3.56 \(\int \csc ^2(a+b x) \csc ^4(2 a+2 b x) \, dx\)
Optimal. Leaf size=72 \[ \frac{\tan ^3(a+b x)}{48 b}+\frac{\tan (a+b x)}{4 b}-\frac{\cot ^5(a+b x)}{80 b}-\frac{\cot ^3(a+b x)}{12 b}-\frac{3 \cot (a+b x)}{8 b} \]
[Out]
(-3*Cot[a + b*x])/(8*b) - Cot[a + b*x]^3/(12*b) - Cot[a + b*x]^5/(80*b) + Tan[a + b*x]/(4*b) + Tan[a + b*x]^3/
(48*b)
________________________________________________________________________________________
Rubi [A] time = 0.070376, antiderivative size = 72, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used =
{4288, 2620, 270} \[ \frac{\tan ^3(a+b x)}{48 b}+\frac{\tan (a+b x)}{4 b}-\frac{\cot ^5(a+b x)}{80 b}-\frac{\cot ^3(a+b x)}{12 b}-\frac{3 \cot (a+b x)}{8 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^4,x]
[Out]
(-3*Cot[a + b*x])/(8*b) - Cot[a + b*x]^3/(12*b) - Cot[a + b*x]^5/(80*b) + Tan[a + b*x]/(4*b) + Tan[a + b*x]^3/
(48*b)
Rule 4288
Int[((f_.)*sin[(a_.) + (b_.)*(x_)])^(n_.)*sin[(c_.) + (d_.)*(x_)]^(p_.), x_Symbol] :> Dist[2^p/f^p, Int[Cos[a
+ b*x]^p*(f*Sin[a + b*x])^(n + p), x], x] /; FreeQ[{a, b, c, d, f, n}, x] && EqQ[b*c - a*d, 0] && EqQ[d/b, 2]
&& IntegerQ[p]
Rule 2620
Int[csc[(e_.) + (f_.)*(x_)]^(m_.)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(1 + x^2)^((
m + n)/2 - 1)/x^m, x], x, Tan[e + f*x]], x] /; FreeQ[{e, f}, x] && IntegersQ[m, n, (m + n)/2]
Rule 270
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]
Rubi steps
\begin{align*} \int \csc ^2(a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac{1}{16} \int \csc ^6(a+b x) \sec ^4(a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^4}{x^6} \, dx,x,\tan (a+b x)\right )}{16 b}\\ &=\frac{\operatorname{Subst}\left (\int \left (4+\frac{1}{x^6}+\frac{4}{x^4}+\frac{6}{x^2}+x^2\right ) \, dx,x,\tan (a+b x)\right )}{16 b}\\ &=-\frac{3 \cot (a+b x)}{8 b}-\frac{\cot ^3(a+b x)}{12 b}-\frac{\cot ^5(a+b x)}{80 b}+\frac{\tan (a+b x)}{4 b}+\frac{\tan ^3(a+b x)}{48 b}\\ \end{align*}
Mathematica [A] time = 0.0535614, size = 90, normalized size = 1.25 \[ \frac{11 \tan (a+b x)}{48 b}-\frac{73 \cot (a+b x)}{240 b}-\frac{\cot (a+b x) \csc ^4(a+b x)}{80 b}-\frac{7 \cot (a+b x) \csc ^2(a+b x)}{120 b}+\frac{\tan (a+b x) \sec ^2(a+b x)}{48 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^2*Csc[2*a + 2*b*x]^4,x]
[Out]
(-73*Cot[a + b*x])/(240*b) - (7*Cot[a + b*x]*Csc[a + b*x]^2)/(120*b) - (Cot[a + b*x]*Csc[a + b*x]^4)/(80*b) +
(11*Tan[a + b*x])/(48*b) + (Sec[a + b*x]^2*Tan[a + b*x])/(48*b)
