3.44 \(\int \csc (a+b x) \csc ^4(2 a+2 b x) \, dx\)
Optimal. Leaf size=89 \[ \frac{35 \sec ^3(a+b x)}{384 b}+\frac{35 \sec (a+b x)}{128 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b} \]
[Out]
(-35*ArcTanh[Cos[a + b*x]])/(128*b) + (35*Sec[a + b*x])/(128*b) + (35*Sec[a + b*x]^3)/(384*b) - (7*Csc[a + b*x
]^2*Sec[a + b*x]^3)/(128*b) - (Csc[a + b*x]^4*Sec[a + b*x]^3)/(64*b)
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Rubi [A] time = 0.07168, antiderivative size = 89, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used =
{4288, 2622, 288, 302, 207} \[ \frac{35 \sec ^3(a+b x)}{384 b}+\frac{35 \sec (a+b x)}{128 b}-\frac{35 \tanh ^{-1}(\cos (a+b x))}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b} \]
Antiderivative was successfully verified.
[In]
Int[Csc[a + b*x]*Csc[2*a + 2*b*x]^4,x]
[Out]
(-35*ArcTanh[Cos[a + b*x]])/(128*b) + (35*Sec[a + b*x])/(128*b) + (35*Sec[a + b*x]^3)/(384*b) - (7*Csc[a + b*x
]^2*Sec[a + b*x]^3)/(128*b) - (Csc[a + b*x]^4*Sec[a + b*x]^3)/(64*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 302
Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]
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 (a+b x) \csc ^4(2 a+2 b x) \, dx &=\frac{1}{16} \int \csc ^5(a+b x) \sec ^4(a+b x) \, dx\\ &=\frac{\operatorname{Subst}\left (\int \frac{x^8}{\left (-1+x^2\right )^3} \, dx,x,\sec (a+b x)\right )}{16 b}\\ &=-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}+\frac{7 \operatorname{Subst}\left (\int \frac{x^6}{\left (-1+x^2\right )^2} \, dx,x,\sec (a+b x)\right )}{64 b}\\ &=-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}+\frac{35 \operatorname{Subst}\left (\int \frac{x^4}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{128 b}\\ &=-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}+\frac{35 \operatorname{Subst}\left (\int \left (1+x^2+\frac{1}{-1+x^2}\right ) \, dx,x,\sec (a+b x)\right )}{128 b}\\ &=\frac{35 \sec (a+b x)}{128 b}+\frac{35 \sec ^3(a+b x)}{384 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}+\frac{35 \operatorname{Subst}\left (\int \frac{1}{-1+x^2} \, dx,x,\sec (a+b x)\right )}{128 b}\\ &=-\frac{35 \tanh ^{-1}(\cos (a+b x))}{128 b}+\frac{35 \sec (a+b x)}{128 b}+\frac{35 \sec ^3(a+b x)}{384 b}-\frac{7 \csc ^2(a+b x) \sec ^3(a+b x)}{128 b}-\frac{\csc ^4(a+b x) \sec ^3(a+b x)}{64 b}\\ \end{align*}
Mathematica [B] time = 0.511508, size = 268, normalized size = 3.01 \[ -\frac{\csc ^{10}(a+b x) \left (658 \cos (2 (a+b x))-228 \cos (3 (a+b x))+140 \cos (4 (a+b x))-76 \cos (5 (a+b x))-210 \cos (6 (a+b x))+76 \cos (7 (a+b x))-315 \cos (3 (a+b x)) \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )-105 \cos (5 (a+b x)) \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+105 \cos (7 (a+b x)) \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+3 \cos (a+b x) \left (-105 \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )+105 \log \left (\cos \left (\frac{1}{2} (a+b x)\right )\right )+76\right )+315 \cos (3 (a+b x)) \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )+105 \cos (5 (a+b x)) \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )-105 \cos (7 (a+b x)) \log \left (\sin \left (\frac{1}{2} (a+b x)\right )\right )-204\right )}{384 b \left (\csc ^2\left (\frac{1}{2} (a+b x)\right )-\sec ^2\left (\frac{1}{2} (a+b x)\right )\right )^3} \]
Antiderivative was successfully verified.
