3.32 \(\int \csc ^5(2 a+2 b x) \sin ^3(a+b x) \, dx\)
Optimal. Leaf size=70 \[ -\frac{15 \csc (a+b x)}{256 b}+\frac{15 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}+\frac{5 \csc (a+b x) \sec ^2(a+b x)}{256 b} \]
[Out]
(15*ArcTanh[Sin[a + b*x]])/(256*b) - (15*Csc[a + b*x])/(256*b) + (5*Csc[a + b*x]*Sec[a + b*x]^2)/(256*b) + (Cs
c[a + b*x]*Sec[a + b*x]^4)/(128*b)
________________________________________________________________________________________
Rubi [A] time = 0.0721454, 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, 2621, 288, 321, 207} \[ -\frac{15 \csc (a+b x)}{256 b}+\frac{15 \tanh ^{-1}(\sin (a+b x))}{256 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}+\frac{5 \csc (a+b x) \sec ^2(a+b x)}{256 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[2*a + 2*b*x]^5*Sin[a + b*x]^3,x]
[Out]
(15*ArcTanh[Sin[a + b*x]])/(256*b) - (15*Csc[a + b*x])/(256*b) + (5*Csc[a + b*x]*Sec[a + b*x]^2)/(256*b) + (Cs
c[a + b*x]*Sec[a + b*x]^4)/(128*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 2621
Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(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 ^5(2 a+2 b x) \sin ^3(a+b x) \, dx &=\frac{1}{32} \int \csc ^2(a+b x) \sec ^5(a+b x) \, dx\\ &=-\frac{\operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^3} \, dx,x,\csc (a+b x)\right )}{32 b}\\ &=\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}-\frac{5 \operatorname{Subst}\left (\int \frac{x^4}{\left (-1+x^2\right )^2} \, dx,x,\csc (a+b x)\right )}{128 b}\\ &=\frac{5 \csc (a+b x) \sec ^2(a+b x)}{256 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}-\frac{15 \operatorname{Subst}\left (\int \frac{x^2}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{256 b}\\ &=-\frac{15 \csc (a+b x)}{256 b}+\frac{5 \csc (a+b x) \sec ^2(a+b x)}{256 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}-\frac{15 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\csc (a+b x)\right )}{256 b}\\ &=\frac{15 \tanh ^{-1}(\sin (a+b x))}{256 b}-\frac{15 \csc (a+b x)}{256 b}+\frac{5 \csc (a+b x) \sec ^2(a+b x)}{256 b}+\frac{\csc (a+b x) \sec ^4(a+b x)}{128 b}\\ \end{align*}
Mathematica [C] time = 0.0345482, size = 29, normalized size = 0.41 \[ -\frac{\csc (a+b x) \text{Hypergeometric2F1}\left (-\frac{1}{2},3,\frac{1}{2},\sin ^2(a+b x)\right )}{32 b} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[2*a + 2*b*x]^5*Sin[a + b*x]^3,x]
[Out]
-(Csc[a + b*x]*Hypergeometric2F1[-1/2, 3, 1/2, Sin[a + b*x]^2])/(32*b)
________________________________________________________________________________________
Maple [A] time = 0.036, size = 76, normalized size = 1.1 \begin{align*}{\frac{1}{128\,b\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{4}}}+{\frac{5}{256\,b\sin \left ( bx+a \right ) \left ( \cos \left ( bx+a \right ) \right ) ^{2}}}-{\frac{15}{256\,b\sin \left ( bx+a \right ) }}+{\frac{15\,\ln \left ( \sec \left ( bx+a \right ) +\tan \left ( bx+a \right ) \right ) }{256\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(2*b*x+2*a)^5*sin(b*x+a)^3,x)
[Out]
1/128/b/sin(b*x+a)/cos(b*x+a)^4+5/256/b/sin(b*x+a)/cos(b*x+a)^2-15/256/b/sin(b*x+a)+15/256/b*ln(sec(b*x+a)+tan
(b*x+a))
________________________________________________________________________________________
Maxima [B] time = 2.06677, size = 2437, normalized size = 34.81 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(2*b*x+2*a)^5*sin(b*x+a)^3,x, algorithm="maxima")
[Out]
1/512*(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))*cos(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
))*cos(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))*c
