\(\int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx\) [310]

Optimal result

Integrand size = 22, antiderivative size = 399 \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b} \]

[Out]

2*I*d*(d*x+c)^3/b^2+1/2*(d*x+c)^4/b-2*(d*x+c)^4*arctanh(exp(2*I*(b*x+a)))/b-6*d^2*(d*x+c)^2*ln(1+exp(2*I*(b*x+
a)))/b^3-3*I*d^3*(d*x+c)*polylog(4,-exp(2*I*(b*x+a)))/b^4+2*I*d*(d*x+c)^3*polylog(2,-exp(2*I*(b*x+a)))/b^2-2*I
*d*(d*x+c)^3*polylog(2,exp(2*I*(b*x+a)))/b^2-3*d^4*polylog(3,-exp(2*I*(b*x+a)))/b^5-3*d^2*(d*x+c)^2*polylog(3,
-exp(2*I*(b*x+a)))/b^3+3*d^2*(d*x+c)^2*polylog(3,exp(2*I*(b*x+a)))/b^3+3*I*d^3*(d*x+c)*polylog(4,exp(2*I*(b*x+
a)))/b^4+6*I*d^3*(d*x+c)*polylog(2,-exp(2*I*(b*x+a)))/b^4+3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5-3/2*d^4*pol
ylog(5,exp(2*I*(b*x+a)))/b^5-2*d*(d*x+c)^3*tan(b*x+a)/b^2+1/2*(d*x+c)^4*tan(b*x+a)^2/b

Rubi [A] (verified)

Time = 1.14 (sec) , antiderivative size = 399, normalized size of antiderivative = 1.00, number of steps used = 25, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.727, Rules used = {2700, 14, 4505, 6873, 12, 6874, 2631, 4268, 2611, 6744, 2320, 6724, 3801, 3800, 2221, 32} \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}+\frac {3 d^4 \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b} \]

[In]

Int[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^3,x]

[Out]

((2*I)*d*(c + d*x)^3)/b^2 + (c + d*x)^4/(2*b) - (2*(c + d*x)^4*ArcTanh[E^((2*I)*(a + b*x))])/b - (6*d^2*(c + d
*x)^2*Log[1 + E^((2*I)*(a + b*x))])/b^3 + ((6*I)*d^3*(c + d*x)*PolyLog[2, -E^((2*I)*(a + b*x))])/b^4 + ((2*I)*
d*(c + d*x)^3*PolyLog[2, -E^((2*I)*(a + b*x))])/b^2 - ((2*I)*d*(c + d*x)^3*PolyLog[2, E^((2*I)*(a + b*x))])/b^
2 - (3*d^4*PolyLog[3, -E^((2*I)*(a + b*x))])/b^5 - (3*d^2*(c + d*x)^2*PolyLog[3, -E^((2*I)*(a + b*x))])/b^3 +
(3*d^2*(c + d*x)^2*PolyLog[3, E^((2*I)*(a + b*x))])/b^3 - ((3*I)*d^3*(c + d*x)*PolyLog[4, -E^((2*I)*(a + b*x))
])/b^4 + ((3*I)*d^3*(c + d*x)*PolyLog[4, E^((2*I)*(a + b*x))])/b^4 + (3*d^4*PolyLog[5, -E^((2*I)*(a + b*x))])/
(2*b^5) - (3*d^4*PolyLog[5, E^((2*I)*(a + b*x))])/(2*b^5) - (2*d*(c + d*x)^3*Tan[a + b*x])/b^2 + ((c + d*x)^4*
Tan[a + b*x]^2)/(2*b)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rule 2221

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x]
- Dist[d*(m/(b*f*g*n*Log[F])), Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

Rule 2631

Int[Log[u_]*((a_.) + (b_.)*(x_))^(m_.), x_Symbol] :> Simp[(a + b*x)^(m + 1)*(Log[u]/(b*(m + 1))), x] - Dist[1/
(b*(m + 1)), Int[SimplifyIntegrand[(a + b*x)^(m + 1)*(D[u, x]/u), x], x], x] /; FreeQ[{a, b, m}, x] && Inverse
FunctionFreeQ[u, x] && NeQ[m, -1]

Rule 2700

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 3800

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[I*((c + d*x)^(m + 1)/(d*(m + 1))), x
] - Dist[2*I, Int[(c + d*x)^m*(E^(2*I*(e + f*x))/(1 + E^(2*I*(e + f*x)))), x], x] /; FreeQ[{c, d, e, f}, x] &&
IGtQ[m, 0]

Rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b*(c + d*x)^m*((b*Tan[e
+ f*x])^(n - 1)/(f*(n - 1))), x] + (-Dist[b*d*(m/(f*(n - 1))), Int[(c + d*x)^(m - 1)*(b*Tan[e + f*x])^(n - 1)
, x], x] - Dist[b^2, Int[(c + d*x)^m*(b*Tan[e + f*x])^(n - 2), x], x]) /; FreeQ[{b, c, d, e, f}, x] && GtQ[n,
1] && GtQ[m, 0]

Rule 4268

Int[csc[(e_.) + (f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^(I*(e + f*
x))]/f), x] + (-Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Log[1 - E^(I*(e + f*x))], x], x] + Dist[d*(m/f), Int[(c +
d*x)^(m - 1)*Log[1 + E^(I*(e + f*x))], x], x]) /; FreeQ[{c, d, e, f}, x] && IGtQ[m, 0]