[In]
Integrate[Csc[a + b*x]*Csc[2*a + 2*b*x]^4,x]
[Out]
-(Csc[a + b*x]^10*(-204 + 658*Cos[2*(a + b*x)] - 228*Cos[3*(a + b*x)] + 140*Cos[4*(a + b*x)] - 76*Cos[5*(a + b
*x)] - 210*Cos[6*(a + b*x)] + 76*Cos[7*(a + b*x)] - 315*Cos[3*(a + b*x)]*Log[Cos[(a + b*x)/2]] - 105*Cos[5*(a
+ b*x)]*Log[Cos[(a + b*x)/2]] + 105*Cos[7*(a + b*x)]*Log[Cos[(a + b*x)/2]] + 3*Cos[a + b*x]*(76 + 105*Log[Cos[
(a + b*x)/2]] - 105*Log[Sin[(a + b*x)/2]]) + 315*Cos[3*(a + b*x)]*Log[Sin[(a + b*x)/2]] + 105*Cos[5*(a + b*x)]
*Log[Sin[(a + b*x)/2]] - 105*Cos[7*(a + b*x)]*Log[Sin[(a + b*x)/2]]))/(384*b*(Csc[(a + b*x)/2]^2 - Sec[(a + b*
x)/2]^2)^3)
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Maple [A] time = 0.037, size = 99, normalized size = 1.1 \begin{align*} -{\frac{1}{64\,b \left ( \sin \left ( bx+a \right ) \right ) ^{4} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}+{\frac{7}{192\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2} \left ( \cos \left ( bx+a \right ) \right ) ^{3}}}-{\frac{35}{384\,b \left ( \sin \left ( bx+a \right ) \right ) ^{2}\cos \left ( bx+a \right ) }}+{\frac{35}{128\,b\cos \left ( bx+a \right ) }}+{\frac{35\,\ln \left ( \csc \left ( bx+a \right ) -\cot \left ( bx+a \right ) \right ) }{128\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csc(b*x+a)*csc(2*b*x+2*a)^4,x)
[Out]
-1/64/b/sin(b*x+a)^4/cos(b*x+a)^3+7/192/b/sin(b*x+a)^2/cos(b*x+a)^3-35/384/b/sin(b*x+a)^2/cos(b*x+a)+35/128/b/
cos(b*x+a)+35/128/b*ln(csc(b*x+a)-cot(b*x+a))
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Maxima [B] time = 1.78768, size = 5192, normalized size = 58.34 \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)*csc(2*b*x+2*a)^4,x, algorithm="maxima")
[Out]
1/768*(4*(105*cos(13*b*x + 13*a) - 70*cos(11*b*x + 11*a) - 329*cos(9*b*x + 9*a) + 204*cos(7*b*x + 7*a) - 329*c
os(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(14*b*x + 14*a) - 420*(cos(12*b*x + 12*a) + 3*cos
(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(13*
b*x + 13*a) + 4*(70*cos(11*b*x + 11*a) + 329*cos(9*b*x + 9*a) - 204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) +
70*cos(3*b*x + 3*a) - 105*cos(b*x + a))*cos(12*b*x + 12*a) + 280*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) -
3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(11*b*x + 11*a) + 12*(329*cos(9*b*x + 9*a)
- 204*cos(7*b*x + 7*a) + 329*cos(5*b*x + 5*a) + 70*cos(3*b*x + 3*a) - 105*cos(b*x + a))*cos(10*b*x + 10*a) - 1
316*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(9*b*x + 9*a) + 1
2*(204*cos(7*b*x + 7*a) - 329*cos(5*b*x + 5*a) - 70*cos(3*b*x + 3*a) + 105*cos(b*x + a))*cos(8*b*x + 8*a) + 81
6*(3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(7*b*x + 7*a) - 84*(47*cos(5*b*x + 5*a)
+ 10*cos(3*b*x + 3*a) - 15*cos(b*x + a))*cos(6*b*x + 6*a) + 1316*(3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*c
os(5*b*x + 5*a) + 420*(2*cos(3*b*x + 3*a) - 3*cos(b*x + a))*cos(4*b*x + 4*a) + 280*(cos(2*b*x + 2*a) - 1)*cos(
3*b*x + 3*a) - 420*cos(2*b*x + 2*a)*cos(b*x + a) + 105*(2*(cos(12*b*x + 12*a) + 3*cos(10*b*x + 10*a) - 3*cos(8
*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(14*b*x + 14*a) - cos(14*b*x
+ 14*a)^2 - 2*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x
+ 2*a) - 1)*cos(12*b*x + 12*a) - cos(12*b*x + 12*a)^2 + 6*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4*
b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a) - 9*cos(10*b*x + 10*a)^2 - 6*(3*cos(6*b*x + 6*a) - 3*cos
(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a) - 9*cos(8*b*x + 8*a)^2 + 6*(3*cos(4*b*x + 4*a) + cos(2*
b*x + 2*a) - 1)*cos(6*b*x + 6*a) - 9*cos(6*b*x + 6*a)^2 - 6*(cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - 9*cos(4*
