3.70 \(\int \csc ^3(a+b x) \csc ^2(2 a+2 b x) \, dx\)
Optimal. Leaf size=70 \[ \frac{15 \sec (a+b x)}{32 b}-\frac{15 \tanh ^{-1}(\cos (a+b x))}{32 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{32 b} \]
[Out]
(-15*ArcTanh[Cos[a + b*x]])/(32*b) + (15*Sec[a + b*x])/(32*b) - (5*Csc[a + b*x]^2*Sec[a + b*x])/(32*b) - (Csc[
a + b*x]^4*Sec[a + b*x])/(16*b)
________________________________________________________________________________________
Rubi [A] time = 0.0726865, antiderivative size = 70, normalized size of antiderivative = 1.,
number of steps used = 6, number of rules used = 5, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used =
{4288, 2622, 288, 321, 207} \[ \frac{15 \sec (a+b x)}{32 b}-\frac{15 \tanh ^{-1}(\cos (a+b x))}{32 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{32 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^2,x]
[Out]
(-15*ArcTanh[Cos[a + b*x]])/(32*b) + (15*Sec[a + b*x])/(32*b) - (5*Csc[a + b*x]^2*Sec[a + b*x])/(32*b) - (Csc[
a + b*x]^4*Sec[a + b*x])/(16*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 2622
Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] && !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])
Rule 288
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] && !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]
Rule 321
Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]
Rule 207
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTanh[(Rt[b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])
Rubi steps
\begin{align*} \int \csc ^3(a+b x) \csc ^2(2 a+2 b x) \, dx &=\frac{1}{4} \int \csc ^5(a+b x) \sec ^2(a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{4 b}\\ &=-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}+\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{32 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}+\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=\frac{15 \sec (a+b x)}{32 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{32 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}+\frac{15 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{32 b}\\ &=-\frac{15 \tanh ^{-1}(\cos (a+b x))}{32 b}+\frac{15 \sec (a+b x)}{32 b}-\frac{5 \csc ^2(a+b x) \sec (a+b x)}{32 b}-\frac{\csc ^4(a+b x) \sec (a+b x)}{16 b}\\ \end{align*}
Mathematica [A] time = 4.63286, size = 129, normalized size = 1.84 \[ -\frac{\csc ^4\left (\frac{1}{2} (a+b x)\right )+14 \csc ^2\left (\frac{1}{2} (a+b x)\right )+\frac{\sec ^2\left (\frac{1}{2} (a+b x)\right ) \left (-14 \tan ^2\left (\frac{1}{2} (a+b x)\right )+\cos (a+b x) \left (\sec ^4\left (\frac{1}{2} (a+b x)\right )-8 \left (-15 \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )+15 \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+8\right )\right )+78\right )}{\tan ^2\left (\frac{1}{2} (a+b x)\right )-1}}{256 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]^3*Csc[2*a + 2*b*x]^2,x]
[Out]
-(14*Csc[(a + b*x)/2]^2 + Csc[(a + b*x)/2]^4 + (Sec[(a + b*x)/2]^2*(78 + Cos[a + b*x]*(-8*(8 + 15*Log[Cos[(a +
b*x)/2]] - 15*Log[Sin[(a + b*x)/2]]) + Sec[(a + b*x)/2]^4) - 14*Tan[(a + b*x)/2]^2))/(-1 + Tan[(a + b*x)/2]^2
))/(256*b)
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Maple [A] time = 0.036, size = 78, normalized size = 1.1 \begin{align*} -{\frac{1}{16\,b \left ( \sin \left ( bx+a \right ) \right ) ^{4}\cos \left ( bx+a \right ) }}-{\frac{5}{32\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}\cos \left ( bx+a \right ) }}+{\frac{15}{32\,b\cos \left ( bx+a \right ) }}+{\frac{15\,\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{32\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)^3*csc(2*b*x+2*a)^2,x)