Rule 4505

Int[Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Modul
e[{u = IntHide[Csc[a + b*x]^n*Sec[a + b*x]^p, x]}, Dist[(c + d*x)^m, u, x] - Dist[d*m, Int[(c + d*x)^(m - 1)*u
, x], x]] /; FreeQ[{a, b, c, d}, x] && IntegersQ[n, p] && GtQ[m, 0] && NeQ[n, p]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rule 6744

Int[((e_.) + (f_.)*(x_))^(m_.)*PolyLog[n_, (d_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(p_.)], x_Symbol] :> Simp
[(e + f*x)^m*(PolyLog[n + 1, d*(F^(c*(a + b*x)))^p]/(b*c*p*Log[F])), x] - Dist[f*(m/(b*c*p*Log[F])), Int[(e +
f*x)^(m - 1)*PolyLog[n + 1, d*(F^(c*(a + b*x)))^p], x], x] /; FreeQ[{F, a, b, c, d, e, f, n, p}, x] && GtQ[m,
0]

Rule 6873

Int[u_, x_Symbol] :> With[{v = NormalizeIntegrand[u, x]}, Int[v, x] /; v =!= u]

Rule 6874

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps \begin{align*} \text {integral}& = \frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-(4 d) \int (c+d x)^3 \left (\frac {\log (\tan (a+b x))}{b}+\frac {\tan ^2(a+b x)}{2 b}\right ) \, dx \\ & = \frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-(4 d) \int \frac {(c+d x)^3 \left (2 \log (\tan (a+b x))+\tan ^2(a+b x)\right )}{2 b} \, dx \\ & = \frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int (c+d x)^3 \left (2 \log (\tan (a+b x))+\tan ^2(a+b x)\right ) \, dx}{b} \\ & = \frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int \left (2 (c+d x)^3 \log (\tan (a+b x))+(c+d x)^3 \tan ^2(a+b x)\right ) \, dx}{b} \\ & = \frac {(c+d x)^4 \log (\tan (a+b x))}{b}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(2 d) \int (c+d x)^3 \tan ^2(a+b x) \, dx}{b}-\frac {(4 d) \int (c+d x)^3 \log (\tan (a+b x)) \, dx}{b} \\ & = -\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\int 2 b (c+d x)^4 \csc (2 a+2 b x) \, dx}{b}+\frac {(2 d) \int (c+d x)^3 \, dx}{b}+\frac {\left (6 d^2\right ) \int (c+d x)^2 \tan (a+b x) \, dx}{b^2} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+2 \int (c+d x)^4 \csc (2 a+2 b x) \, dx-\frac {\left (12 i d^2\right ) \int \frac {e^{2 i (a+b x)} (c+d x)^2}{1+e^{2 i (a+b x)}} \, dx}{b^2} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {(4 d) \int (c+d x)^3 \log \left (1-e^{i (2 a+2 b x)}\right ) \, dx}{b}+\frac {(4 d) \int (c+d x)^3 \log \left (1+e^{i (2 a+2 b x)}\right ) \, dx}{b}+\frac {\left (12 d^3\right ) \int (c+d x) \log \left (1+e^{2 i (a+b x)}\right ) \, dx}{b^3} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}-\frac {\left (6 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (2 a+2 b x)}\right ) \, dx}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (2 a+2 b x)}\right ) \, dx}{b^2}-\frac {\left (6 i d^4\right ) \int \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right ) \, dx}{b^4} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\left (6 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,-e^{i (2 a+2 b x)}\right ) \, dx}{b^3}-\frac {\left (6 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,e^{i (2 a+2 b x)}\right ) \, dx}{b^3}-\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{2 i (a+b x)}\right )}{b^5} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\left (3 i d^4\right ) \int \operatorname {PolyLog}\left (4,-e^{i (2 a+2 b x)}\right ) \, dx}{b^4}-\frac {\left (3 i d^4\right ) \int \operatorname {PolyLog}\left (4,e^{i (2 a+2 b x)}\right ) \, dx}{b^4} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b}+\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{2 b^5}-\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{i (2 a+2 b x)}\right )}{2 b^5} \\ & = \frac {2 i d (c+d x)^3}{b^2}+\frac {(c+d x)^4}{2 b}-\frac {2 (c+d x)^4 \text {arctanh}\left (e^{2 i (a+b x)}\right )}{b}-\frac {6 d^2 (c+d x)^2 \log \left (1+e^{2 i (a+b x)}\right )}{b^3}+\frac {6 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{2 i (a+b x)}\right )}{b^2}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{2 i (a+b x)}\right )}{b^2}-\frac {3 d^4 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^5}-\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{2 i (a+b x)}\right )}{b^3}+\frac {3 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{2 i (a+b x)}\right )}{b^3}-\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{2 i (a+b x)}\right )}{b^4}+\frac {3 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{2 i (a+b x)}\right )}{b^4}+\frac {3 d^4 \operatorname {PolyLog}\left (5,-e^{2 i (a+b x)}\right )}{2 b^5}-\frac {3 d^4 \operatorname {PolyLog}\left (5,e^{2 i (a+b x)}\right )}{2 b^5}-\frac {2 d (c+d x)^3 \tan (a+b x)}{b^2}+\frac {(c+d x)^4 \tan ^2(a+b x)}{2 b} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(2227\) vs. \(2(399)=798\).