b*x + 4*a)^2 - cos(2*b*x + 2*a)^2 + 2*(sin(12*b*x + 12*a) + 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(
6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(14*b*x + 14*a) - sin(14*b*x + 14*a)^2 - 2*(3*sin(10*
b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(12*b*x + 12
*a) - sin(12*b*x + 12*a)^2 + 6*(3*sin(8*b*x + 8*a) + 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a
))*sin(10*b*x + 10*a) - 9*sin(10*b*x + 10*a)^2 - 6*(3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a)
)*sin(8*b*x + 8*a) - 9*sin(8*b*x + 8*a)^2 + 6*(3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 9*sin
(6*b*x + 6*a)^2 - 9*sin(4*b*x + 4*a)^2 - 6*sin(4*b*x + 4*a)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2 + 2*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) - 105
*(2*(cos(12*b*x + 12*a) + 3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a)
+ cos(2*b*x + 2*a) - 1)*cos(14*b*x + 14*a) - cos(14*b*x + 14*a)^2 - 2*(3*cos(10*b*x + 10*a) - 3*cos(8*b*x + 8*
a) - 3*cos(6*b*x + 6*a) + 3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(12*b*x + 12*a) - cos(12*b*x + 12*a)^2
+ 6*(3*cos(8*b*x + 8*a) + 3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(10*b*x + 10*a)
- 9*cos(10*b*x + 10*a)^2 - 6*(3*cos(6*b*x + 6*a) - 3*cos(4*b*x + 4*a) - cos(2*b*x + 2*a) + 1)*cos(8*b*x + 8*a)
- 9*cos(8*b*x + 8*a)^2 + 6*(3*cos(4*b*x + 4*a) + cos(2*b*x + 2*a) - 1)*cos(6*b*x + 6*a) - 9*cos(6*b*x + 6*a)^
2 - 6*(cos(2*b*x + 2*a) - 1)*cos(4*b*x + 4*a) - 9*cos(4*b*x + 4*a)^2 - cos(2*b*x + 2*a)^2 + 2*(sin(12*b*x + 12
*a) + 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*
sin(14*b*x + 14*a) - sin(14*b*x + 14*a)^2 - 2*(3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a)
+ 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(12*b*x + 12*a) - sin(12*b*x + 12*a)^2 + 6*(3*sin(8*b*x + 8*a) + 3
*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(10*b*x + 10*a) - 9*sin(10*b*x + 10*a)^2 - 6*(3*
sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 9*sin(8*b*x + 8*a)^2 + 6*(3*sin(4
*b*x + 4*a) + sin(2*b*x + 2*a))*sin(6*b*x + 6*a) - 9*sin(6*b*x + 6*a)^2 - 9*sin(4*b*x + 4*a)^2 - 6*sin(4*b*x +
4*a)*sin(2*b*x + 2*a) - sin(2*b*x + 2*a)^2 + 2*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*(105*sin(13*b*x + 13*a) - 70*sin(11*b*x + 11*a) - 329*s
in(9*b*x + 9*a) + 204*sin(7*b*x + 7*a) - 329*sin(5*b*x + 5*a) - 70*sin(3*b*x + 3*a) + 105*sin(b*x + a))*sin(14
*b*x + 14*a) - 420*(sin(12*b*x + 12*a) + 3*sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*si
n(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(13*b*x + 13*a) + 4*(70*sin(11*b*x + 11*a) + 329*sin(9*b*x + 9*a) - 204*
sin(7*b*x + 7*a) + 329*sin(5*b*x + 5*a) + 70*sin(3*b*x + 3*a) - 105*sin(b*x + a))*sin(12*b*x + 12*a) + 280*(3*
sin(10*b*x + 10*a) - 3*sin(8*b*x + 8*a) - 3*sin(6*b*x + 6*a) + 3*sin(4*b*x + 4*a) + sin(2*b*x + 2*a))*sin(11*b
*x + 11*a) + 12*(329*sin(9*b*x + 9*a) - 204*sin(7*b*x + 7*a) + 329*sin(5*b*x + 5*a) + 70*sin(3*b*x + 3*a) - 10
5*sin(b*x + a))*sin(10*b*x + 10*a) - 1316*(3*sin(8*b*x + 8*a) + 3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(
2*b*x + 2*a))*sin(9*b*x + 9*a) + 12*(204*sin(7*b*x + 7*a) - 329*sin(5*b*x + 5*a) - 70*sin(3*b*x + 3*a) + 105*s
in(b*x + a))*sin(8*b*x + 8*a) + 816*(3*sin(6*b*x + 6*a) - 3*sin(4*b*x + 4*a) - sin(2*b*x + 2*a))*sin(7*b*x + 7
*a) - 84*(47*sin(5*b*x + 5*a) + 10*sin(3*b*x + 3*a) - 15*sin(b*x + a))*sin(6*b*x + 6*a) + 1316*(3*sin(4*b*x +
4*a) + sin(2*b*x + 2*a))*sin(5*b*x + 5*a) + 420*(2*sin(3*b*x + 3*a) - 3*sin(b*x + a))*sin(4*b*x + 4*a) + 280*s
in(3*b*x + 3*a)*sin(2*b*x + 2*a) - 420*sin(2*b*x + 2*a)*sin(b*x + a) + 420*cos(b*x + a))/(b*cos(14*b*x + 14*a)
^2 + b*cos(12*b*x + 12*a)^2 + 9*b*cos(10*b*x + 10*a)^2 + 9*b*cos(8*b*x + 8*a)^2 + 9*b*cos(6*b*x + 6*a)^2 + 9*b