________________________________________________________________________________________
Maple [A] time = 0.038, size = 87, normalized size = 1.2 \begin{align*}{\frac{1}{16\,b} \left ( -{\frac{1}{5\, \left ( \sin \left ( bx+a \right ) \right ) ^{5} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}+{\frac{8}{15\, \left ( \sin \left ( bx+a \right ) \right ) ^{3} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}-{\frac{16}{15\, \left ( \sin \left ( bx+a \right ) \right ) ^{3}\cos \left ( bx+a \right ) }}+{\frac{64}{15\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }}-{\frac{128\,\cot \left ( bx+a \right ) }{15}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^2*csc(2*b*x+2*a)^4,x)
[Out]
1/16/b*(-1/5/sin(b*x+a)^5/cos(b*x+a)^3+8/15/sin(b*x+a)^3/cos(b*x+a)^3-16/15/sin(b*x+a)^3/cos(b*x+a)+64/15/sin(
b*x+a)/cos(b*x+a)-128/15*cot(b*x+a))
________________________________________________________________________________________
Maxima [B] time = 1.25661, size = 1656, normalized size = 23. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^4,x, algorithm="maxima")
[Out]
16/15*(2*(3*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*cos(16*b*x + 16*a) - 4*(3*sin(6*b*x + 6*a)
- sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*cos(14*b*x + 14*a) - 4*(3*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) - sin(2*
b*x + 2*a))*cos(12*b*x + 12*a) + 12*(3*sin(6*b*x + 6*a) - sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*cos(10*b*x + 10
*a) - (6*cos(6*b*x + 6*a) - 2*cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + 1)*sin(16*b*x + 16*a) + 2*(6*cos(6*b*x +
6*a) - 2*cos(4*b*x + 4*a) - 2*cos(2*b*x + 2*a) + 1)*sin(14*b*x + 14*a) + 2*(6*cos(6*b*x + 6*a) - 2*cos(4*b*x
+ 4*a) - 2*cos(2*b*x + 2*a) + 1)*sin(12*b*x + 12*a) - 6*(6*cos(6*b*x + 6*a) - 2*cos(4*b*x + 4*a) - 2*cos(2*b*x
+ 2*a) + 1)*sin(10*b*x + 10*a))/(b*cos(16*b*x + 16*a)^2 + 4*b*cos(14*b*x + 14*a)^2 + 4*b*cos(12*b*x + 12*a)^2
+ 36*b*cos(10*b*x + 10*a)^2 + 36*b*cos(6*b*x + 6*a)^2 + 4*b*cos(4*b*x + 4*a)^2 + 4*b*cos(2*b*x + 2*a)^2 + b*s
in(16*b*x + 16*a)^2 + 4*b*sin(14*b*x + 14*a)^2 + 4*b*sin(12*b*x + 12*a)^2 + 36*b*sin(10*b*x + 10*a)^2 + 36*b*s
in(6*b*x + 6*a)^2 + 4*b*sin(4*b*x + 4*a)^2 + 8*b*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) + 4*b*sin(2*b*x + 2*a)^2 -
2*(2*b*cos(14*b*x + 14*a) + 2*b*cos(12*b*x + 12*a) - 6*b*cos(10*b*x + 10*a) + 6*b*cos(6*b*x + 6*a) - 2*b*cos(4
*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(16*b*x + 16*a) + 4*(2*b*cos(12*b*x + 12*a) - 6*b*cos(10*b*x + 10*a
) + 6*b*cos(6*b*x + 6*a) - 2*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(14*b*x + 14*a) - 4*(6*b*cos(10
*b*x + 10*a) - 6*b*cos(6*b*x + 6*a) + 2*b*cos(4*b*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(12*b*x + 12*a) - 12