Time = 7.65 (sec) , antiderivative size = 2227, normalized size of antiderivative = 5.58 \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Result too large to show} \]

[In]

Integrate[(c + d*x)^4*Csc[a + b*x]*Sec[a + b*x]^3,x]

[Out]

-((c^2*d^2*E^(I*a)*Csc[a]*((2*b^3*x^3)/E^((2*I)*a) + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 - E^((-I)*(a + b*x
))] + (3*I)*b^2*(1 - E^((-2*I)*a))*x^2*Log[1 + E^((-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, -E^((
-I)*(a + b*x))] - 6*b*(1 - E^((-2*I)*a))*x*PolyLog[2, E^((-I)*(a + b*x))] + (6*I)*(1 - E^((-2*I)*a))*PolyLog[3
, -E^((-I)*(a + b*x))] + (6*I)*(1 - E^((-2*I)*a))*PolyLog[3, E^((-I)*(a + b*x))]))/b^3) - (c*d^3*E^(I*a)*Csc[a
]*((b^4*x^4)/E^((2*I)*a) + (2*I)*b^3*(1 - E^((-2*I)*a))*x^3*Log[1 - E^((-I)*(a + b*x))] + (2*I)*b^3*(1 - E^((-
2*I)*a))*x^3*Log[1 + E^((-I)*(a + b*x))] - 6*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[2, -E^((-I)*(a + b*x))] - 6*b^
2*(1 - E^((-2*I)*a))*x^2*PolyLog[2, E^((-I)*(a + b*x))] + (12*I)*b*(1 - E^((-2*I)*a))*x*PolyLog[3, -E^((-I)*(a
+ b*x))] + (12*I)*b*(1 - E^((-2*I)*a))*x*PolyLog[3, E^((-I)*(a + b*x))] + 12*(1 - E^((-2*I)*a))*PolyLog[4, -E
^((-I)*(a + b*x))] + 12*(1 - E^((-2*I)*a))*PolyLog[4, E^((-I)*(a + b*x))]))/b^4 - (d^4*E^(I*a)*Csc[a]*((2*b^5*
x^5)/E^((2*I)*a) + (5*I)*b^4*(1 - E^((-2*I)*a))*x^4*Log[1 - E^((-I)*(a + b*x))] + (5*I)*b^4*(1 - E^((-2*I)*a))
*x^4*Log[1 + E^((-I)*(a + b*x))] - 20*b^3*(1 - E^((-2*I)*a))*x^3*PolyLog[2, -E^((-I)*(a + b*x))] - 20*b^3*(1 -
E^((-2*I)*a))*x^3*PolyLog[2, E^((-I)*(a + b*x))] + (60*I)*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[3, -E^((-I)*(a +
b*x))] + (60*I)*b^2*(1 - E^((-2*I)*a))*x^2*PolyLog[3, E^((-I)*(a + b*x))] + 120*b*(1 - E^((-2*I)*a))*x*PolyLo
g[4, -E^((-I)*(a + b*x))] + 120*b*(1 - E^((-2*I)*a))*x*PolyLog[4, E^((-I)*(a + b*x))] - (120*I)*(1 - E^((-2*I)
*a))*PolyLog[5, -E^((-I)*(a + b*x))] - (120*I)*(1 - E^((-2*I)*a))*PolyLog[5, E^((-I)*(a + b*x))]))/(10*b^5) +
(x*(5*c^4 + 10*c^3*d*x + 10*c^2*d^2*x^2 + 5*c*d^3*x^3 + d^4*x^4)*Csc[a]*Sec[a])/5 - ((I/2)*c^2*d^2*(2*b^2*x^2*
(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2, -E^((-2*I
)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^3*E^(I*a)) - ((I/2)*d^4*
(2*b^2*x^2*(2*b*x - (3*I)*(1 + E^((2*I)*a))*Log[1 + E^((-2*I)*(a + b*x))]) + 6*b*(1 + E^((2*I)*a))*x*PolyLog[2