[Out]
-1/16/b/sin(b*x+a)^4/cos(b*x+a)-5/32/b/sin(b*x+a)^2/cos(b*x+a)+15/32/b/cos(b*x+a)+15/32/b*ln(csc(b*x+a)-cot(b*
x+a))
________________________________________________________________________________________
Maxima [B] time = 1.34277, size = 3020, normalized size = 43.14 \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)^3*csc(2*b*x+2*a)^2,x, algorithm="maxima")
[Out]
1/64*(4*(15*cos(9*b*x + 9*a) - 40*cos(7*b*x + 7*a) + 18*cos(5*b*x + 5*a) - 40*cos(3*b*x + 3*a) + 15*cos(b*x +
a))*cos(10*b*x + 10*a) - 60*(3*cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) - 2*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a)
- 1)*cos(9*b*x + 9*a) + 12*(40*cos(7*b*x + 7*a) - 18*cos(5*b*x + 5*a) + 40*cos(3*b*x + 3*a) - 15*cos(b*x + a)
)*cos(8*b*x + 8*a) - 160*(2*cos(6*b*x + 6*a) + 2*cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*cos(7*b*x + 7*a) +
8*(18*cos(5*b*x + 5*a) - 40*cos(3*b*x + 3*a) + 15*cos(b*x + a))*cos(6*b*x + 6*a) + 72*(2*cos(4*b*x + 4*a) - 3
*cos(2*b*x + 2*a) + 1)*cos(5*b*x + 5*a) - 40*(8*cos(3*b*x + 3*a) - 3*cos(b*x + a))*cos(4*b*x + 4*a) + 160*(3*c
os(2*b*x + 2*a) - 1)*cos(3*b*x + 3*a) - 180*cos(2*b*x + 2*a)*cos(b*x + a) + 15*(2*(3*cos(8*b*x + 8*a) - 2*cos(
6*b*x + 6*a) - 2*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) - cos(10*b*x + 10*a)^2 + 6*(2*c
os(6*b*x + 6*a) + 2*cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 9*cos(8*b*x + 8*a)^2 - 4*(2*
cos(4*b*x + 4*a) - 3*cos(2*b*x + 2*a) + 1)*cos(6*b*x + 6*a) - 4*cos(6*b*x + 6*a)^2 + 4*(3*cos(2*b*x + 2*a) - 1
)*cos(4*b*x + 4*a) - 4*cos(4*b*x + 4*a)^2 - 9*cos(2*b*x + 2*a)^2 + 2*(3*sin(8*b*x + 8*a) - 2*sin(6*b*x + 6*a)
- 2*sin(4*b*x + 4*a) + 3*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - sin(10*b*x + 10*a)^2 + 6*(2*sin(6*b*x + 6*a) +
2*sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 9*sin(8*b*x + 8*a)^2 - 4*(2*sin(4*b*x + 4*a) - 3*
sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 4*sin(6*b*x + 6*a)^2 - 4*sin(4*b*x + 4*a)^2 + 12*sin(4*b*x + 4*a)*sin(2*b
*x + 2*a) - 9*sin(2*b*x + 2*a)^2 + 6*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 + 2*cos(b*x)*cos(a) + cos(a)^2 + sin
(b*x)^2 - 2*sin(b*x)*sin(a) + sin(a)^2) - 15*(2*(3*cos(8*b*x + 8*a) - 2*cos(6*b*x + 6*a) - 2*cos(4*b*x + 4*a)
+ 3*cos(2*b*x + 2*a) - 1)*cos(10*b*x + 10*a) - cos(10*b*x + 10*a)^2 + 6*(2*cos(6*b*x + 6*a) + 2*cos(4*b*x + 4*
a) - 3*cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 9*cos(8*b*x + 8*a)^2 - 4*(2*cos(4*b*x + 4*a) - 3*cos(2*b*x + 2
*a) + 1)*cos(6*b*x + 6*a) - 4*cos(6*b*x + 6*a)^2 + 4*(3*cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - 4*cos(4*b*x +
4*a)^2 - 9*cos(2*b*x + 2*a)^2 + 2*(3*sin(8*b*x + 8*a) - 2*sin(6*b*x + 6*a) - 2*sin(4*b*x + 4*a) + 3*sin(2*b*x
+ 2*a))*sin(10*b*x + 10*a) - sin(10*b*x + 10*a)^2 + 6*(2*sin(6*b*x + 6*a) + 2*sin(4*b*x + 4*a) - 3*sin(2*b*x
+ 2*a))*sin(8*b*x + 8*a) - 9*sin(8*b*x + 8*a)^2 - 4*(2*sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*sin(6*b*x + 6*a)
- 4*sin(6*b*x + 6*a)^2 - 4*sin(4*b*x + 4*a)^2 + 12*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - 9*sin(2*b*x + 2*a)^2 +
6*cos(2*b*x + 2*a) - 1)*log(cos(b*x)^2 - 2*cos(b*x)*cos(a) + cos(a)^2 + sin(b*x)^2 + 2*sin(b*x)*sin(a) + sin(
a)^2) + 4*(15*sin(9*b*x + 9*a) - 40*sin(7*b*x + 7*a) + 18*sin(5*b*x + 5*a) - 40*sin(3*b*x + 3*a) + 15*sin(b*x
+ a))*sin(10*b*x + 10*a) - 60*(3*sin(8*b*x + 8*a) - 2*sin(6*b*x + 6*a) - 2*sin(4*b*x + 4*a) + 3*sin(2*b*x + 2*
a))*sin(9*b*x + 9*a) + 12*(40*sin(7*b*x + 7*a) - 18*sin(5*b*x + 5*a) + 40*sin(3*b*x + 3*a) - 15*sin(b*x + a))*
