\(\int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx\) [112]

Optimal result

Integrand size = 22, antiderivative size = 416 \[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=-\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^4 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^5}+\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {12 d^4 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^5}-\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {12 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}+\frac {12 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5} \]

[Out]

-12*d^2*(d*x+c)^2*arctanh(exp(I*(b*x+a)))/b^3+(d*x+c)^4*arctanh(exp(I*(b*x+a)))/b-2*d*(d*x+c)^3*csc(b*x+a)/b^2
-1/2*(d*x+c)^4*cot(b*x+a)*csc(b*x+a)/b+12*I*d^3*(d*x+c)*polylog(2,-exp(I*(b*x+a)))/b^4-12*I*d^3*(d*x+c)*polylo
g(2,exp(I*(b*x+a)))/b^4+12*I*d^3*(d*x+c)*polylog(4,-exp(I*(b*x+a)))/b^4-12*I*d^3*(d*x+c)*polylog(4,exp(I*(b*x+
a)))/b^4-12*d^4*polylog(3,-exp(I*(b*x+a)))/b^5+6*d^2*(d*x+c)^2*polylog(3,-exp(I*(b*x+a)))/b^3+12*d^4*polylog(3
,exp(I*(b*x+a)))/b^5-6*d^2*(d*x+c)^2*polylog(3,exp(I*(b*x+a)))/b^3-2*I*d*(d*x+c)^3*polylog(2,-exp(I*(b*x+a)))/
b^2+2*I*d*(d*x+c)^3*polylog(2,exp(I*(b*x+a)))/b^2-12*d^4*polylog(5,-exp(I*(b*x+a)))/b^5+12*d^4*polylog(5,exp(I
*(b*x+a)))/b^5

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.00, number of steps used = 31, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4500, 4268, 2611, 6744, 2320, 6724, 4271} \[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=-\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {12 d^4 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^5}+\frac {12 d^4 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^5}-\frac {12 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}+\frac {12 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b} \]

[In]

Int[(c + d*x)^4*Cot[a + b*x]^2*Csc[a + b*x],x]

[Out]

(-12*d^2*(c + d*x)^2*ArcTanh[E^(I*(a + b*x))])/b^3 + ((c + d*x)^4*ArcTanh[E^(I*(a + b*x))])/b - (2*d*(c + d*x)
^3*Csc[a + b*x])/b^2 - ((c + d*x)^4*Cot[a + b*x]*Csc[a + b*x])/(2*b) + ((12*I)*d^3*(c + d*x)*PolyLog[2, -E^(I*
(a + b*x))])/b^4 - ((2*I)*d*(c + d*x)^3*PolyLog[2, -E^(I*(a + b*x))])/b^2 - ((12*I)*d^3*(c + d*x)*PolyLog[2, E
^(I*(a + b*x))])/b^4 + ((2*I)*d*(c + d*x)^3*PolyLog[2, E^(I*(a + b*x))])/b^2 - (12*d^4*PolyLog[3, -E^(I*(a + b
*x))])/b^5 + (6*d^2*(c + d*x)^2*PolyLog[3, -E^(I*(a + b*x))])/b^3 + (12*d^4*PolyLog[3, E^(I*(a + b*x))])/b^5 -
(6*d^2*(c + d*x)^2*PolyLog[3, E^(I*(a + b*x))])/b^3 + ((12*I)*d^3*(c + d*x)*PolyLog[4, -E^(I*(a + b*x))])/b^4
- ((12*I)*d^3*(c + d*x)*PolyLog[4, E^(I*(a + b*x))])/b^4 - (12*d^4*PolyLog[5, -E^(I*(a + b*x))])/b^5 + (12*d^
4*PolyLog[5, E^(I*(a + b*x))])/b^5

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 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 4271

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

Rule 4500

Int[Cot[(a_.) + (b_.)*(x_)]^(p_)*Csc[(a_.) + (b_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Int[(c + d*
x)^m*Csc[a + b*x]*Cot[a + b*x]^(p - 2), x] + Int[(c + d*x)^m*Csc[a + b*x]^3*Cot[a + b*x]^(p - 2), x] /; FreeQ[
{a, b, c, d, m}, x] && IGtQ[p/2, 0]