, -E^((-2*I)*(a + b*x))] - (3*I)*(1 + E^((2*I)*a))*PolyLog[3, -E^((-2*I)*(a + b*x))])*Sec[a])/(b^5*E^(I*a)) -
((I/2)*c*d^3*E^(I*a)*((2*b^4*x^4)/E^((2*I)*a) - (4*I)*b^3*(1 + E^((-2*I)*a))*x^3*Log[1 + E^((-2*I)*(a + b*x))]
+ 6*b^2*(1 + E^((-2*I)*a))*x^2*PolyLog[2, -E^((-2*I)*(a + b*x))] - (6*I)*b*(1 + E^((-2*I)*a))*x*PolyLog[3, -E
^((-2*I)*(a + b*x))] - 3*(1 + E^((-2*I)*a))*PolyLog[4, -E^((-2*I)*(a + b*x))])*Sec[a])/b^4 - ((I/20)*d^4*E^(I*
a)*((4*b^5*x^5)/E^((2*I)*a) - (10*I)*b^4*(1 + E^((-2*I)*a))*x^4*Log[1 + E^((-2*I)*(a + b*x))] + 20*b^3*(1 + E^
((-2*I)*a))*x^3*PolyLog[2, -E^((-2*I)*(a + b*x))] - (30*I)*b^2*(1 + E^((-2*I)*a))*x^2*PolyLog[3, -E^((-2*I)*(a
+ b*x))] - 30*b*(1 + E^((-2*I)*a))*x*PolyLog[4, -E^((-2*I)*(a + b*x))] + (15*I)*(1 + E^((-2*I)*a))*PolyLog[5,
-E^((-2*I)*(a + b*x))])*Sec[a])/b^5 + ((c + d*x)^4*Sec[a + b*x]^2)/(2*b) - (c^4*Sec[a]*(Cos[a]*Log[Cos[a]*Cos
[b*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (6*c^2*d^2*Sec[a]*(Cos[a]*Log[Cos[a]*Cos[b
*x] - Sin[a]*Sin[b*x]] + b*x*Sin[a]))/(b^3*(Cos[a]^2 + Sin[a]^2)) + (c^4*Csc[a]*(-(b*x*Cos[a]) + Log[Cos[b*x]*
Sin[a] + Cos[a]*Sin[b*x]]*Sin[a]))/(b*(Cos[a]^2 + Sin[a]^2)) - (2*c^3*d*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]])
- (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^(
(2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTan[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLo
g[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2])*Sec[a])/(b^2*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]
) - (6*c*d^3*Csc[a]*((b^2*x^2)/E^(I*ArcTan[Cot[a]]) - (Cot[a]*(I*b*x*(-Pi - 2*ArcTan[Cot[a]]) - Pi*Log[1 + E^(
(-2*I)*b*x)] - 2*(b*x - ArcTan[Cot[a]])*Log[1 - E^((2*I)*(b*x - ArcTan[Cot[a]]))] + Pi*Log[Cos[b*x]] - 2*ArcTa
n[Cot[a]]*Log[Sin[b*x - ArcTan[Cot[a]]]] + I*PolyLog[2, E^((2*I)*(b*x - ArcTan[Cot[a]]))]))/Sqrt[1 + Cot[a]^2]
)*Sec[a])/(b^4*Sqrt[Csc[a]^2*(Cos[a]^2 + Sin[a]^2)]) - (2*Sec[a]*Sec[a + b*x]*(c^3*d*Sin[b*x] + 3*c^2*d^2*x*Si
n[b*x] + 3*c*d^3*x^2*Sin[b*x] + d^4*x^3*Sin[b*x]))/b^2 - (2*c^3*d*Csc[a]*Sec[a]*(b^2*E^(I*ArcTan[Tan[a]])*x^2
+ ((I*b*x*(-Pi + 2*ArcTan[Tan[a]]) - Pi*Log[1 + E^((-2*I)*b*x)] - 2*(b*x + ArcTan[Tan[a]])*Log[1 - E^((2*I)*(b
*x + ArcTan[Tan[a]]))] + Pi*Log[Cos[b*x]] + 2*ArcTan[Tan[a]]*Log[Sin[b*x + ArcTan[Tan[a]]]] + I*PolyLog[2, E^(
(2*I)*(b*x + ArcTan[Tan[a]]))])*Tan[a])/Sqrt[1 + Tan[a]^2]))/(b^2*Sqrt[Sec[a]^2*(Cos[a]^2 + Sin[a]^2)])