*(6*b*cos(6*b*x + 6*a) - 2*b*cos(4*b*x + 4*a) - 2*b*cos(2*b*x + 2*a) + b)*cos(10*b*x + 10*a) - 12*(2*b*cos(4*b
*x + 4*a) + 2*b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) + 4*(2*b*cos(2*b*x + 2*a) - b)*cos(4*b*x + 4*a) - 4*b*c
os(2*b*x + 2*a) - 4*(b*sin(14*b*x + 14*a) + b*sin(12*b*x + 12*a) - 3*b*sin(10*b*x + 10*a) + 3*b*sin(6*b*x + 6*
a) - b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(16*b*x + 16*a) + 8*(b*sin(12*b*x + 12*a) - 3*b*sin(10*b*x +
10*a) + 3*b*sin(6*b*x + 6*a) - b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - 8*(3*b*sin(10*b*x
+ 10*a) - 3*b*sin(6*b*x + 6*a) + b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(12*b*x + 12*a) - 24*(3*b*sin(6*
b*x + 6*a) - b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 24*(b*sin(4*b*x + 4*a) + b*sin(2*b*
x + 2*a))*sin(6*b*x + 6*a) + b)
________________________________________________________________________________________
Fricas [A] time = 0.481648, size = 228, normalized size = 3.17 \begin{align*} -\frac{128 \, \cos \left (b x + a\right )^{8} - 320 \, \cos \left (b x + a\right )^{6} + 240 \, \cos \left (b x + a\right )^{4} - 40 \, \cos \left (b x + a\right )^{2} - 5}{240 \,{\left (b \cos \left (b x + a\right )^{7} - 2 \, b \cos \left (b x + a\right )^{5} + b \cos \left (b x + a\right )^{3}\right )} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^4,x, algorithm="fricas")
[Out]
-1/240*(128*cos(b*x + a)^8 - 320*cos(b*x + a)^6 + 240*cos(b*x + a)^4 - 40*cos(b*x + a)^2 - 5)/((b*cos(b*x + a)
^7 - 2*b*cos(b*x + a)^5 + b*cos(b*x + a)^3)*sin(b*x + a))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**2*csc(2*b*x+2*a)**4,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 3.42267, size = 4651, normalized size = 64.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^2*csc(2*b*x+2*a)^4,x, algorithm="giac")
[Out]
-1/1920*(5*(108*tan(b*x + 4*a)^2*tan(1/2*a)^34 - 18*tan(b*x + 4*a)*tan(1/2*a)^35 + tan(1/2*a)^36 + 12240*tan(b
*x + 4*a)^2*tan(1/2*a)^32 - 3774*tan(b*x + 4*a)*tan(1/2*a)^33 + 342*tan(1/2*a)^34 - 85632*tan(b*x + 4*a)^2*tan
(1/2*a)^30 + 75456*tan(b*x + 4*a)*tan(1/2*a)^31 - 9783*tan(1/2*a)^32 + 58608*tan(b*x + 4*a)^2*tan(1/2*a)^28 -
253152*tan(b*x + 4*a)*tan(1/2*a)^29 + 86064*tan(1/2*a)^30 + 631440*tan(b*x + 4*a)^2*tan(1/2*a)^26 - 389448*tan
(b*x + 4*a)*tan(1/2*a)^27 - 85500*tan(1/2*a)^28 - 679344*tan(b*x + 4*a)^2*tan(1/2*a)^24 + 2047464*tan(b*x + 4*
a)*tan(1/2*a)^25 - 702936*tan(1/2*a)^26 - 2420352*tan(b*x + 4*a)^2*tan(1/2*a)^22 + 1465056*tan(b*x + 4*a)*tan(
1/2*a)^23 + 675636*tan(1/2*a)^24 + 1394928*tan(b*x + 4*a)^2*tan(1/2*a)^20 - 6297216*tan(b*x + 4*a)*tan(1/2*a)^
21 + 2768976*tan(1/2*a)^22 + 5321736*tan(b*x + 4*a)^2*tan(1/2*a)^18 - 5657724*tan(b*x + 4*a)*tan(1/2*a)^19 - 5