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]

Rubi steps \begin{align*} \text {integral}& = -\int (c+d x)^4 \csc (a+b x) \, dx+\int (c+d x)^4 \csc ^3(a+b x) \, dx \\ & = \frac {2 (c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {1}{2} \int (c+d x)^4 \csc (a+b x) \, dx+\frac {(4 d) \int (c+d x)^3 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}-\frac {(4 d) \int (c+d x)^3 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}+\frac {\left (6 d^2\right ) \int (c+d x)^2 \csc (a+b x) \, dx}{b^2} \\ & = -\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}-\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}+\frac {4 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {(2 d) \int (c+d x)^3 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b}+\frac {(2 d) \int (c+d x)^3 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b}+\frac {\left (12 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (12 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (12 d^3\right ) \int (c+d x) \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (12 d^3\right ) \int (c+d x) \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^3} \\ & = -\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {12 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}-\frac {\left (6 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac {\left (6 i d^2\right ) \int (c+d x)^2 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^2}-\frac {\left (24 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac {\left (24 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (12 i d^4\right ) \int \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right ) \, dx}{b^4}+\frac {\left (12 i d^4\right ) \int \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = -\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}+\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}-\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {24 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}+\frac {\left (12 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (12 d^3\right ) \int (c+d x) \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right ) \, dx}{b^3}-\frac {\left (12 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}+\frac {\left (12 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}-\frac {\left (24 i d^4\right ) \int \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right ) \, dx}{b^4}+\frac {\left (24 i d^4\right ) \int \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = -\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^4 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^5}+\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {12 d^4 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^5}-\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {\left (24 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}+\frac {\left (24 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}+\frac {\left (12 i d^4\right ) \int \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right ) \, dx}{b^4}-\frac {\left (12 i d^4\right ) \int \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right ) \, dx}{b^4} \\ & = -\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^4 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^5}+\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {12 d^4 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^5}-\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {24 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}+\frac {24 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5}+\frac {\left (12 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}-\frac {\left (12 d^4\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(4,x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5} \\ & = -\frac {12 d^2 (c+d x)^2 \text {arctanh}\left (e^{i (a+b x)}\right )}{b^3}+\frac {(c+d x)^4 \text {arctanh}\left (e^{i (a+b x)}\right )}{b}-\frac {2 d (c+d x)^3 \csc (a+b x)}{b^2}-\frac {(c+d x)^4 \cot (a+b x) \csc (a+b x)}{2 b}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^4}-\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^2}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^4}+\frac {2 i d (c+d x)^3 \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )}{b^2}-\frac {12 d^4 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^5}+\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^3}+\frac {12 d^4 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^5}-\frac {6 d^2 (c+d x)^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )}{b^3}+\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^4}-\frac {12 i d^3 (c+d x) \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )}{b^4}-\frac {12 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )}{b^5}+\frac {12 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{b^5} \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(966\) vs. \(2(416)=832\).