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1728 vs. \(2 (361 ) = 722\).

Time = 1.03 (sec) , antiderivative size = 1729, normalized size of antiderivative = 4.33

method	result	size
risch	\(\text {Expression too large to display}\)	\(1729\)

[In]

int((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x,method=_RETURNVERBOSE)

[Out]

2*(b*d^4*x^4*exp(2*I*(b*x+a))+4*b*c*d^3*x^3*exp(2*I*(b*x+a))+6*b*c^2*d^2*x^2*exp(2*I*(b*x+a))+4*b*c^3*d*x*exp(
2*I*(b*x+a))-2*I*d^4*x^3*exp(2*I*(b*x+a))+b*c^4*exp(2*I*(b*x+a))-6*I*c*d^3*x^2*exp(2*I*(b*x+a))-6*I*c^2*d^2*x*
exp(2*I*(b*x+a))-2*I*d^4*x^3-2*I*c^3*d*exp(2*I*(b*x+a))-6*I*c*d^3*x^2-6*I*c^2*d^2*x-2*I*c^3*d)/b^2/(exp(2*I*(b
*x+a))+1)^2-3*d^4*polylog(3,-exp(2*I*(b*x+a)))/b^5+24*I/b^3*d^3*c*x*a-12*I/b^2*d^3*c*polylog(2,-exp(I*(b*x+a))
)*x^2+6*I/b^2*d^3*c*polylog(2,-exp(2*I*(b*x+a)))*x^2-12*I/b^2*d^3*c*polylog(2,exp(I*(b*x+a)))*x^2-12*I/b^2*c^2
*d^2*polylog(2,-exp(I*(b*x+a)))*x+6*I/b^2*c^2*d^2*polylog(2,-exp(2*I*(b*x+a)))*x-12*I/b^2*c^2*d^2*polylog(2,ex
p(I*(b*x+a)))*x-12/b^3*d^3*c*ln(exp(2*I*(b*x+a))+1)*x+4/b*c^3*d*ln(1-exp(I*(b*x+a)))*x+4/b^4*d^3*c*ln(1-exp(I*
(b*x+a)))*a^3+12*I/b^4*d^3*c*a^2+6*I/b^4*d^3*c*polylog(2,-exp(2*I*(b*x+a)))+2*I/b^2*d^4*polylog(2,-exp(2*I*(b*
x+a)))*x^3+3/2*d^4*polylog(5,-exp(2*I*(b*x+a)))/b^5-4*I/b^2*c^3*d*polylog(2,exp(I*(b*x+a)))+12*I/b^2*d^3*c*x^2
+24*I/b^4*d^3*c*polylog(4,-exp(I*(b*x+a)))-3*I/b^4*d^3*c*polylog(4,-exp(2*I*(b*x+a)))+24*I/b^4*d^3*c*polylog(4
,exp(I*(b*x+a)))+6*I/b^4*d^4*polylog(2,-exp(2*I*(b*x+a)))*x-4*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)))*x^3-4*I/b^2
*d^4*polylog(2,exp(I*(b*x+a)))*x^3-12*I/b^4*a^2*d^4*x+24*I/b^4*d^4*polylog(4,-exp(I*(b*x+a)))*x-3*I/b^4*d^4*po
lylog(4,-exp(2*I*(b*x+a)))*x+24*I/b^4*d^4*polylog(4,exp(I*(b*x+a)))*x-24*d^4*polylog(5,-exp(I*(b*x+a)))/b^5-24
*d^4*polylog(5,exp(I*(b*x+a)))/b^5-1/b^5*d^4*ln(1-exp(I*(b*x+a)))*a^4+1/b^5*a^4*d^4*ln(exp(I*(b*x+a))-1)+1/b*d
^4*ln(exp(I*(b*x+a))+1)*x^4-1/b*d^4*ln(exp(2*I*(b*x+a))+1)*x^4+1/b*d^4*ln(1-exp(I*(b*x+a)))*x^4+12/b^3*d^4*pol
ylog(3,exp(I*(b*x+a)))*x^2-6/b^3*d^4*ln(exp(2*I*(b*x+a))+1)*x^2+12/b^3*d^4*polylog(3,-exp(I*(b*x+a)))*x^2-3/b^
3*d^4*polylog(3,-exp(2*I*(b*x+a)))*x^2+12/b^3*c^2*d^2*polylog(3,-exp(I*(b*x+a)))-3/b^3*c^2*d^2*polylog(3,-exp(
2*I*(b*x+a)))+12/b^3*c^2*d^2*polylog(3,exp(I*(b*x+a)))+12/b^3*c^2*d^2*ln(exp(I*(b*x+a)))-6/b^3*c^2*d^2*ln(exp(
2*I*(b*x+a))+1)+12/b^5*a^2*d^4*ln(exp(I*(b*x+a)))+4*I/b^2*d^4*x^3-8*I/b^5*a^3*d^4+6/b^3*a^2*c^2*d^2*ln(exp(I*(
b*x+a))-1)-6/b*c^2*d^2*ln(exp(2*I*(b*x+a))+1)*x^2+6/b*c^2*d^2*ln(1-exp(I*(b*x+a)))*x^2-6/b^3*c^2*d^2*ln(1-exp(
I*(b*x+a)))*a^2+4/b*d^3*c*ln(exp(I*(b*x+a))+1)*x^3-4/b*d^3*c*ln(exp(2*I*(b*x+a))+1)*x^3+4/b*d^3*c*ln(1-exp(I*(
b*x+a)))*x^3+1/b*c^4*ln(exp(I*(b*x+a))-1)+1/b*c^4*ln(exp(I*(b*x+a))+1)-1/b*c^4*ln(exp(2*I*(b*x+a))+1)-4*I/b^2*
c^3*d*polylog(2,-exp(I*(b*x+a)))+2*I/b^2*c^3*d*polylog(2,-exp(2*I*(b*x+a)))+24/b^3*d^3*c*polylog(3,-exp(I*(b*x
+a)))*x-6/b^3*d^3*c*polylog(3,-exp(2*I*(b*x+a)))*x+24/b^3*d^3*c*polylog(3,exp(I*(b*x+a)))*x+4/b*c^3*d*ln(exp(I
*(b*x+a))+1)*x-4/b^4*a^3*c*d^3*ln(exp(I*(b*x+a))-1)-4/b^2*a*c^3*d*ln(exp(I*(b*x+a))-1)-24/b^4*a*c*d^3*ln(exp(I
*(b*x+a)))-4/b*c^3*d*ln(exp(2*I*(b*x+a))+1)*x+6/b*c^2*d^2*ln(exp(I*(b*x+a))+1)*x^2+4/b^2*c^3*d*ln(1-exp(I*(b*x
+a)))*a

Fricas [F(-2)]

Exception generated. \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Exception raised: TypeError} \]

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   Too many variables

Sympy [F(-1)]

Timed out. \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Timed out} \]

[In]

integrate((d*x+c)**4*csc(b*x+a)*sec(b*x+a)**3,x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 8853 vs. \(2 (352) = 704\).