*cos(4*b*x + 4*a)^2 + b*cos(2*b*x + 2*a)^2 + b*sin(14*b*x + 14*a)^2 + b*sin(12*b*x + 12*a)^2 + 9*b*sin(10*b*x
+ 10*a)^2 + 9*b*sin(8*b*x + 8*a)^2 + 9*b*sin(6*b*x + 6*a)^2 + 9*b*sin(4*b*x + 4*a)^2 + 6*b*sin(4*b*x + 4*a)*si
n(2*b*x + 2*a) + b*sin(2*b*x + 2*a)^2 - 2*(b*cos(12*b*x + 12*a) + 3*b*cos(10*b*x + 10*a) - 3*b*cos(8*b*x + 8*a
) - 3*b*cos(6*b*x + 6*a) + 3*b*cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) - b)*cos(14*b*x + 14*a) + 2*(3*b*cos(10*b
*x + 10*a) - 3*b*cos(8*b*x + 8*a) - 3*b*cos(6*b*x + 6*a) + 3*b*cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) - b)*cos(
12*b*x + 12*a) - 6*(3*b*cos(8*b*x + 8*a) + 3*b*cos(6*b*x + 6*a) - 3*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) +
b)*cos(10*b*x + 10*a) + 6*(3*b*cos(6*b*x + 6*a) - 3*b*cos(4*b*x + 4*a) - b*cos(2*b*x + 2*a) + b)*cos(8*b*x + 8
*a) - 6*(3*b*cos(4*b*x + 4*a) + b*cos(2*b*x + 2*a) - b)*cos(6*b*x + 6*a) + 6*(b*cos(2*b*x + 2*a) - b)*cos(4*b*
x + 4*a) - 2*b*cos(2*b*x + 2*a) - 2*(b*sin(12*b*x + 12*a) + 3*b*sin(10*b*x + 10*a) - 3*b*sin(8*b*x + 8*a) - 3*
b*sin(6*b*x + 6*a) + 3*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(14*b*x + 14*a) + 2*(3*b*sin(10*b*x + 10*a)
- 3*b*sin(8*b*x + 8*a) - 3*b*sin(6*b*x + 6*a) + 3*b*sin(4*b*x + 4*a) + b*sin(2*b*x + 2*a))*sin(12*b*x + 12*a)
- 6*(3*b*sin(8*b*x + 8*a) + 3*b*sin(6*b*x + 6*a) - 3*b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(10*b*x + 10
*a) + 6*(3*b*sin(6*b*x + 6*a) - 3*b*sin(4*b*x + 4*a) - b*sin(2*b*x + 2*a))*sin(8*b*x + 8*a) - 6*(3*b*sin(4*b*x
+ 4*a) + b*sin(2*b*x + 2*a))*sin(6*b*x + 6*a) + b)
________________________________________________________________________________________
Fricas [A] time = 0.518579, size = 417, normalized size = 4.69 \begin{align*} \frac{210 \, \cos \left (b x + a\right )^{6} - 350 \, \cos \left (b x + a\right )^{4} + 112 \, \cos \left (b x + a\right )^{2} - 105 \,{\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 105 \,{\left (\cos \left (b x + a\right )^{7} - 2 \, \cos \left (b x + a\right )^{5} + \cos \left (b x + a\right )^{3}\right )} \log \left (-\frac{1}{2} \, \cos \left (b x + a\right ) + \frac{1}{2}\right ) + 16}{768 \,{\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 )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csc(b*x+a)*csc(2*b*x+2*a)^4,x, algorithm="fricas")
[Out]
1/768*(210*cos(b*x + a)^6 - 350*cos(b*x + a)^4 + 112*cos(b*x + a)^2 - 105*(cos(b*x + a)^7 - 2*cos(b*x + a)^5 +
cos(b*x + a)^3)*log(1/2*cos(b*x + a) + 1/2) + 105*(cos(b*x + a)^7 - 2*cos(b*x + a)^5 + cos(b*x + a)^3)*log(-1
/2*cos(b*x + a) + 1/2) + 16)/(b*cos(b*x + a)^7 - 2*b*cos(b*x + a)^5 + b*cos(b*x + a)^3)
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \csc{\left (a + b x \right )} \csc ^{4}{\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)*csc(2*b*x+2*a)**4,x)
[Out]
Integral(csc(a + b*x)*csc(2*a + 2*b*x)**4, x)
________________________________________________________________________________________
Giac [B] time = 12.2181, size = 7393, normalized size = 83.07 \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)*csc(2*b*x+2*a)^4,x, algorithm="giac")
[Out]
-1/3072*(256*(36*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^35 - 6*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^36 - 1848*tan(1/2*b*x
+ 2*a)^5*tan(1/2*a)^33 + 1134*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^34 - 180*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^35 + 9*
tan(1/2*b*x + 2*a)^2*tan(1/2*a)^36 + 39276*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^31 - 42894*tan(1/2*b*x + 2*a)^4*tan
(1/2*a)^32 + 14532*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^33 - 2052*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^34 + 144*tan(1/2*