14818*tan(1/2*a)^20 + 1394928*tan(b*x + 4*a)^2*tan(1/2*a)^16 + 5657724*tan(b*x + 4*a)*tan(1/2*a)^17 - 4173820*
tan(1/2*a)^18 - 2420352*tan(b*x + 4*a)^2*tan(1/2*a)^14 + 6297216*tan(b*x + 4*a)*tan(1/2*a)^15 - 514818*tan(1/2
*a)^16 - 679344*tan(b*x + 4*a)^2*tan(1/2*a)^12 - 1465056*tan(b*x + 4*a)*tan(1/2*a)^13 + 2768976*tan(1/2*a)^14
+ 631440*tan(b*x + 4*a)^2*tan(1/2*a)^10 - 2047464*tan(b*x + 4*a)*tan(1/2*a)^11 + 675636*tan(1/2*a)^12 + 58608*
tan(b*x + 4*a)^2*tan(1/2*a)^8 + 389448*tan(b*x + 4*a)*tan(1/2*a)^9 - 702936*tan(1/2*a)^10 - 85632*tan(b*x + 4*
a)^2*tan(1/2*a)^6 + 253152*tan(b*x + 4*a)*tan(1/2*a)^7 - 85500*tan(1/2*a)^8 + 12240*tan(b*x + 4*a)^2*tan(1/2*a
)^4 - 75456*tan(b*x + 4*a)*tan(1/2*a)^5 + 86064*tan(1/2*a)^6 + 108*tan(b*x + 4*a)^2*tan(1/2*a)^2 + 3774*tan(b*
x + 4*a)*tan(1/2*a)^3 - 9783*tan(1/2*a)^4 + 18*tan(b*x + 4*a)*tan(1/2*a) + 342*tan(1/2*a)^2 + 1)/((27*tan(1/2*
a)^15 - 270*tan(1/2*a)^13 + 981*tan(1/2*a)^11 - 1540*tan(1/2*a)^9 + 981*tan(1/2*a)^7 - 270*tan(1/2*a)^5 + 27*t
an(1/2*a)^3)*(6*tan(b*x + 4*a)*tan(1/2*a)^5 - tan(1/2*a)^6 - 20*tan(b*x + 4*a)*tan(1/2*a)^3 + 15*tan(1/2*a)^4
+ 6*tan(b*x + 4*a)*tan(1/2*a) - 15*tan(1/2*a)^2 + 1)^3) + 8*(90*tan(b*x + 4*a)^4*tan(1/2*a)^60 - 8100*tan(b*x
+ 4*a)^4*tan(1/2*a)^58 + 1800*tan(b*x + 4*a)^3*tan(1/2*a)^59 + 20*tan(b*x + 4*a)^2*tan(1/2*a)^60 + 337950*tan(
b*x + 4*a)^4*tan(1/2*a)^56 - 151800*tan(b*x + 4*a)^3*tan(1/2*a)^57 + 13200*tan(b*x + 4*a)^2*tan(1/2*a)^58 + 15
0*tan(b*x + 4*a)*tan(1/2*a)^59 + 3*tan(1/2*a)^60 - 8558760*tan(b*x + 4*a)^4*tan(1/2*a)^54 + 5856480*tan(b*x +
4*a)^3*tan(1/2*a)^55 - 1068900*tan(b*x + 4*a)^2*tan(1/2*a)^56 + 44050*tan(b*x + 4*a)*tan(1/2*a)^57 + 270*tan(1
/2*a)^58 + 143179650*tan(b*x + 4*a)^4*tan(1/2*a)^52 - 133967520*tan(b*x + 4*a)^3*tan(1/2*a)^53 + 37650400*tan(
b*x + 4*a)^2*tan(1/2*a)^54 - 3349740*tan(b*x + 4*a)*tan(1/2*a)^55 + 56265*tan(1/2*a)^56 - 1610662860*tan(b*x +
4*a)^4*tan(1/2*a)^50 + 1962664200*tan(b*x + 4*a)^3*tan(1/2*a)^51 - 761504220*tan(b*x + 4*a)^2*tan(1/2*a)^52 +
105669900*tan(b*x + 4*a)*tan(1/2*a)^53 - 3921700*tan(1/2*a)^54 + 11902135590*tan(b*x + 4*a)^4*tan(1/2*a)^48 -
18613230840*tan(b*x + 4*a)^3*tan(1/2*a)^49 + 9508215600*tan(b*x + 4*a)^2*tan(1/2*a)^50 - 1840189110*tan(b*x +
4*a)*tan(1/2*a)^51 + 108499095*tan(1/2*a)^52 - 53242909200*tan(b*x + 4*a)^4*tan(1/2*a)^46 + 109725269760*tan(
b*x + 4*a)^3*tan(1/2*a)^47 - 73309882260*tan(b*x + 4*a)^2*tan(1/2*a)^48 + 18891597150*tan(b*x + 4*a)*tan(1/2*a
)^49 - 1563175302*tan(1/2*a)^50 + 107646377490*tan(b*x + 4*a)^4*tan(1/2*a)^44 - 347626321920*tan(b*x + 4*a)^3*