sin(8*b*x + 8*a) - 160*(2*sin(6*b*x + 6*a) + 2*sin(4*b*x + 4*a) - 3*sin(2*b*x + 2*a))*sin(7*b*x + 7*a) + 8*(18
*sin(5*b*x + 5*a) - 40*sin(3*b*x + 3*a) + 15*sin(b*x + a))*sin(6*b*x + 6*a) + 72*(2*sin(4*b*x + 4*a) - 3*sin(2
*b*x + 2*a))*sin(5*b*x + 5*a) - 40*(8*sin(3*b*x + 3*a) - 3*sin(b*x + a))*sin(4*b*x + 4*a) + 480*sin(3*b*x + 3*
a)*sin(2*b*x + 2*a) - 180*sin(2*b*x + 2*a)*sin(b*x + a) + 60*cos(b*x + a))/(b*cos(10*b*x + 10*a)^2 + 9*b*cos(8
*b*x + 8*a)^2 + 4*b*cos(6*b*x + 6*a)^2 + 4*b*cos(4*b*x + 4*a)^2 + 9*b*cos(2*b*x + 2*a)^2 + b*sin(10*b*x + 10*a
)^2 + 9*b*sin(8*b*x + 8*a)^2 + 4*b*sin(6*b*x + 6*a)^2 + 4*b*sin(4*b*x + 4*a)^2 - 12*b*sin(4*b*x + 4*a)*sin(2*b
*x + 2*a) + 9*b*sin(2*b*x + 2*a)^2 - 2*(3*b*cos(8*b*x + 8*a) - 2*b*cos(6*b*x + 6*a) - 2*b*cos(4*b*x + 4*a) + 3
*b*cos(2*b*x + 2*a) - b)*cos(10*b*x + 10*a) - 6*(2*b*cos(6*b*x + 6*a) + 2*b*cos(4*b*x + 4*a) - 3*b*cos(2*b*x +
2*a) + b)*cos(8*b*x + 8*a) + 4*(2*b*cos(4*b*x + 4*a) - 3*b*cos(2*b*x + 2*a) + b)*cos(6*b*x + 6*a) - 4*(3*b*co
s(2*b*x + 2*a) - b)*cos(4*b*x + 4*a) - 6*b*cos(2*b*x + 2*a) - 2*(3*b*sin(8*b*x + 8*a) - 2*b*sin(6*b*x + 6*a) -
2*b*sin(4*b*x + 4*a) + 3*b*sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 6*(2*b*sin(6*b*x + 6*a) + 2*b*sin(4*b*x + 4
*a) - 3*b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) + 4*(2*b*sin(4*b*x + 4*a) - 3*b*sin(2*b*x + 2*a))*sin(6*b*x + 6*a
) + b)
________________________________________________________________________________________
Fricas [B] time = 0.508706, size = 374, normalized size = 5.34 \begin{align*} \frac{30 \, \cos \left (b x + a\right )^{4} - 50 \, \cos \left (b x + a\right )^{2} - 15 \,{\left (\cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 15 \,{\left (\cos \left (b x + a\right )^{5} - 2 \, \cos \left (b x + a\right )^{3} + \cos \left (b x + a\right )\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 16}{64 \,{\left (b \cos \left (b x + a\right )^{5} - 2 \, b \cos \left (b x + a\right )^{3} + b \cos \left (b x + a\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)^3*csc(2*b*x+2*a)^2,x, algorithm="fricas")
[Out]
1/64*(30*cos(b*x + a)^4 - 50*cos(b*x + a)^2 - 15*(cos(b*x + a)^5 - 2*cos(b*x + a)^3 + cos(b*x + a))*log(1/2*co
s(b*x + a) + 1/2) + 15*(cos(b*x + a)^5 - 2*cos(b*x + a)^3 + cos(b*x + a))*log(-1/2*cos(b*x + a) + 1/2) + 16)/(
b*cos(b*x + a)^5 - 2*b*cos(b*x + a)^3 + b*cos(b*x + a))
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc ^{3}{\left (a + b x \right )} \csc ^{2}{\left (2 a + 2 b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)**3*csc(2*b*x+2*a)**2,x)
[Out]
Integral(csc(a + b*x)**3*csc(2*a + 2*b*x)**2, x)
________________________________________________________________________________________
Giac [B] time = 4.4033, size = 5096, normalized size = 72.8 \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)^3*csc(2*b*x+2*a)^2,x, algorithm="giac")
[Out]
-1/256*(128*(6*tan(1/2*b*x + 2*a)*tan(1/2*a)^11 - tan(1/2*a)^12 - 2*tan(1/2*b*x + 2*a)*tan(1/2*a)^9 + 12*tan(1
/2*a)^10 - 36*tan(1/2*b*x + 2*a)*tan(1/2*a)^7 + 27*tan(1/2*a)^8 - 36*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 - 2*tan(1
/2*b*x + 2*a)*tan(1/2*a)^3 - 27*tan(1/2*a)^4 + 6*tan(1/2*b*x + 2*a)*tan(1/2*a) - 12*tan(1/2*a)^2 + 1)/((tan(1/
2*b*x + 2*a)^2*tan(1/2*a)^6 - 15*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^4 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 - tan(
1/2*a)^6 + 15*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^2 - 40*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 15*tan(1/2*a)^4 - tan(1