os(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))*cos(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))*cos(6*b*x + 6*a) + 72*(2*sin(4*b*x + 4*a) + 3*sin(2*
b*x + 2*a))*cos(5*b*x + 5*a) - 40*(8*sin(3*b*x + 3*a) + 3*sin(b*x + a))*cos(4*b*x + 4*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 + 1
0*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*si
n(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*a)^2 + cos(a)^2 - 2
*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 + 2*cos(b*x + 2*a)*sin(a) + sin(a)^2)/(cos(b*x + 2*a)^2 + cos(a)^2 +
2*cos(a)*sin(b*x + 2*a) + sin(b*x + 2*a)^2 - 2*cos(b*x + 2*a)*sin(a) + sin(a)^2)) - 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))*sin(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)*sin(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))*sin(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)*sin(7*b*x + 7*a) - 8*(18*cos(5*b*x + 5*a) + 40*co
s(3*b*x + 3*a) + 15*cos(b*x + a))*sin(6*b*x + 6*a) - 72*(2*cos(4*b*x + 4*a) + 3*cos(2*b*x + 2*a) + 1)*sin(5*b*
x + 5*a) + 40*(8*cos(3*b*x + 3*a) + 3*cos(b*x + a))*sin(4*b*x + 4*a) - 160*(3*cos(2*b*x + 2*a) + 1)*sin(3*b*x
+ 3*a) + 480*cos(3*b*x + 3*a)*sin(2*b*x + 2*a) + 180*cos(b*x + a)*sin(2*b*x + 2*a) - 180*cos(2*b*x + 2*a)*sin(
b*x + a) - 60*sin(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*co
s(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*cos(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))*s
in(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 [A] time = 0.509329, size = 262, normalized size = 3.74 \begin{align*} \frac{15 \, \cos \left (b x + a\right )^{4} \log \left (\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 15 \, \cos \left (b x + a\right )^{4} \log \left (-\sin \left (b x + a\right ) + 1\right ) \sin \left (b x + a\right ) - 30 \, \cos \left (b x + a\right )^{4} + 10 \, \cos \left (b x + a\right )^{2} + 4}{512 \, b \cos \left (b x + a\right )^{4} \sin \left (b x + a\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(2*b*x+2*a)^5*sin(b*x+a)^3,x, algorithm="fricas")
[Out]
1/512*(15*cos(b*x + a)^4*log(sin(b*x + a) + 1)*sin(b*x + a) - 15*cos(b*x + a)^4*log(-sin(b*x + a) + 1)*sin(b*x
+ a) - 30*cos(b*x + a)^4 + 10*cos(b*x + a)^2 + 4)/(b*cos(b*x + a)^4*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(2*b*x+2*a)**5*sin(b*x+a)**3,x)
[Out]
Timed out
________________________________________________________________________________________
Giac [B] time = 5.78996, size = 5443, normalized size = 77.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(2*b*x+2*a)^5*sin(b*x+a)^3,x, algorithm="giac")
[Out]
-1/256*(4*(tan(1/2*b*x + 2*a)*tan(1/2*a)^12 - 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^10 + 6*tan(1/2*a)^11 - 27*tan(1
/2*b*x + 2*a)*tan(1/2*a)^8 - 2*tan(1/2*a)^9 - 36*tan(1/2*a)^7 + 27*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 - 36*tan(1/
2*a)^5 + 12*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - 2*tan(1/2*a)^3 - tan(1/2*b*x + 2*a) + 6*tan(1/2*a))/((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))*(3*tan(1/2*a)^5 - 10*tan(1/2*a)^3 + 3*tan(1/2*a))
) + 2*(9*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^48 - 54*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^46 + 90*tan(1/2*b*x + 2*a)^6*
tan(1/2*a)^47 - tan(1/2*b*x + 2*a)^5*tan(1/2*a)^48 - 16938*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^44 + 10878*tan(1/2*
b*x + 2*a)^6*tan(1/2*a)^45 - 2058*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^46 + 162*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^47