b*x + 2*a)*tan(1/2*a)^35 - 5*tan(1/2*a)^36 - 433836*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^29 + 709068*tan(1/2*b*x +
2*a)^4*tan(1/2*a)^30 - 376632*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^31 + 83367*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^32 -
8364*tan(1/2*b*x + 2*a)*tan(1/2*a)^33 + 342*tan(1/2*a)^34 + 2430348*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^27 - 57423
96*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^28 + 4466808*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^29 - 1422120*tan(1/2*b*x + 2*a
)^2*tan(1/2*a)^30 + 193068*tan(1/2*b*x + 2*a)*tan(1/2*a)^31 - 9639*tan(1/2*a)^32 - 5123196*tan(1/2*b*x + 2*a)^
5*tan(1/2*a)^25 + 20329092*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^26 - 24275368*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^27 +
11529468*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^28 - 2196396*tan(1/2*b*x + 2*a)*tan(1/2*a)^29 + 141660*tan(1/2*a)^30
- 1201860*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^23 - 18742620*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^24 + 50327784*tan(1/2*
b*x + 2*a)^3*tan(1/2*a)^25 - 40521528*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^26 + 12145596*tan(1/2*b*x + 2*a)*tan(1/2
*a)^27 - 1116072*tan(1/2*a)^28 + 15332100*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^21 - 42937380*tan(1/2*b*x + 2*a)^4*t
an(1/2*a)^22 + 10264680*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^23 + 37504740*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^24 - 254
25036*tan(1/2*b*x + 2*a)*tan(1/2*a)^25 + 4162356*tan(1/2*a)^26 - 11041020*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^19 +
84945240*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^20 - 153596328*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^21 + 85465080*tan(1/2
*b*x + 2*a)^2*tan(1/2*a)^22 - 5642820*tan(1/2*b*x + 2*a)*tan(1/2*a)^23 - 3694080*tan(1/2*a)^24 - 11041020*tan(
1/2*b*x + 2*a)^5*tan(1/2*a)^17 + 112912560*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^19 - 170450298*tan(1/2*b*x + 2*a)^2
*tan(1/2*a)^20 + 76700868*tan(1/2*b*x + 2*a)*tan(1/2*a)^21 - 8751060*tan(1/2*a)^22 + 15332100*tan(1/2*b*x + 2*
a)^5*tan(1/2*a)^15 - 84945240*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^16 + 112912560*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^1
7 - 55767060*tan(1/2*b*x + 2*a)*tan(1/2*a)^19 + 16730118*tan(1/2*a)^20 - 1201860*tan(1/2*b*x + 2*a)^5*tan(1/2*
a)^13 + 42937380*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^14 - 153596328*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^15 + 170450298
*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^16 - 55767060*tan(1/2*b*x + 2*a)*tan(1/2*a)^17 - 5123196*tan(1/2*b*x + 2*a)^5
*tan(1/2*a)^11 + 18742620*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^12 + 10264680*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^13 - 8
5465080*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^14 + 76700868*tan(1/2*b*x + 2*a)*tan(1/2*a)^15 - 16730118*tan(1/2*a)^1
6 + 2430348*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^9 - 20329092*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^10 + 50327784*tan(1/2
*b*x + 2*a)^3*tan(1/2*a)^11 - 37504740*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^12 - 5642820*tan(1/2*b*x + 2*a)*tan(1/2
*a)^13 + 8751060*tan(1/2*a)^14 - 433836*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^7 + 5742396*tan(1/2*b*x + 2*a)^4*tan(1
/2*a)^8 - 24275368*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^9 + 40521528*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^10 - 25425036*
tan(1/2*b*x + 2*a)*tan(1/2*a)^11 + 3694080*tan(1/2*a)^12 + 39276*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^5 - 709068*ta