/2*b*x + 2*a)^2 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a) - 15*tan(1/2*a)^2 + 1)*(tan(1/2*a)^6 - 15*tan(1/2*a)^4 + 15
*tan(1/2*a)^2 - 1)) + (108*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^45 - 54*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^46 + 12*tan
(1/2*b*x + 2*a)^5*tan(1/2*a)^47 - tan(1/2*b*x + 2*a)^4*tan(1/2*a)^48 + 4428*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^43
- 2880*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^44 + 536*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^45 + 48*tan(1/2*b*x + 2*a)^4*
tan(1/2*a)^46 - 12*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^47 - 153216*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^41 + 210486*tan
(1/2*b*x + 2*a)^6*tan(1/2*a)^42 - 95832*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^43 + 16740*tan(1/2*b*x + 2*a)^4*tan(1/
2*a)^44 - 536*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^45 - 54*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^46 + 2486912*tan(1/2*b*x
+ 2*a)^7*tan(1/2*a)^39 - 4777344*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^40 + 3337020*tan(1/2*b*x + 2*a)^5*tan(1/2*a)
^41 - 1046736*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^42 + 142488*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^43 - 5472*tan(1/2*b*
x + 2*a)^2*tan(1/2*a)^44 - 108*tan(1/2*b*x + 2*a)*tan(1/2*a)^45 - 20891364*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^37
+ 54465762*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^38 - 52518388*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^39 + 23995332*tan(1/2
*b*x + 2*a)^4*tan(1/2*a)^40 - 5409324*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^41 + 546366*tan(1/2*b*x + 2*a)^2*tan(1/2
*a)^42 - 16092*tan(1/2*b*x + 2*a)*tan(1/2*a)^43 - 162*tan(1/2*a)^44 + 95841468*tan(1/2*b*x + 2*a)^7*tan(1/2*a)
^35 - 336486592*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^36 + 436639944*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^37 - 272279280*
tan(1/2*b*x + 2*a)^4*tan(1/2*a)^38 + 87480580*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^39 - 14094720*tan(1/2*b*x + 2*a)
^2*tan(1/2*a)^40 + 977472*tan(1/2*b*x + 2*a)*tan(1/2*a)^41 - 17280*tan(1/2*a)^42 - 234315648*tan(1/2*b*x + 2*a
)^7*tan(1/2*a)^33 + 1144864350*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^34 - 2010477960*tan(1/2*b*x + 2*a)^5*tan(1/2*a)
^35 + 1680996460*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^36 - 729548040*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^37 + 164195226
*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^38 - 17453120*tan(1/2*b*x + 2*a)*tan(1/2*a)^39 + 642384*tan(1/2*a)^40 + 21164
1984*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^31 - 1888388352*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^32 + 4935828252*tan(1/2*b
*x + 2*a)^5*tan(1/2*a)^33 - 5726288880*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^34 + 3352125960*tan(1/2*b*x + 2*a)^3*ta
n(1/2*a)^35 - 1012326432*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^36 + 147189588*tan(1/2*b*x + 2*a)*tan(1/2*a)^37 - 790
2336*tan(1/2*a)^38 + 314808792*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^29 + 96589076*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^3
0 - 4440981672*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^31 + 9447759099*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^32 - 8215340892
*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^33 + 3430559358*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^34 - 671389308*tan(1/2*b*x +
2*a)*tan(1/2*a)^35 + 48570730*tan(1/2*a)^36 - 939066728*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^27 + 4808309376*tan(1/