- tan(1/2*b*x + 2*a)^3*tan(1/2*a)^48 + 648690*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^42 - 772902*tan(1/2*b*x + 2*a)^6
*tan(1/2*a)^43 + 320922*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^44 - 59994*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^45 + 5718*t
an(1/2*b*x + 2*a)^3*tan(1/2*a)^46 - 306*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^47 + 9*tan(1/2*b*x + 2*a)*tan(1/2*a)^4
8 - 11649780*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^40 + 20073870*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^41 - 12606642*tan(1
/2*b*x + 2*a)^5*tan(1/2*a)^42 + 3790962*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^43 - 601830*tan(1/2*b*x + 2*a)^3*tan(1
/2*a)^44 + 53562*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^45 - 2646*tan(1/2*b*x + 2*a)*tan(1/2*a)^46 + 54*tan(1/2*a)^47
+ 122301894*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^38 - 281501690*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^39 + 240373332*tan
(1/2*b*x + 2*a)^5*tan(1/2*a)^40 - 99232506*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^41 + 21805614*tan(1/2*b*x + 2*a)^3*
tan(1/2*a)^42 - 2617650*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^43 + 166230*tan(1/2*b*x + 2*a)*tan(1/2*a)^44 - 4446*ta
n(1/2*a)^45 - 762446542*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^36 + 2299471746*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^37 - 2
589365766*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^38 + 1410179550*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^39 - 404516268*tan(1
/2*b*x + 2*a)^3*tan(1/2*a)^40 + 62438634*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^41 - 4933038*tan(1/2*b*x + 2*a)*tan(1
/2*a)^42 + 159462*tan(1/2*a)^43 + 2624819022*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^34 - 10699256970*tan(1/2*b*x + 2*
a)^6*tan(1/2*a)^35 + 16002520222*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^36 - 11500345350*tan(1/2*b*x + 2*a)^4*tan(1/2
*a)^37 + 4307714970*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^38 - 849637038*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^39 + 835422
84*tan(1/2*b*x + 2*a)*tan(1/2*a)^40 - 3248766*tan(1/2*a)^41 - 4363726131*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^32 +
26510577474*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^33 - 55167038478*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^34 + 53489761470*
tan(1/2*b*x + 2*a)^4*tan(1/2*a)^35 - 26616485090*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^36 + 6857208774*tan(1/2*b*x +
2*a)^2*tan(1/2*a)^37 - 856321050*tan(1/2*b*x + 2*a)*tan(1/2*a)^38 + 40894922*tan(1/2*a)^39 + 148294628*tan(1/
2*b*x + 2*a)^7*tan(1/2*a)^30 - 23718089916*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^31 + 91329113691*tan(1/2*b*x + 2*a)
^5*tan(1/2*a)^32 - 132573864918*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^33 + 92023852050*tan(1/2*b*x + 2*a)^3*tan(1/2*
a)^34 - 32055103710*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^35 + 5284344754*tan(1/2*b*x + 2*a)*tan(1/2*a)^36 - 3220931
22*tan(1/2*a)^37 + 11061475644*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^28 - 36130112340*tan(1/2*b*x + 2*a)^6*tan(1/2*a
)^29 - 3422040164*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^30 + 118445882004*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^31 - 15245
2905381*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^32 + 79837550214*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^33 - 18462316050*tan(
1/2*b*x + 2*a)*tan(1/2*a)^34 + 1522201770*tan(1/2*a)^35 - 14430929868*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^26 + 104