n(1/2*b*x + 2*a)^4*tan(1/2*a)^6 + 4466808*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^7 - 11529468*tan(1/2*b*x + 2*a)^2*ta
n(1/2*a)^8 + 12145596*tan(1/2*b*x + 2*a)*tan(1/2*a)^9 - 4162356*tan(1/2*a)^10 - 1848*tan(1/2*b*x + 2*a)^5*tan(
1/2*a)^3 + 42894*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^4 - 376632*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^5 + 1422120*tan(1/
2*b*x + 2*a)^2*tan(1/2*a)^6 - 2196396*tan(1/2*b*x + 2*a)*tan(1/2*a)^7 + 1116072*tan(1/2*a)^8 + 36*tan(1/2*b*x
+ 2*a)^5*tan(1/2*a) - 1134*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^2 + 14532*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^3 - 83367
*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^4 + 193068*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 - 141660*tan(1/2*a)^6 + 6*tan(1/2*
b*x + 2*a)^4 - 180*tan(1/2*b*x + 2*a)^3*tan(1/2*a) + 2052*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^2 - 8364*tan(1/2*b*x
+ 2*a)*tan(1/2*a)^3 + 9639*tan(1/2*a)^4 - 9*tan(1/2*b*x + 2*a)^2 + 144*tan(1/2*b*x + 2*a)*tan(1/2*a) - 342*ta
n(1/2*a)^2 + 5)/((tan(1/2*a)^18 - 45*tan(1/2*a)^16 + 720*tan(1/2*a)^14 - 4728*tan(1/2*a)^12 + 10890*tan(1/2*a)
^10 - 10890*tan(1/2*a)^8 + 4728*tan(1/2*a)^6 - 720*tan(1/2*a)^4 + 45*tan(1/2*a)^2 - 1)*(tan(1/2*b*x + 2*a)^2*t
an(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*ta
n(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)^3) + 3*(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)^4
8 + 8316*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^43 - 6120*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^44 + 1400*tan(1/2*b*x + 2*a
)^5*tan(1/2*a)^45 - 24*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 - 253008*tan
(1/2*b*x + 2*a)^7*tan(1/2*a)^41 + 354774*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^42 - 165384*tan(1/2*b*x + 2*a)^5*tan(
1/2*a)^43 + 30180*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^44 - 1400*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 + 3918128*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^39 - 7606080*tan(1/2*b*x + 2*a)^6*tan(1/2*a
)^40 + 5365548*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^41 - 1698920*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^42 + 235368*tan(1/
2*b*x + 2*a)^3*tan(1/2*a)^43 - 10008*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
- 32664372*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^37 + 85352514*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^38 - 82649988*tan(1/
2*b*x + 2*a)^5*tan(1/2*a)^39 + 37937604*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^40 - 8596476*tan(1/2*b*x + 2*a)^3*tan(
1/2*a)^41 + 879006*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^42 - 27756*tan(1/2*b*x + 2*a)*tan(1/2*a)^43 - 162*tan(1/2*a
)^44 + 150470268*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^35 - 527449208*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^36 + 684424728
*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^37 - 427365432*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^38 + 137583540*tan(1/2*b*x + 2
*a)^3*tan(1/2*a)^39 - 22235328*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^40 + 1556784*tan(1/2*b*x + 2*a)*tan(1/2*a)^41 -
28944*tan(1/2*a)^42 - 369285568*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^33 + 1800957150*tan(1/2*b*x + 2*a)^6*tan(1/2*
a)^34 - 3157829352*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^35 + 2638274028*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^36 - 114501
5352*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^37 + 257818938*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^38 - 27438800*tan(1/2*b*x
+ 2*a)*tan(1/2*a)^39 + 1015632*tan(1/2*a)^40 + 332678976*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^31 - 2973661056*tan(1