2*b*x + 2*a)^6*tan(1/2*a)^28 - 6655212048*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^29 - 466539808*tan(1/2*b*x + 2*a)^4*
tan(1/2*a)^30 + 7397865576*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^31 - 5649253632*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^32
+ 1635149568*tan(1/2*b*x + 2*a)*tan(1/2*a)^33 - 162842112*tan(1/2*a)^34 + 564924672*tan(1/2*b*x + 2*a)^7*tan(1
/2*a)^25 - 6391344852*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^26 + 19663795984*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^27 - 24
029264184*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^28 + 11050166160*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^29 + 310355316*tan(
1/2*b*x + 2*a)^2*tan(1/2*a)^30 - 1477416960*tan(1/2*b*x + 2*a)*tan(1/2*a)^31 + 267141600*tan(1/2*a)^32 + 56492
4672*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^23 - 11879791560*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^25 + 31957055712*tan(1/2
*b*x + 2*a)^4*tan(1/2*a)^26 - 32793412368*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^27 + 14381865792*tan(1/2*b*x + 2*a)^
2*tan(1/2*a)^28 - 2181545880*tan(1/2*b*x + 2*a)*tan(1/2*a)^29 - 17553920*tan(1/2*a)^30 - 939066728*tan(1/2*b*x
+ 2*a)^7*tan(1/2*a)^21 + 6391344852*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^22 - 11879791560*tan(1/2*b*x + 2*a)^5*tan
(1/2*a)^23 + 19856454696*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^25 - 19258200036*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^26 +
6571946504*tan(1/2*b*x + 2*a)*tan(1/2*a)^27 - 680794644*tan(1/2*a)^28 + 314808792*tan(1/2*b*x + 2*a)^7*tan(1/
2*a)^19 - 4808309376*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^20 + 19663795984*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^21 - 319
57055712*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^22 + 19856454696*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^23 - 4012160256*tan(
1/2*b*x + 2*a)*tan(1/2*a)^25 + 925883136*tan(1/2*a)^26 + 211641984*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^17 - 965890
76*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^18 - 6655212048*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^19 + 24029264184*tan(1/2*b*
x + 2*a)^4*tan(1/2*a)^20 - 32793412368*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^21 + 19258200036*tan(1/2*b*x + 2*a)^2*t
an(1/2*a)^22 - 4012160256*tan(1/2*b*x + 2*a)*tan(1/2*a)^23 - 234315648*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^15 + 18
88388352*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^16 - 4440981672*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^17 + 466539808*tan(1/
2*b*x + 2*a)^4*tan(1/2*a)^18 + 11050166160*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^19 - 14381865792*tan(1/2*b*x + 2*a)
^2*tan(1/2*a)^20 + 6571946504*tan(1/2*b*x + 2*a)*tan(1/2*a)^21 - 925883136*tan(1/2*a)^22 + 95841468*tan(1/2*b*
x + 2*a)^7*tan(1/2*a)^13 - 1144864350*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^14 + 4935828252*tan(1/2*b*x + 2*a)^5*tan
(1/2*a)^15 - 9447759099*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^16 + 7397865576*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^17 - 3
10355316*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^18 - 2181545880*tan(1/2*b*x + 2*a)*tan(1/2*a)^19 + 680794644*tan(1/2*
a)^20 - 20891364*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^11 + 336486592*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^12 - 201047796
0*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^13 + 5726288880*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^14 - 8215340892*tan(1/2*b*x