tan(1/2*a)^45 + 324785269440*tan(b*x + 4*a)^2*tan(1/2*a)^46 - 112787150040*tan(b*x + 4*a)*tan(1/2*a)^47 + 1260
2922605*tan(1/2*a)^48 + 100499354940*tan(b*x + 4*a)^4*tan(1/2*a)^42 + 243954351720*tan(b*x + 4*a)^3*tan(1/2*a)
^43 - 614500276860*tan(b*x + 4*a)^2*tan(1/2*a)^44 + 342870428760*tan(b*x + 4*a)*tan(1/2*a)^45 - 54544776360*ta
n(1/2*a)^46 - 720524857050*tan(b*x + 4*a)^4*tan(1/2*a)^40 + 1552622504040*tan(b*x + 4*a)^3*tan(1/2*a)^41 - 592
670717680*tan(b*x + 4*a)^2*tan(1/2*a)^42 - 221864573610*tan(b*x + 4*a)*tan(1/2*a)^43 + 98588733975*tan(1/2*a)^
44 + 545843494440*tan(b*x + 4*a)^4*tan(1/2*a)^38 - 3402462596640*tan(b*x + 4*a)^3*tan(1/2*a)^39 + 420803115486
0*tan(b*x + 4*a)^2*tan(1/2*a)^40 - 1502042255310*tan(b*x + 4*a)*tan(1/2*a)^41 + 99097279550*tan(1/2*a)^42 + 13
94520156090*tan(b*x + 4*a)^4*tan(1/2*a)^36 - 1060535913120*tan(b*x + 4*a)^3*tan(1/2*a)^37 - 3399607142880*tan(
b*x + 4*a)^2*tan(1/2*a)^38 + 3367956274060*tan(b*x + 4*a)*tan(1/2*a)^39 - 681790695915*tan(1/2*a)^40 - 2195406
541260*tan(b*x + 4*a)^4*tan(1/2*a)^34 + 8940274641000*tan(b*x + 4*a)^3*tan(1/2*a)^35 - 8188018122860*tan(b*x +
4*a)^2*tan(1/2*a)^36 + 893802691860*tan(b*x + 4*a)*tan(1/2*a)^37 + 583218519780*tan(1/2*a)^38 - 793687329810*
tan(b*x + 4*a)^4*tan(1/2*a)^32 - 4368381430680*tan(b*x + 4*a)^3*tan(1/2*a)^33 + 13505898965040*tan(b*x + 4*a)^
2*tan(1/2*a)^34 - 8980226310150*tan(b*x + 4*a)*tan(1/2*a)^35 + 1327591075915*tan(1/2*a)^36 + 3207851661600*tan
(b*x + 4*a)^4*tan(1/2*a)^30 - 10136128700160*tan(b*x + 4*a)^3*tan(1/2*a)^31 + 4668559700220*tan(b*x + 4*a)^2*t
an(1/2*a)^32 + 4628264277390*tan(b*x + 4*a)*tan(1/2*a)^33 - 2309792723910*tan(1/2*a)^34 - 793687329810*tan(b*x
+ 4*a)^4*tan(1/2*a)^28 + 10136128700160*tan(b*x + 4*a)^3*tan(1/2*a)^29 - 19695904506240*tan(b*x + 4*a)^2*tan(
1/2*a)^30 + 10330657752240*tan(b*x + 4*a)*tan(1/2*a)^31 - 756295285575*tan(1/2*a)^32 - 2195406541260*tan(b*x +
4*a)^4*tan(1/2*a)^26 + 4368381430680*tan(b*x + 4*a)^3*tan(1/2*a)^27 + 4668559700220*tan(b*x + 4*a)^2*tan(1/2*
a)^28 - 10330657752240*tan(b*x + 4*a)*tan(1/2*a)^29 + 3368788208080*tan(1/2*a)^30 + 1394520156090*tan(b*x + 4*
a)^4*tan(1/2*a)^24 - 8940274641000*tan(b*x + 4*a)^3*tan(1/2*a)^25 + 13505898965040*tan(b*x + 4*a)^2*tan(1/2*a)
^26 - 4628264277390*tan(b*x + 4*a)*tan(1/2*a)^27 - 756295285575*tan(1/2*a)^28 + 545843494440*tan(b*x + 4*a)^4*
tan(1/2*a)^22 + 1060535913120*tan(b*x + 4*a)^3*tan(1/2*a)^23 - 8188018122860*tan(b*x + 4*a)^2*tan(1/2*a)^24 +
8980226310150*tan(b*x + 4*a)*tan(1/2*a)^25 - 2309792723910*tan(1/2*a)^26 - 720524857050*tan(b*x + 4*a)^4*tan(1
/2*a)^20 + 3402462596640*tan(b*x + 4*a)^3*tan(1/2*a)^21 - 3399607142880*tan(b*x + 4*a)^2*tan(1/2*a)^22 - 89380