533629188*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^27 - 231729549276*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^28 + 180365650620*
tan(1/2*b*x + 2*a)^4*tan(1/2*a)^29 + 5258310492*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^30 - 71047415988*tan(1/2*b*x +
2*a)^2*tan(1/2*a)^31 + 30651378381*tan(1/2*b*x + 2*a)*tan(1/2*a)^32 - 3833122290*tan(1/2*a)^33 - 62534029428*
tan(1/2*b*x + 2*a)^6*tan(1/2*a)^25 + 304139029068*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^26 - 522734622092*tan(1/2*b*
x + 2*a)^4*tan(1/2*a)^27 + 386477003556*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^28 - 109057608828*tan(1/2*b*x + 2*a)^2
*tan(1/2*a)^29 - 750175580*tan(1/2*b*x + 2*a)*tan(1/2*a)^30 + 3377366748*tan(1/2*a)^31 + 14430929868*tan(1/2*b
*x + 2*a)^7*tan(1/2*a)^22 - 62534029428*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^23 + 313058642364*tan(1/2*b*x + 2*a)^4
*tan(1/2*a)^25 - 506043183348*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^26 + 313110921612*tan(1/2*b*x + 2*a)^2*tan(1/2*a
)^27 - 77490791364*tan(1/2*b*x + 2*a)*tan(1/2*a)^28 + 5273542932*tan(1/2*a)^29 - 11061475644*tan(1/2*b*x + 2*a
)^7*tan(1/2*a)^20 + 104533629188*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^21 - 304139029068*tan(1/2*b*x + 2*a)^5*tan(1/
2*a)^22 + 313058642364*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^23 - 186855789276*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^25 +
100600780788*tan(1/2*b*x + 2*a)*tan(1/2*a)^26 - 14857714436*tan(1/2*a)^27 - 148294628*tan(1/2*b*x + 2*a)^7*tan
(1/2*a)^18 - 36130112340*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^19 + 231729549276*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^20
- 522734622092*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^21 + 506043183348*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^22 - 18685578
9276*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^23 + 8802017172*tan(1/2*a)^25 + 4363726131*tan(1/2*b*x + 2*a)^7*tan(1/2*a
)^16 - 23718089916*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^17 + 3422040164*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^18 + 180365
650620*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^19 - 386477003556*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^20 + 313110921612*tan
(1/2*b*x + 2*a)^2*tan(1/2*a)^21 - 100600780788*tan(1/2*b*x + 2*a)*tan(1/2*a)^22 + 8802017172*tan(1/2*a)^23 - 2
624819022*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^14 + 26510577474*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^15 - 91329113691*ta
n(1/2*b*x + 2*a)^5*tan(1/2*a)^16 + 118445882004*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^17 - 5258310492*tan(1/2*b*x +
2*a)^3*tan(1/2*a)^18 - 109057608828*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^19 + 77490791364*tan(1/2*b*x + 2*a)*tan(1/
2*a)^20 - 14857714436*tan(1/2*a)^21 + 762446542*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^12 - 10699256970*tan(1/2*b*x +
2*a)^6*tan(1/2*a)^13 + 55167038478*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^14 - 132573864918*tan(1/2*b*x + 2*a)^4*tan
(1/2*a)^15 + 152452905381*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^16 - 71047415988*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^17
+ 750175580*tan(1/2*b*x + 2*a)*tan(1/2*a)^18 + 5273542932*tan(1/2*a)^19 - 122301894*tan(1/2*b*x + 2*a)^7*tan(1
/2*a)^10 + 2299471746*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^11 - 16002520222*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^12 + 53