Time = 8.09 (sec) , antiderivative size = 966, normalized size of antiderivative = 2.32 \[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=\frac {-b^4 c^4 \log \left (1-e^{i (a+b x)}\right )+12 b^2 c^2 d^2 \log \left (1-e^{i (a+b x)}\right )-4 b^4 c^3 d x \log \left (1-e^{i (a+b x)}\right )+24 b^2 c d^3 x \log \left (1-e^{i (a+b x)}\right )-6 b^4 c^2 d^2 x^2 \log \left (1-e^{i (a+b x)}\right )+12 b^2 d^4 x^2 \log \left (1-e^{i (a+b x)}\right )-4 b^4 c d^3 x^3 \log \left (1-e^{i (a+b x)}\right )-b^4 d^4 x^4 \log \left (1-e^{i (a+b x)}\right )+b^4 c^4 \log \left (1+e^{i (a+b x)}\right )-12 b^2 c^2 d^2 \log \left (1+e^{i (a+b x)}\right )+4 b^4 c^3 d x \log \left (1+e^{i (a+b x)}\right )-24 b^2 c d^3 x \log \left (1+e^{i (a+b x)}\right )+6 b^4 c^2 d^2 x^2 \log \left (1+e^{i (a+b x)}\right )-12 b^2 d^4 x^2 \log \left (1+e^{i (a+b x)}\right )+4 b^4 c d^3 x^3 \log \left (1+e^{i (a+b x)}\right )+b^4 d^4 x^4 \log \left (1+e^{i (a+b x)}\right )-4 i b d (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,-e^{i (a+b x)}\right )+4 i b d (c+d x) \left (-6 d^2+b^2 (c+d x)^2\right ) \operatorname {PolyLog}\left (2,e^{i (a+b x)}\right )+12 b^2 c^2 d^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )-24 d^4 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )+24 b^2 c d^3 x \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )+12 b^2 d^4 x^2 \operatorname {PolyLog}\left (3,-e^{i (a+b x)}\right )-12 b^2 c^2 d^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )+24 d^4 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )-24 b^2 c d^3 x \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )-12 b^2 d^4 x^2 \operatorname {PolyLog}\left (3,e^{i (a+b x)}\right )+24 i b c d^3 \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )+24 i b d^4 x \operatorname {PolyLog}\left (4,-e^{i (a+b x)}\right )-24 i b c d^3 \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )-24 i b d^4 x \operatorname {PolyLog}\left (4,e^{i (a+b x)}\right )-24 d^4 \operatorname {PolyLog}\left (5,-e^{i (a+b x)}\right )+24 d^4 \operatorname {PolyLog}\left (5,e^{i (a+b x)}\right )}{2 b^5}-\frac {\csc ^2(a+b x) \left (b c^4 \cos (a+b x)+4 b c^3 d x \cos (a+b x)+6 b c^2 d^2 x^2 \cos (a+b x)+4 b c d^3 x^3 \cos (a+b x)+b d^4 x^4 \cos (a+b x)+4 c^3 d \sin (a+b x)+12 c^2 d^2 x \sin (a+b x)+12 c d^3 x^2 \sin (a+b x)+4 d^4 x^3 \sin (a+b x)\right )}{2 b^2} \]

[In]

Integrate[(c + d*x)^4*Cot[a + b*x]^2*Csc[a + b*x],x]

[Out]

