d*x + 1/2*c)^8 + 360*b^4*tan(1/2*d*x + 1/2*c)^8 - 120*a^3*b*tan(1/2*d*x + 1/2*c)^7 -
150*a*b^3*tan(1/2*d*x + 1/2*c)^7 - 480*a^4*tan(1/2*d*x + 1/2*c)^6 - 1200*a^2*b^2*tan(1/2*d*x + 1/2*c)^6 - 720
*b^4*tan(1/2*d*x + 1/2*c)^6 + 720*a^4*tan(1/2*d*x + 1/2*c)^4 + 1600*a^2*b^2*tan(1/2*d*x + 1/2*c)^4 + 1120*b^4*
tan(1/2*d*x + 1/2*c)^4 + 120*a^3*b*tan(1/2*d*x + 1/2*c)^3 + 150*a*b^3*tan(1/2*d*x + 1/2*c)^3 - 480*a^4*tan(1/2
*d*x + 1/2*c)^2 - 1040*a^2*b^2*tan(1/2*d*x + 1/2*c)^2 - 560*b^4*tan(1/2*d*x + 1/2*c)^2 - 60*a^3*b*tan(1/2*d*x
+ 1/2*c) - 135*a*b^3*tan(1/2*d*x + 1/2*c) + 120*a^4 + 280*a^2*b^2 + 184*b^4)/((tan(1/2*d*x + 1/2*c)^2 - 1)^5*b
^5))/d

Mupad [B] (verification not implemented)

Time = 25.10 (sec) , antiderivative size = 2979, normalized size of antiderivative = 11.37 \[ \int \frac {\sec ^6(c+d x)}{a \cos (c+d x)+b \sin (c+d x)} \, dx=\text {Too large to display} \]

[In]

int(1/(cos(c + d*x)^6*(a*cos(c + d*x) + b*sin(c + d*x))),x)

[Out]

(atan(((((a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^
14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b
^7 + 128*a^13*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c
/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + (((a^2 + b^2)^5)^(1/2)*(32
*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b^6)*1i)/b^6 + (((a^2 + b^2)^5)^(
1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*
(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^
15) - (((a^2 + b^2)^5)^(1/2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(
2*b^15) - (28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 + (((a^2 + b^2)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*
x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b^6)*1i)/b^6)/((32*a^16 + 120*a^2*b^14 + 655*a^4*b^12 + 15
49*a^6*b^10 + 2069*a^8*b^8 + 1695*a^10*b^6 + 856*a^12*b^4 + 248*a^14*b^2)/b^14 + (((a^2 + b^2)^5)^(1/2)*(((225
*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17
+ 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((a^
2 + b^2)^5)^(1/2)*((28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*
b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + (((a^2 + b^2)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*
a*b^19 + 128*a^3*b^17))/(2*b^15)))/b^6))/b^6))/b^6 - (((a^2 + b^2)^5)^(1/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11
+ 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^
13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((a^2 + b^2)^5)^(1/2)*((tan(c/2
+ (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 + 44*a^4*b^14 +
16*a^6*b^12)/b^14 + (((a^2 + b^2)^5)^(1/2)*(32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b
^15)))/b^6))/b^6))/b^6 - (tan(c/2 + (d*x)/2)*(128*a^17 + 450*a^3*b^14 + 2550*a^5*b^12 + 6230*a^7*b^10 + 8530*a
^9*b^8 + 7088*a^11*b^6 + 3584*a^13*b^4 + 1024*a^15*b^2))/b^15))*((a^2 + b^2)^5)^(1/2)*2i)/(b^6*d) - ((2*(15*a^
4 + 23*b^4 + 35*a^2*b^2))/(15*b^5) + (tan(c/2 + (d*x)/2)^3*(5*a*b^2 + 4*a^3))/(2*b^4) - (tan(c/2 + (d*x)/2)^7*
(5*a*b^2 + 4*a^3))/(2*b^4) + (tan(c/2 + (d*x)/2)^9*(9*a*b^2 + 4*a^3))/(4*b^4) + (2*tan(c/2 + (d*x)/2)^8*(a^4 +
3*b^4 + 3*a^2*b^2))/b^5 - (4*tan(c/2 + (d*x)/2)^6*(2*a^4 + 3*b^4 + 5*a^2*b^2))/b^5 - (4*tan(c/2 + (d*x)/2)^2*
(6*a^4 + 7*b^4 + 13*a^2*b^2))/(3*b^5) + (4*tan(c/2 + (d*x)/2)^4*(9*a^4 + 14*b^4 + 20*a^2*b^2))/(3*b^5) - (tan(
c/2 + (d*x)/2)*(9*a*b^2 + 4*a^3))/(4*b^4))/(d*(5*tan(c/2 + (d*x)/2)^2 - 10*tan(c/2 + (d*x)/2)^4 + 10*tan(c/2 +
(d*x)/2)^6 - 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^10 - 1)) + (atan(((((15*a*b^4)/8 + a^5 + (5*a^3*b^2)
/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(
64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^1
5) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*((28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/
2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(1
92*a*b^19 + 128*a^3*b^17))/(2*b^15))*((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2))/b^6))/b^6)*1i)/b^6 + (((15*a*b^4)/8
+ a^5 + (5*a^3*b^2)/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (
tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 1
28*a^13*b^5))/(2*b^15) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16
+ 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 + 44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((32*a^2*b^3 + (t
an(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15))*((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2))/b^6))/b^6)*1i)/
b^6)/((32*a^16 + 120*a^2*b^14 + 655*a^4*b^12 + 1549*a^6*b^10 + 2069*a^8*b^8 + 1695*a^10*b^6 + 856*a^12*b^4 + 2
48*a^14*b^2)/b^14 + (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 + 320*a^8*b^9 + 16
0*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^13 + 3160*a^7*b^11
+ 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*((28*a^2*b^16
+ 44*a^4*b^14 + 16*a^6*b^12)/b^14 - (tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b
^12))/(2*b^15) + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15))*((15*a*b^4)/8 + a^5
+ (5*a^3*b^2)/2))/b^6))/b^6))/b^6 - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2)*(((225*a^4*b^13)/2 + 300*a^6*b^11 +
320*a^8*b^9 + 160*a^10*b^7 + 32*a^12*b^5)/b^14 + (tan(c/2 + (d*x)/2)*(64*a*b^17 + 834*a^3*b^15 + 2385*a^5*b^1
3 + 3160*a^7*b^11 + 2240*a^9*b^9 + 832*a^11*b^7 + 128*a^13*b^5))/(2*b^15) - (((15*a*b^4)/8 + a^5 + (5*a^3*b^2)
/2)*((tan(c/2 + (d*x)/2)*(128*a*b^18 + 384*a^3*b^16 + 384*a^5*b^14 + 128*a^7*b^12))/(2*b^15) - (28*a^2*b^16 +
44*a^4*b^14 + 16*a^6*b^12)/b^14 + ((32*a^2*b^3 + (tan(c/2 + (d*x)/2)*(192*a*b^19 + 128*a^3*b^17))/(2*b^15))*((
15*a*b^4)/8 + a^5 + (5*a^3*b^2)/2))/b^6))/b^6))/b^6 - (tan(c/2 + (d*x)/2)*(128*a^17 + 450*a^3*b^14 + 2550*a^5*
b^12 + 6230*a^7*b^10 + 8530*a^9*b^8 + 7088*a^11*b^6 + 3584*a^13*b^4 + 1024*a^15*b^2))/b^15))*((15*a*b^4)/8 + a
^5 + (5*a^3*b^2)/2)*2i)/(b^6*d)