Time = 4.27 (sec) , antiderivative size = 8853, normalized size of antiderivative = 22.19 \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x, algorithm="maxima")

[Out]

-1/2*(c^4*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2)) - 4*a*c^3*d*(1/(sin(b*x + a
)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b + 6*a^2*c^2*d^2*(1/(sin(b*x + a)^2 - 1) + log(sin(
b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^2 - 4*a^3*c*d^3*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - l
og(sin(b*x + a)^2))/b^3 + a^4*d^4*(1/(sin(b*x + a)^2 - 1) + log(sin(b*x + a)^2 - 1) - log(sin(b*x + a)^2))/b^4
+ 2*(24*b^3*c^3*d - 72*a*b^2*c^2*d^2 + 72*a^2*b*c*d^3 - 24*a^3*d^4 + 4*(3*(b*x + a)^4*d^4 + 9*b^2*c^2*d^2 - 1
8*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x
+ a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a) + (3*(b*x + a)^4*d
^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3
+ (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x +
a))*cos(4*b*x + 4*a) + 2*(3*(b*x + a)^4*d^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*
(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(
a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-3*I*(b*x + a)^4*d^4 - 9*I*b^2*c^2*d^2 + 18
*I*a*b*c*d^3 - 9*I*a^2*d^4 + 8*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 9*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 + (-I*a^
2 - I)*d^4)*(b*x + a)^2 + 6*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 + 3*(-I*a^2 - I)*b*c*d^3 + (I*a^3 + 3*I*a)*d^4)*
(b*x + a))*sin(4*b*x + 4*a) - 2*(-3*I*(b*x + a)^4*d^4 - 9*I*b^2*c^2*d^2 + 18*I*a*b*c*d^3 - 9*I*a^2*d^4 + 8*(-I
*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 9*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 + (-I*a^2 - I)*d^4)*(b*x + a)^2 + 6*(-I*b^
3*c^3*d + 3*I*a*b^2*c^2*d^2 + 3*(-I*a^2 - I)*b*c*d^3 + (I*a^3 + 3*I*a)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arcta
n2(sin(2*b*x + 2*a), cos(2*b*x + 2*a) + 1) - 6*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2
*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a
) + ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 +
4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 +
4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2
*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (I*(b*x + a)^4*d^4 + 4*(I*b*c*d^3 - I*a*d^4)
*(b*x + a)^3 + 6*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^2 + 4*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2
+ 3*I*a^2*b*c*d^3 - I*a^3*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*(I*(b*x + a)^4*d^4 + 4*(I*b*c*d^3 - I*a*d^4)*(b
*x + a)^3 + 6*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a)^2 + 4*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3
*I*a^2*b*c*d^3 - I*a^3*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), cos(b*x + a) + 1) + 6*((b*x + a
)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d
- 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a) + ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 +
6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^
4)*(b*x + a))*cos(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b
*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*cos(2*b*x
+ 2*a) - (-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a
^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*sin(4*b*x
+ 4*a) - 2*(-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*
a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*sin(2*b*x
+ 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 24*((b*x + a)^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*
(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 12*(-I*(b*x + a)^4*d^4 - 2*b^3*c^3*d + 6*a
*b^2*c^2*d^2 - 6*a^2*b*c*d^3 + 2*a^3*d^4 + 2*(-2*I*b*c*d^3 + (2*I*a + 1)*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2
+ (2*I*a + 1)*b*c*d^3 + (-I*a^2 - a)*d^4)*(b*x + a)^2 + 2*(-2*I*b^3*c^3*d + 3*(2*I*a + 1)*b^2*c^2*d^2 + 6*(-I*
a^2 - a)*b*c*d^3 + (2*I*a^3 + 3*a^2)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - 12*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 2*(b
*x + a)^3*d^4 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a
*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 2*(b*x + a)^3*d^4 + 3*(a^2 + 1)*b*c*d^3 -
(a^3 + 3*a)*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a))*
cos(4*b*x + 4*a) + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 2*(b*x + a)^3*d^4 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4
+ 4*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a))*cos(2*b*x + 2*a)