(-(b^4*c^4*Log[1 - E^(I*(a + b*x))]) + 12*b^2*c^2*d^2*Log[1 - E^(I*(a + b*x))] - 4*b^4*c^3*d*x*Log[1 - E^(I*(a
+ b*x))] + 24*b^2*c*d^3*x*Log[1 - E^(I*(a + b*x))] - 6*b^4*c^2*d^2*x^2*Log[1 - E^(I*(a + b*x))] + 12*b^2*d^4*
x^2*Log[1 - E^(I*(a + b*x))] - 4*b^4*c*d^3*x^3*Log[1 - E^(I*(a + b*x))] - b^4*d^4*x^4*Log[1 - E^(I*(a + b*x))]
+ b^4*c^4*Log[1 + E^(I*(a + b*x))] - 12*b^2*c^2*d^2*Log[1 + E^(I*(a + b*x))] + 4*b^4*c^3*d*x*Log[1 + E^(I*(a
+ b*x))] - 24*b^2*c*d^3*x*Log[1 + E^(I*(a + b*x))] + 6*b^4*c^2*d^2*x^2*Log[1 + E^(I*(a + b*x))] - 12*b^2*d^4*x
^2*Log[1 + E^(I*(a + b*x))] + 4*b^4*c*d^3*x^3*Log[1 + E^(I*(a + b*x))] + b^4*d^4*x^4*Log[1 + E^(I*(a + b*x))]
- (4*I)*b*d*(c + d*x)*(-6*d^2 + b^2*(c + d*x)^2)*PolyLog[2, -E^(I*(a + b*x))] + (4*I)*b*d*(c + d*x)*(-6*d^2 +
b^2*(c + d*x)^2)*PolyLog[2, E^(I*(a + b*x))] + 12*b^2*c^2*d^2*PolyLog[3, -E^(I*(a + b*x))] - 24*d^4*PolyLog[3,
-E^(I*(a + b*x))] + 24*b^2*c*d^3*x*PolyLog[3, -E^(I*(a + b*x))] + 12*b^2*d^4*x^2*PolyLog[3, -E^(I*(a + b*x))]
- 12*b^2*c^2*d^2*PolyLog[3, E^(I*(a + b*x))] + 24*d^4*PolyLog[3, E^(I*(a + b*x))] - 24*b^2*c*d^3*x*PolyLog[3,
E^(I*(a + b*x))] - 12*b^2*d^4*x^2*PolyLog[3, E^(I*(a + b*x))] + (24*I)*b*c*d^3*PolyLog[4, -E^(I*(a + b*x))] +
(24*I)*b*d^4*x*PolyLog[4, -E^(I*(a + b*x))] - (24*I)*b*c*d^3*PolyLog[4, E^(I*(a + b*x))] - (24*I)*b*d^4*x*Pol
yLog[4, E^(I*(a + b*x))] - 24*d^4*PolyLog[5, -E^(I*(a + b*x))] + 24*d^4*PolyLog[5, E^(I*(a + b*x))])/(2*b^5) -
(Csc[a + b*x]^2*(b*c^4*Cos[a + b*x] + 4*b*c^3*d*x*Cos[a + b*x] + 6*b*c^2*d^2*x^2*Cos[a + b*x] + 4*b*c*d^3*x^3
*Cos[a + b*x] + b*d^4*x^4*Cos[a + b*x] + 4*c^3*d*Sin[a + b*x] + 12*c^2*d^2*x*Sin[a + b*x] + 12*c*d^3*x^2*Sin[a
+ b*x] + 4*d^4*x^3*Sin[a + b*x]))/(2*b^2)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1672 vs. \(2 (380 ) = 760\).

Time = 1.49 (sec) , antiderivative size = 1673, normalized size of antiderivative = 4.02

method	result	size
risch	\(\text {Expression too large to display}\)	\(1673\)

[In]

int((d*x+c)^4*cot(b*x+a)^2*csc(b*x+a),x,method=_RETURNVERBOSE)

[Out]