2691860*tan(b*x + 4*a)*tan(1/2*a)^23 + 1327591075915*tan(1/2*a)^24 + 100499354940*tan(b*x + 4*a)^4*tan(1/2*a)^
18 - 1552622504040*tan(b*x + 4*a)^3*tan(1/2*a)^19 + 4208031154860*tan(b*x + 4*a)^2*tan(1/2*a)^20 - 33679562740
60*tan(b*x + 4*a)*tan(1/2*a)^21 + 583218519780*tan(1/2*a)^22 + 107646377490*tan(b*x + 4*a)^4*tan(1/2*a)^16 - 2
43954351720*tan(b*x + 4*a)^3*tan(1/2*a)^17 - 592670717680*tan(b*x + 4*a)^2*tan(1/2*a)^18 + 1502042255310*tan(b
*x + 4*a)*tan(1/2*a)^19 - 681790695915*tan(1/2*a)^20 - 53242909200*tan(b*x + 4*a)^4*tan(1/2*a)^14 + 3476263219
20*tan(b*x + 4*a)^3*tan(1/2*a)^15 - 614500276860*tan(b*x + 4*a)^2*tan(1/2*a)^16 + 221864573610*tan(b*x + 4*a)*
tan(1/2*a)^17 + 99097279550*tan(1/2*a)^18 + 11902135590*tan(b*x + 4*a)^4*tan(1/2*a)^12 - 109725269760*tan(b*x
+ 4*a)^3*tan(1/2*a)^13 + 324785269440*tan(b*x + 4*a)^2*tan(1/2*a)^14 - 342870428760*tan(b*x + 4*a)*tan(1/2*a)^
15 + 98588733975*tan(1/2*a)^16 - 1610662860*tan(b*x + 4*a)^4*tan(1/2*a)^10 + 18613230840*tan(b*x + 4*a)^3*tan(
1/2*a)^11 - 73309882260*tan(b*x + 4*a)^2*tan(1/2*a)^12 + 112787150040*tan(b*x + 4*a)*tan(1/2*a)^13 - 545447763
60*tan(1/2*a)^14 + 143179650*tan(b*x + 4*a)^4*tan(1/2*a)^8 - 1962664200*tan(b*x + 4*a)^3*tan(1/2*a)^9 + 950821
5600*tan(b*x + 4*a)^2*tan(1/2*a)^10 - 18891597150*tan(b*x + 4*a)*tan(1/2*a)^11 + 12602922605*tan(1/2*a)^12 - 8
558760*tan(b*x + 4*a)^4*tan(1/2*a)^6 + 133967520*tan(b*x + 4*a)^3*tan(1/2*a)^7 - 761504220*tan(b*x + 4*a)^2*ta
n(1/2*a)^8 + 1840189110*tan(b*x + 4*a)*tan(1/2*a)^9 - 1563175302*tan(1/2*a)^10 + 337950*tan(b*x + 4*a)^4*tan(1
/2*a)^4 - 5856480*tan(b*x + 4*a)^3*tan(1/2*a)^5 + 37650400*tan(b*x + 4*a)^2*tan(1/2*a)^6 - 105669900*tan(b*x +
4*a)*tan(1/2*a)^7 + 108499095*tan(1/2*a)^8 - 8100*tan(b*x + 4*a)^4*tan(1/2*a)^2 + 151800*tan(b*x + 4*a)^3*tan
(1/2*a)^3 - 1068900*tan(b*x + 4*a)^2*tan(1/2*a)^4 + 3349740*tan(b*x + 4*a)*tan(1/2*a)^5 - 3921700*tan(1/2*a)^6
+ 90*tan(b*x + 4*a)^4 - 1800*tan(b*x + 4*a)^3*tan(1/2*a) + 13200*tan(b*x + 4*a)^2*tan(1/2*a)^2 - 44050*tan(b*
x + 4*a)*tan(1/2*a)^3 + 56265*tan(1/2*a)^4 + 20*tan(b*x + 4*a)^2 - 150*tan(b*x + 4*a)*tan(1/2*a) + 270*tan(1/2
*a)^2 + 3)/((tan(1/2*a)^30 - 75*tan(1/2*a)^28 + 2325*tan(1/2*a)^26 - 38255*tan(1/2*a)^24 + 356925*tan(1/2*a)^2
2 - 1880175*tan(1/2*a)^20 + 5430385*tan(1/2*a)^18 - 9069075*tan(1/2*a)^16 + 9069075*tan(1/2*a)^14 - 5430385*ta
n(1/2*a)^12 + 1880175*tan(1/2*a)^10 - 356925*tan(1/2*a)^8 + 38255*tan(1/2*a)^6 - 2325*tan(1/2*a)^4 + 75*tan(1/
2*a)^2 - 1)*(tan(b*x + 4*a)*tan(1/2*a)^6 - 15*tan(b*x + 4*a)*tan(1/2*a)^4 + 6*tan(1/2*a)^5 + 15*tan(b*x + 4*a)
*tan(1/2*a)^2 - 20*tan(1/2*a)^3 - tan(b*x + 4*a) + 6*tan(1/2*a))^5))/b