+ (I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 2*I*(b*x + a)^3*d^4 + 3*(I*a^2 + I)*b*c*d^3 + (-I*a^3 - 3*I*a)*d^4 + 4*(I
*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + (I*a^2 + I)*d^4)*(b*x + a))*sin(4*b*x + 4
*a) + 2*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 2*I*(b*x + a)^3*d^4 + 3*(I*a^2 + I)*b*c*d^3 + (-I*a^3 - 3*I*a)*d^4
+ 4*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + (I*a^2 + I)*d^4)*(b*x + a))*sin(2*b
*x + 2*a))*dilog(-e^(2*I*b*x + 2*I*a)) + 24*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a
^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + (b^3*c^3*d - 3*
a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 -
2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x +
a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(
2*b*x + 2*a) - (-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I*a^3*d^4 + 3*(-I*b*c
*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(
-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I*a^3*d^4 + 3*(-I*b*c*d^3 + I*a*d^4)*
(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*
a)) + 24*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x +
a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*
x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*
cos(4*b*x + 4*a) + 2*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 + (b*x + a)^3*d^4 - a^3*d^4 + 3*(b*c*d^3 - a
*d^4)*(b*x + a)^2 + 3*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*b^3*c^3*d + 3*I*
a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I*a^3*d^4 + 3*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I
*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 -
3*I*a^2*b*c*d^3 - I*(b*x + a)^3*d^4 + I*a^3*d^4 + 3*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^2 + 3*(-I*b^2*c^2*d^2 + 2
*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(e^(I*b*x + I*a)) - 2*(3*I*(b*x + a)^4*d^4 + 9*I*b
^2*c^2*d^2 - 18*I*a*b*c*d^3 + 9*I*a^2*d^4 + 8*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^3 + 9*(I*b^2*c^2*d^2 - 2*I*a*b*c
*d^3 + (I*a^2 + I)*d^4)*(b*x + a)^2 + 6*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*(I*a^2 + I)*b*c*d^3 + (-I*a^3 - 3
*I*a)*d^4)*(b*x + a) + (3*I*(b*x + a)^4*d^4 + 9*I*b^2*c^2*d^2 - 18*I*a*b*c*d^3 + 9*I*a^2*d^4 + 8*(I*b*c*d^3 -
I*a*d^4)*(b*x + a)^3 + 9*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + (I*a^2 + I)*d^4)*(b*x + a)^2 + 6*(I*b^3*c^3*d - 3*I*
a*b^2*c^2*d^2 + 3*(I*a^2 + I)*b*c*d^3 + (-I*a^3 - 3*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(3*I*(b*x + a)^4
*d^4 + 9*I*b^2*c^2*d^2 - 18*I*a*b*c*d^3 + 9*I*a^2*d^4 + 8*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^3 + 9*(I*b^2*c^2*d^2
- 2*I*a*b*c*d^3 + (I*a^2 + I)*d^4)*(b*x + a)^2 + 6*(I*b^3*c^3*d - 3*I*a*b^2*c^2*d^2 + 3*(I*a^2 + I)*b*c*d^3 +
(-I*a^3 - 3*I*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (3*(b*x + a)^4*d^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2
*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^
3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(3*(b*x + a)^4*
d^4 + 9*b^2*c^2*d^2 - 18*a*b*c*d^3 + 9*a^2*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 9*(b^2*c^2*d^2 - 2*a*b*c*d^
3 + (a^2 + 1)*d^4)*(b*x + a)^2 + 6*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 + 1)*b*c*d^3 - (a^3 + 3*a)*d^4)*(b*x
+ a))*sin(2*b*x + 2*a))*log(cos(2*b*x + 2*a)^2 + sin(2*b*x + 2*a)^2 + 2*cos(2*b*x + 2*a) + 1) - 3*(-I*(b*x + a
)^4*d^4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 +
4*(-I*b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a) + (-I*(b*x + a)^4*d^4 + 4*(-I*b*c
*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3
*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*(b*x + a)^4*d^4 + 4*(-I*b*
c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d +
3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)^4*d^4 + 4*(b*c*d^3 -
a*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3
*a^2*b*c*d^3 - a^3*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6
*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)
*(b*x + a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 + 2*cos(b*x + a) + 1) - 3*(-I*(b*x + a)^4*d^
4 + 4*(-I*b*c*d^3 + I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*
b^3*c^3*d + 3*I*a*b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a) + (-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3 +
I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*b
^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*(b*x + a)^4*d^4 + 4*(-I*b*c*d^3
+ I*a*d^4)*(b*x + a)^3 + 6*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*a^2*d^4)*(b*x + a)^2 + 4*(-I*b^3*c^3*d + 3*I*a*