1/b^2/(exp(2*I*(b*x+a))-1)^2*(d^4*x^4*b*exp(3*I*(b*x+a))+4*c*d^3*x^3*b*exp(3*I*(b*x+a))+6*c^2*d^2*x^2*b*exp(3*
I*(b*x+a))+d^4*x^4*b*exp(I*(b*x+a))+4*c^3*d*x*b*exp(3*I*(b*x+a))+4*c*d^3*x^3*b*exp(I*(b*x+a))-4*I*c^3*d*exp(3*
I*(b*x+a))+c^4*b*exp(3*I*(b*x+a))+6*c^2*d^2*x^2*b*exp(I*(b*x+a))-12*I*c^2*d^2*x*exp(3*I*(b*x+a))+4*c^3*d*x*b*e
xp(I*(b*x+a))-4*I*d^4*x^3*exp(3*I*(b*x+a))+4*I*c^3*d*exp(I*(b*x+a))+c^4*b*exp(I*(b*x+a))+12*I*c*d^3*x^2*exp(I*
(b*x+a))+12*I*c^2*d^2*x*exp(I*(b*x+a))+4*I*d^4*x^3*exp(I*(b*x+a))-12*I*c*d^3*x^2*exp(3*I*(b*x+a)))-1/2/b^5*d^4
*ln(exp(I*(b*x+a))+1)*a^4-1/2/b*d^4*ln(1-exp(I*(b*x+a)))*x^4+1/2/b*d^4*ln(exp(I*(b*x+a))+1)*x^4+1/b^5*d^4*a^4*
arctanh(exp(I*(b*x+a)))-12/b^5*d^4*a^2*arctanh(exp(I*(b*x+a)))-6/b^3*c^2*d^2*polylog(3,exp(I*(b*x+a)))+6/b^3*c
^2*d^2*polylog(3,-exp(I*(b*x+a)))-6/b^3*d^4*ln(exp(I*(b*x+a))+1)*x^2-6/b^3*d^4*polylog(3,exp(I*(b*x+a)))*x^2+6
/b^3*d^4*polylog(3,-exp(I*(b*x+a)))*x^2-12/b^3*c^2*d^2*arctanh(exp(I*(b*x+a)))+6/b^3*d^4*ln(1-exp(I*(b*x+a)))*
x^2+1/2/b^5*d^4*ln(1-exp(I*(b*x+a)))*a^4-6/b^5*d^4*ln(1-exp(I*(b*x+a)))*a^2+6/b^5*d^4*ln(exp(I*(b*x+a))+1)*a^2
-12*I/b^4*c*d^3*polylog(2,exp(I*(b*x+a)))+12*I/b^4*c*d^3*polylog(2,-exp(I*(b*x+a)))+2*I/b^2*d^4*polylog(2,exp(
I*(b*x+a)))*x^3-2*I/b^2*d^4*polylog(2,-exp(I*(b*x+a)))*x^3-12*I/b^4*d^4*polylog(2,exp(I*(b*x+a)))*x+12*I/b^4*d
^4*polylog(2,-exp(I*(b*x+a)))*x-12*I/b^4*d^4*polylog(4,exp(I*(b*x+a)))*x+12*I/b^4*d^4*polylog(4,-exp(I*(b*x+a)
))*x-12*I/b^4*c*d^3*polylog(4,exp(I*(b*x+a)))+12*I/b^4*c*d^3*polylog(4,-exp(I*(b*x+a)))+2*I/b^2*c^3*d*polylog(
2,exp(I*(b*x+a)))-2*I/b^2*c^3*d*polylog(2,-exp(I*(b*x+a)))+1/b*c^4*arctanh(exp(I*(b*x+a)))+6*I/b^2*c^2*d^2*pol
ylog(2,exp(I*(b*x+a)))*x-6*I/b^2*c^2*d^2*polylog(2,-exp(I*(b*x+a)))*x+6*I/b^2*c*d^3*polylog(2,exp(I*(b*x+a)))*
x^2-6*I/b^2*c*d^3*polylog(2,-exp(I*(b*x+a)))*x^2-12*d^4*polylog(5,-exp(I*(b*x+a)))/b^5+12*d^4*polylog(5,exp(I*
(b*x+a)))/b^5-12*d^4*polylog(3,-exp(I*(b*x+a)))/b^5+12*d^4*polylog(3,exp(I*(b*x+a)))/b^5+2/b*c^3*d*ln(exp(I*(b
*x+a))+1)*x-2/b^2*c^3*d*ln(1-exp(I*(b*x+a)))*a+2/b^2*c^3*d*ln(exp(I*(b*x+a))+1)*a+12/b^4*c*d^3*ln(1-exp(I*(b*x
+a)))*a-12/b^4*c*d^3*ln(exp(I*(b*x+a))+1)*a+2/b^4*c*d^3*ln(exp(I*(b*x+a))+1)*a^3-3/b*c^2*d^2*ln(1-exp(I*(b*x+a
)))*x^2+3/b*c^2*d^2*ln(exp(I*(b*x+a))+1)*x^2-2/b*c*d^3*ln(1-exp(I*(b*x+a)))*x^3-2/b^4*c*d^3*ln(1-exp(I*(b*x+a)
))*a^3+24/b^4*c*d^3*a*arctanh(exp(I*(b*x+a)))+2/b*c*d^3*ln(exp(I*(b*x+a))+1)*x^3-2/b*c^3*d*ln(1-exp(I*(b*x+a))
)*x+3/b^3*c^2*d^2*ln(1-exp(I*(b*x+a)))*a^2-3/b^3*c^2*d^2*ln(exp(I*(b*x+a))+1)*a^2-12/b^3*c*d^3*polylog(3,exp(I
*(b*x+a)))*x+12/b^3*c*d^3*polylog(3,-exp(I*(b*x+a)))*x+12/b^3*c*d^3*ln(1-exp(I*(b*x+a)))*x-12/b^3*c*d^3*ln(exp
(I*(b*x+a))+1)*x+6/b^3*c^2*d^2*a^2*arctanh(exp(I*(b*x+a)))-4/b^4*c*d^3*a^3*arctanh(exp(I*(b*x+a)))-4/b^2*c^3*d
*a*arctanh(exp(I*(b*x+a)))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 2770 vs. \(2 (370) = 740\).