/2*b*x + 2*a)^6*tan(1/2*a)^32 + 7767810012*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^33 - 9003276360*tan(1/2*b*x + 2*a)^
4*tan(1/2*a)^34 + 5266355688*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^35 - 1589641352*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^3
6 + 231030852*tan(1/2*b*x + 2*a)*tan(1/2*a)^37 - 12400752*tan(1/2*a)^38 + 498198168*tan(1/2*b*x + 2*a)^7*tan(1
/2*a)^29 + 141213844*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^30 - 6978686184*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^31 + 1486
0730619*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^32 - 12919461276*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^33 + 5392538622*tan(1
/2*b*x + 2*a)^2*tan(1/2*a)^34 - 1054828476*tan(1/2*b*x + 2*a)*tan(1/2*a)^35 + 76246378*tan(1/2*a)^36 - 1473564
360*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^27 + 7565433072*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^28 - 10492595088*tan(1/2*b
*x + 2*a)^5*tan(1/2*a)^29 - 707654192*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^30 + 11624633640*tan(1/2*b*x + 2*a)^3*ta
n(1/2*a)^31 - 8882796672*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^32 + 2571218880*tan(1/2*b*x + 2*a)*tan(1/2*a)^33 - 25
6007232*tan(1/2*a)^34 + 883677600*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^25 - 10018470420*tan(1/2*b*x + 2*a)^6*tan(1/
2*a)^26 + 30880710896*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^27 - 37780516920*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^28 + 17
392225680*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^29 + 478350836*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^30 - 2321948736*tan(1
/2*b*x + 2*a)*tan(1/2*a)^31 + 420191712*tan(1/2*a)^32 + 883677600*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^23 - 1862376
5160*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^25 + 50161528368*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^26 - 51515325744*tan(1/2
*b*x + 2*a)^3*tan(1/2*a)^27 + 22607976720*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^28 - 3433884120*tan(1/2*b*x + 2*a)*t
an(1/2*a)^29 - 26975424*tan(1/2*a)^30 - 1473564360*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^21 + 10018470420*tan(1/2*b*
x + 2*a)^6*tan(1/2*a)^22 - 18623765160*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^23 + 31169987784*tan(1/2*b*x + 2*a)^3*t
an(1/2*a)^25 - 30241452708*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^26 + 10324445736*tan(1/2*b*x + 2*a)*tan(1/2*a)^27 -
1070628372*tan(1/2*a)^28 + 498198168*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^19 - 7565433072*tan(1/2*b*x + 2*a)^6*tan
(1/2*a)^20 + 30880710896*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^21 - 50161528368*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^22 +
31169987784*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^23 - 6296940000*tan(1/2*b*x + 2*a)*tan(1/2*a)^25 + 1453140000*tan
(1/2*a)^26 + 332678976*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^17 - 141213844*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^18 - 104
92595088*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^19 + 37780516920*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^20 - 51515325744*tan
(1/2*b*x + 2*a)^3*tan(1/2*a)^21 + 30241452708*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^22 - 6296940000*tan(1/2*b*x + 2*
a)*tan(1/2*a)^23 - 369285568*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^15 + 2973661056*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^1
6 - 6978686184*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^17 + 707654192*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^18 + 17392225680