489761470*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^13 - 92023852050*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^14 + 79837550214*ta
n(1/2*b*x + 2*a)^2*tan(1/2*a)^15 - 30651378381*tan(1/2*b*x + 2*a)*tan(1/2*a)^16 + 3377366748*tan(1/2*a)^17 + 1
1649780*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^8 - 281501690*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^9 + 2589365766*tan(1/2*b
*x + 2*a)^5*tan(1/2*a)^10 - 11500345350*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^11 + 26616485090*tan(1/2*b*x + 2*a)^3*
tan(1/2*a)^12 - 32055103710*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^13 + 18462316050*tan(1/2*b*x + 2*a)*tan(1/2*a)^14
- 3833122290*tan(1/2*a)^15 - 648690*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^6 + 20073870*tan(1/2*b*x + 2*a)^6*tan(1/2*
a)^7 - 240373332*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^8 + 1410179550*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^9 - 4307714970
*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^10 + 6857208774*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^11 - 5284344754*tan(1/2*b*x +
2*a)*tan(1/2*a)^12 + 1522201770*tan(1/2*a)^13 + 16938*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^4 - 772902*tan(1/2*b*x
+ 2*a)^6*tan(1/2*a)^5 + 12606642*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^6 - 99232506*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^
7 + 404516268*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^8 - 849637038*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^9 + 856321050*tan(
1/2*b*x + 2*a)*tan(1/2*a)^10 - 322093122*tan(1/2*a)^11 + 54*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^2 + 10878*tan(1/2*
b*x + 2*a)^6*tan(1/2*a)^3 - 320922*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^4 + 3790962*tan(1/2*b*x + 2*a)^4*tan(1/2*a)
^5 - 21805614*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^6 + 62438634*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^7 - 83542284*tan(1/
2*b*x + 2*a)*tan(1/2*a)^8 + 40894922*tan(1/2*a)^9 - 9*tan(1/2*b*x + 2*a)^7 + 90*tan(1/2*b*x + 2*a)^6*tan(1/2*a
) + 2058*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^2 - 59994*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^3 + 601830*tan(1/2*b*x + 2*
a)^3*tan(1/2*a)^4 - 2617650*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^5 + 4933038*tan(1/2*b*x + 2*a)*tan(1/2*a)^6 - 3248
766*tan(1/2*a)^7 + tan(1/2*b*x + 2*a)^5 + 162*tan(1/2*b*x + 2*a)^4*tan(1/2*a) - 5718*tan(1/2*b*x + 2*a)^3*tan(
1/2*a)^2 + 53562*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^3 - 166230*tan(1/2*b*x + 2*a)*tan(1/2*a)^4 + 159462*tan(1/2*a
)^5 + tan(1/2*b*x + 2*a)^3 - 306*tan(1/2*b*x + 2*a)^2*tan(1/2*a) + 2646*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - 4446
*tan(1/2*a)^3 - 9*tan(1/2*b*x + 2*a) + 54*tan(1/2*a))/((tan(1/2*a)^24 - 60*tan(1/2*a)^22 + 1410*tan(1/2*a)^20
- 16204*tan(1/2*a)^18 + 92655*tan(1/2*a)^16 - 245880*tan(1/2*a)^14 + 336156*tan(1/2*a)^12 - 245880*tan(1/2*a)^
10 + 92655*tan(1/2*a)^8 - 16204*tan(1/2*a)^6 + 1410*tan(1/2*a)^4 - 60*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*t
an(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)^4) - 15*log(abs(tan(1/2*b*x + 2*a)*tan(1/2*a)^3 + 3
*tan(1/2*b*x + 2*a)*tan(1/2*a)^2 - tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*tan(1/2*a)^2 - tan(1/2*b
*x + 2*a) + 3*tan(1/2*a) - 1)) + 15*log(abs(tan(1/2*b*x + 2*a)*tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a)^
2 + tan(1/2*a)^3 - 3*tan(1/2*b*x + 2*a)*tan(1/2*a) + 3*tan(1/2*a)^2 + tan(1/2*b*x + 2*a) - 3*tan(1/2*a) - 1)))
/b