Time = 0.35 (sec) , antiderivative size = 2770, normalized size of antiderivative = 6.66 \[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*cot(b*x+a)^2*csc(b*x+a),x, algorithm="fricas")

[Out]

1/4*(2*(b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 6*b^4*c^2*d^2*x^2 + 4*b^4*c^3*d*x + b^4*c^4)*cos(b*x + a) - 4*(I*b^3*d
^4*x^3 + 3*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 6*I*b*c*d^3 + (-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - I*b^3*c^3*d + 6
*I*b*c*d^3 - 3*I*(b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^2 + 3*I*(b^3*c^2*d^2 - 2*b*d^4)*x)*dilog(cos(b*x + a)
+ I*sin(b*x + a)) - 4*(-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - I*b^3*c^3*d + 6*I*b*c*d^3 + (I*b^3*d^4*x^3 + 3*I*
b^3*c*d^3*x^2 + I*b^3*c^3*d - 6*I*b*c*d^3 + 3*I*(b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^2 - 3*I*(b^3*c^2*d^2 -
2*b*d^4)*x)*dilog(cos(b*x + a) - I*sin(b*x + a)) - 4*(I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 6*I*b
*c*d^3 + (-I*b^3*d^4*x^3 - 3*I*b^3*c*d^3*x^2 - I*b^3*c^3*d + 6*I*b*c*d^3 - 3*I*(b^3*c^2*d^2 - 2*b*d^4)*x)*cos(
b*x + a)^2 + 3*I*(b^3*c^2*d^2 - 2*b*d^4)*x)*dilog(-cos(b*x + a) + I*sin(b*x + a)) - 4*(-I*b^3*d^4*x^3 - 3*I*b^
3*c*d^3*x^2 - I*b^3*c^3*d + 6*I*b*c*d^3 + (I*b^3*d^4*x^3 + 3*I*b^3*c*d^3*x^2 + I*b^3*c^3*d - 6*I*b*c*d^3 + 3*I
*(b^3*c^2*d^2 - 2*b*d^4)*x)*cos(b*x + a)^2 - 3*I*(b^3*c^2*d^2 - 2*b*d^4)*x)*dilog(-cos(b*x + a) - I*sin(b*x +
a)) - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 - 12*b^2*c^2*d^2 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 - (b^4*d^4*x
^4 + 4*b^4*c*d^3*x^3 + b^4*c^4 - 12*b^2*c^2*d^2 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3
)*x)*cos(b*x + a)^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*log(cos(b*x + a) + I*sin(b*x + a) + 1) - (b^4*d^4*x^4 + 4
*b^4*c*d^3*x^3 + b^4*c^4 - 12*b^2*c^2*d^2 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 +
b^4*c^4 - 12*b^2*c^2*d^2 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*cos(b*x + a)^2 +
4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*log(cos(b*x + a) - I*sin(b*x + a) + 1) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 2)
*b^2*c^2*d^2 - 4*(a^3 - 6*a)*b*c*d^3 + (a^4 - 12*a^2)*d^4 - (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 2)*b^2*c^2*d^2
- 4*(a^3 - 6*a)*b*c*d^3 + (a^4 - 12*a^2)*d^4)*cos(b*x + a)^2)*log(-1/2*cos(b*x + a) + 1/2*I*sin(b*x + a) + 1/
2) + (b^4*c^4 - 4*a*b^3*c^3*d + 6*(a^2 - 2)*b^2*c^2*d^2 - 4*(a^3 - 6*a)*b*c*d^3 + (a^4 - 12*a^2)*d^4 - (b^4*c^