b^2*c^2*d^2 - 3*I*a^2*b*c*d^3 + I*a^3*d^4)*(b*x + a))*cos(2*b*x + 2*a) + ((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4
)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b
*c*d^3 - a^3*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 + 4*(b*c*d^3 - a*d^4)*(b*x + a)^3 + 6*(b^2*
c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*a^2*b*c*d^3 - a^3*d^4)*(b*x
+ a))*sin(2*b*x + 2*a))*log(cos(b*x + a)^2 + sin(b*x + a)^2 - 2*cos(b*x + a) + 1) - 18*(-I*d^4*cos(4*b*x + 4*a
) - 2*I*d^4*cos(2*b*x + 2*a) + d^4*sin(4*b*x + 4*a) + 2*d^4*sin(2*b*x + 2*a) - I*d^4)*polylog(5, -e^(2*I*b*x +
2*I*a)) - 144*(I*d^4*cos(4*b*x + 4*a) + 2*I*d^4*cos(2*b*x + 2*a) - d^4*sin(4*b*x + 4*a) - 2*d^4*sin(2*b*x + 2
*a) + I*d^4)*polylog(5, -e^(I*b*x + I*a)) - 144*(I*d^4*cos(4*b*x + 4*a) + 2*I*d^4*cos(2*b*x + 2*a) - d^4*sin(4
*b*x + 4*a) - 2*d^4*sin(2*b*x + 2*a) + I*d^4)*polylog(5, e^(I*b*x + I*a)) + 12*(2*b*c*d^3 + 3*(b*x + a)*d^4 -
2*a*d^4 + (2*b*c*d^3 + 3*(b*x + a)*d^4 - 2*a*d^4)*cos(4*b*x + 4*a) + 2*(2*b*c*d^3 + 3*(b*x + a)*d^4 - 2*a*d^4)
*cos(2*b*x + 2*a) - (-2*I*b*c*d^3 - 3*I*(b*x + a)*d^4 + 2*I*a*d^4)*sin(4*b*x + 4*a) - 2*(-2*I*b*c*d^3 - 3*I*(b
*x + a)*d^4 + 2*I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, -e^(2*I*b*x + 2*I*a)) - 144*(b*c*d^3 + (b*x + a)*d^4 - a
*d^4 + (b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(4*b*x + 4*a) + 2*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*
a) + (I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*sin(4*b*x + 4*a) + 2*(I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*sin(
2*b*x + 2*a))*polylog(4, -e^(I*b*x + I*a)) - 144*(b*c*d^3 + (b*x + a)*d^4 - a*d^4 + (b*c*d^3 + (b*x + a)*d^4 -
a*d^4)*cos(4*b*x + 4*a) + 2*(b*c*d^3 + (b*x + a)*d^4 - a*d^4)*cos(2*b*x + 2*a) + (I*b*c*d^3 + I*(b*x + a)*d^4
- I*a*d^4)*sin(4*b*x + 4*a) + 2*(I*b*c*d^3 + I*(b*x + a)*d^4 - I*a*d^4)*sin(2*b*x + 2*a))*polylog(4, e^(I*b*x
+ I*a)) - 6*(3*I*b^2*c^2*d^2 - 6*I*a*b*c*d^3 + 6*I*(b*x + a)^2*d^4 + 3*(I*a^2 + I)*d^4 + 8*(I*b*c*d^3 - I*a*d
^4)*(b*x + a) + (3*I*b^2*c^2*d^2 - 6*I*a*b*c*d^3 + 6*I*(b*x + a)^2*d^4 + 3*(I*a^2 + I)*d^4 + 8*(I*b*c*d^3 - I*
a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(3*I*b^2*c^2*d^2 - 6*I*a*b*c*d^3 + 6*I*(b*x + a)^2*d^4 + 3*(I*a^2 + I)*
d^4 + 8*(I*b*c*d^3 - I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (3*b^2*c^2*d^2 - 6*a*b*c*d^3 + 6*(b*x + a)^2*d^4 +
3*(a^2 + 1)*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(3*b^2*c^2*d^2 - 6*a*b*c*d^3 + 6*(b*x +
a)^2*d^4 + 3*(a^2 + 1)*d^4 + 8*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, -e^(2*I*b*x + 2*I*a)
) - 72*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a) +
(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*cos(4*b*
x + 4*a) + 2*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x +
a))*cos(2*b*x + 2*a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a)
)*sin(4*b*x + 4*a) + 2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))
*sin(2*b*x + 2*a))*polylog(3, -e^(I*b*x + I*a)) - 72*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a
^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a) + (-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 - I*a^2*d^4
+ 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*cos(4*b*x + 4*a) + 2*(-I*b^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4
- I*a^2*d^4 + 2*(-I*b*c*d^3 + I*a*d^4)*(b*x + a))*cos(2*b*x + 2*a) + (b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2
*d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + (b*x + a)^2*
d^4 + a^2*d^4 + 2*(b*c*d^3 - a*d^4)*(b*x + a))*sin(2*b*x + 2*a))*polylog(3, e^(I*b*x + I*a)) - 24*(I*(b*x + a)
^3*d^4 + 3*(I*b*c*d^3 - I*a*d^4)*(b*x + a)^2 + 3*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*(b*x + a))*sin(4*
b*x + 4*a) - 12*((b*x + a)^4*d^4 - 2*I*b^3*c^3*d + 6*I*a*b^2*c^2*d^2 - 6*I*a^2*b*c*d^3 + 2*I*a^3*d^4 + 2*(2*b*
c*d^3 - (2*a - I)*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - (2*a - I)*b*c*d^3 + (a^2 - I*a)*d^4)*(b*x + a)^2 + 2*(2*
b^3*c^3*d - 3*(2*a - I)*b^2*c^2*d^2 + 6*(a^2 - I*a)*b*c*d^3 - (2*a^3 - 3*I*a^2)*d^4)*(b*x + a))*sin(2*b*x + 2*
a))/(-6*I*b^4*cos(4*b*x + 4*a) - 12*I*b^4*cos(2*b*x + 2*a) + 6*b^4*sin(4*b*x + 4*a) + 12*b^4*sin(2*b*x + 2*a)
- 6*I*b^4))/b

Giac [F]

\[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \csc \left (b x + a\right ) \sec \left (b x + a\right )^{3} \,d x } \]

[In]

integrate((d*x+c)^4*csc(b*x+a)*sec(b*x+a)^3,x, algorithm="giac")

[Out]

integrate((d*x + c)^4*csc(b*x + a)*sec(b*x + a)^3, x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \csc (a+b x) \sec ^3(a+b x) \, dx=\text {Hanged} \]

[In]

int((c + d*x)^4/(cos(a + b*x)^3*sin(a + b*x)),x)

[Out]

\text{Hanged}