+ 2*a)^3*tan(1/2*a)^15 + 5649253632*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^16 - 1477416960*tan(1/2*b*x + 2*a)*tan(1/2
*a)^17 + 17553920*tan(1/2*a)^18 + 2486912*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^9 - 54465762*tan(1/2*b*x + 2*a)^6*ta
n(1/2*a)^10 + 436639944*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^11 - 1680996460*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^12 + 3
352125960*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^13 - 3430559358*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^14 + 1635149568*tan(
1/2*b*x + 2*a)*tan(1/2*a)^15 - 267141600*tan(1/2*a)^16 - 153216*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^7 + 4777344*ta
n(1/2*b*x + 2*a)^6*tan(1/2*a)^8 - 52518388*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^9 + 272279280*tan(1/2*b*x + 2*a)^4*
tan(1/2*a)^10 - 729548040*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^11 + 1012326432*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^12 -
671389308*tan(1/2*b*x + 2*a)*tan(1/2*a)^13 + 162842112*tan(1/2*a)^14 + 4428*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^5
- 210486*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^6 + 3337020*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^7 - 23995332*tan(1/2*b*x
+ 2*a)^4*tan(1/2*a)^8 + 87480580*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^9 - 164195226*tan(1/2*b*x + 2*a)^2*tan(1/2*a
)^10 + 147189588*tan(1/2*b*x + 2*a)*tan(1/2*a)^11 - 48570730*tan(1/2*a)^12 + 108*tan(1/2*b*x + 2*a)^7*tan(1/2*
a)^3 + 2880*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^4 - 95832*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^5 + 1046736*tan(1/2*b*x
+ 2*a)^4*tan(1/2*a)^6 - 5409324*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^7 + 14094720*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^8
- 17453120*tan(1/2*b*x + 2*a)*tan(1/2*a)^9 + 7902336*tan(1/2*a)^10 + 54*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^2 + 5
36*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^3 - 16740*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^4 + 142488*tan(1/2*b*x + 2*a)^3*t
an(1/2*a)^5 - 546366*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^6 + 977472*tan(1/2*b*x + 2*a)*tan(1/2*a)^7 - 642384*tan(1
/2*a)^8 + 12*tan(1/2*b*x + 2*a)^5*tan(1/2*a) - 48*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^2 - 536*tan(1/2*b*x + 2*a)^3
*tan(1/2*a)^3 + 5472*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^4 - 16092*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 + 17280*tan(1/2
*a)^6 + tan(1/2*b*x + 2*a)^4 - 12*tan(1/2*b*x + 2*a)^3*tan(1/2*a) + 54*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^2 - 108
*tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 162*tan(1/2*a)^4)/((81*tan(1/2*a)^20 - 1080*tan(1/2*a)^18 + 5724*tan(1/2*a)
^16 - 15240*tan(1/2*a)^14 + 21286*tan(1/2*a)^12 - 15240*tan(1/2*a)^10 + 5724*tan(1/2*a)^8 - 1080*tan(1/2*a)^6
+ 81*tan(1/2*a)^4)*(3*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 - tan(1/2*b*x + 2*a)*tan(1/2*a)^6 - 10*tan(1/2*b*x + 2
*a)^2*tan(1/2*a)^3 + 15*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 - 3*tan(1/2*a)^5 + 3*tan(1/2*b*x + 2*a)^2*tan(1/2*a) -
15*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 + 10*tan(1/2*a)^3 + tan(1/2*b*x + 2*a) - 3*tan(1/2*a))^4) + 120*log(abs(ta
n(1/2*b*x + 2*a)*tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*tan(1/2*a)^2 - 1)) - 120*log(abs(3*tan(1/2
*b*x + 2*a)*tan(1/2*a)^2 - tan(1/2*a)^3 - tan(1/2*b*x + 2*a) + 3*tan(1/2*a))))/b