*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^19 - 22607976720*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^20 + 10324445736*tan(1/2*b*x
+ 2*a)*tan(1/2*a)^21 - 1453140000*tan(1/2*a)^22 + 150470268*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^13 - 1800957150*t
an(1/2*b*x + 2*a)^6*tan(1/2*a)^14 + 7767810012*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^15 - 14860730619*tan(1/2*b*x +
2*a)^4*tan(1/2*a)^16 + 11624633640*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^17 - 478350836*tan(1/2*b*x + 2*a)^2*tan(1/2
*a)^18 - 3433884120*tan(1/2*b*x + 2*a)*tan(1/2*a)^19 + 1070628372*tan(1/2*a)^20 - 32664372*tan(1/2*b*x + 2*a)^
7*tan(1/2*a)^11 + 527449208*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^12 - 3157829352*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^13
+ 9003276360*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^14 - 12919461276*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^15 + 8882796672
*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^16 - 2321948736*tan(1/2*b*x + 2*a)*tan(1/2*a)^17 + 26975424*tan(1/2*a)^18 + 3
918128*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^9 - 85352514*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^10 + 684424728*tan(1/2*b*x
+ 2*a)^5*tan(1/2*a)^11 - 2638274028*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^12 + 5266355688*tan(1/2*b*x + 2*a)^3*tan(
1/2*a)^13 - 5392538622*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^14 + 2571218880*tan(1/2*b*x + 2*a)*tan(1/2*a)^15 - 4201
91712*tan(1/2*a)^16 - 253008*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^7 + 7606080*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^8 - 8
2649988*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^9 + 427365432*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^10 - 1145015352*tan(1/2*
b*x + 2*a)^3*tan(1/2*a)^11 + 1589641352*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^12 - 1054828476*tan(1/2*b*x + 2*a)*tan
(1/2*a)^13 + 256007232*tan(1/2*a)^14 + 8316*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^5 - 354774*tan(1/2*b*x + 2*a)^6*ta
n(1/2*a)^6 + 5365548*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^7 - 37937604*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^8 + 13758354
0*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^9 - 257818938*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^10 + 231030852*tan(1/2*b*x + 2
*a)*tan(1/2*a)^11 - 76246378*tan(1/2*a)^12 + 108*tan(1/2*b*x + 2*a)^7*tan(1/2*a)^3 + 6120*tan(1/2*b*x + 2*a)^6
*tan(1/2*a)^4 - 165384*tan(1/2*b*x + 2*a)^5*tan(1/2*a)^5 + 1698920*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^6 - 8596476
*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^7 + 22235328*tan(1/2*b*x + 2*a)^2*tan(1/2*a)^8 - 27438800*tan(1/2*b*x + 2*a)*
tan(1/2*a)^9 + 12400752*tan(1/2*a)^10 + 54*tan(1/2*b*x + 2*a)^6*tan(1/2*a)^2 + 1400*tan(1/2*b*x + 2*a)^5*tan(1
/2*a)^3 - 30180*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^4 + 235368*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^5 - 879006*tan(1/2*
b*x + 2*a)^2*tan(1/2*a)^6 + 1556784*tan(1/2*b*x + 2*a)*tan(1/2*a)^7 - 1015632*tan(1/2*a)^8 + 12*tan(1/2*b*x +
2*a)^5*tan(1/2*a) + 24*tan(1/2*b*x + 2*a)^4*tan(1/2*a)^2 - 1400*tan(1/2*b*x + 2*a)^3*tan(1/2*a)^3 + 10008*tan(
1/2*b*x + 2*a)^2*tan(1/2*a)^4 - 27756*tan(1/2*b*x + 2*a)*tan(1/2*a)^5 + 28944*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*t
an(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)*ta
n(1/2*a)^2 + 10*tan(1/2*a)^3 + tan(1/2*b*x + 2*a) - 3*tan(1/2*a))^4) + 840*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) + 3*tan(1/2*a)^2 - 1)) - 840*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