4 - 4*a*b^3*c^3*d + 6*(a^2 - 2)*b^2*c^2*d^2 - 4*(a^3 - 6*a)*b*c*d^3 + (a^4 - 12*a^2)*d^4)*cos(b*x + a)^2)*log(
-1/2*cos(b*x + a) - 1/2*I*sin(b*x + a) + 1/2) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2
*d^2 + 4*(a^3 - 6*a)*b*c*d^3 - (a^4 - 12*a^2)*d^4 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 - (b^4*d^4*x^4 + 4*b^4*c*d
^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 6*a)*b*c*d^3 - (a^4 - 12*a^2)*d^4 + 6*(b^4*c^2*d^2 - 2*b
^2*d^4)*x^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*cos(b*x + a)^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*log(-cos(b*x + a)
+ I*sin(b*x + a) + 1) + (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d - 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 6*a)*b*
c*d^3 - (a^4 - 12*a^2)*d^4 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 - (b^4*d^4*x^4 + 4*b^4*c*d^3*x^3 + 4*a*b^3*c^3*d
- 6*a^2*b^2*c^2*d^2 + 4*(a^3 - 6*a)*b*c*d^3 - (a^4 - 12*a^2)*d^4 + 6*(b^4*c^2*d^2 - 2*b^2*d^4)*x^2 + 4*(b^4*c^
3*d - 6*b^2*c*d^3)*x)*cos(b*x + a)^2 + 4*(b^4*c^3*d - 6*b^2*c*d^3)*x)*log(-cos(b*x + a) - I*sin(b*x + a) + 1)
+ 24*(d^4*cos(b*x + a)^2 - d^4)*polylog(5, cos(b*x + a) + I*sin(b*x + a)) + 24*(d^4*cos(b*x + a)^2 - d^4)*poly
log(5, cos(b*x + a) - I*sin(b*x + a)) - 24*(d^4*cos(b*x + a)^2 - d^4)*polylog(5, -cos(b*x + a) + I*sin(b*x + a
)) - 24*(d^4*cos(b*x + a)^2 - d^4)*polylog(5, -cos(b*x + a) - I*sin(b*x + a)) - 24*(-I*b*d^4*x - I*b*c*d^3 + (
I*b*d^4*x + I*b*c*d^3)*cos(b*x + a)^2)*polylog(4, cos(b*x + a) + I*sin(b*x + a)) - 24*(I*b*d^4*x + I*b*c*d^3 +
(-I*b*d^4*x - I*b*c*d^3)*cos(b*x + a)^2)*polylog(4, cos(b*x + a) - I*sin(b*x + a)) - 24*(-I*b*d^4*x - I*b*c*d
^3 + (I*b*d^4*x + I*b*c*d^3)*cos(b*x + a)^2)*polylog(4, -cos(b*x + a) + I*sin(b*x + a)) - 24*(I*b*d^4*x + I*b*
c*d^3 + (-I*b*d^4*x - I*b*c*d^3)*cos(b*x + a)^2)*polylog(4, -cos(b*x + a) - I*sin(b*x + a)) + 12*(b^2*d^4*x^2
+ 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4 - (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4)*cos(b*x + a)^2)*po
lylog(3, cos(b*x + a) + I*sin(b*x + a)) + 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4 - (b^2*d^4*x^2
+ 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4)*cos(b*x + a)^2)*polylog(3, cos(b*x + a) - I*sin(b*x + a)) - 12*(b^2*d^
4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4 - (b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4)*cos(b*x + a)
^2)*polylog(3, -cos(b*x + a) + I*sin(b*x + a)) - 12*(b^2*d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4 - (b^2*
d^4*x^2 + 2*b^2*c*d^3*x + b^2*c^2*d^2 - 2*d^4)*cos(b*x + a)^2)*polylog(3, -cos(b*x + a) - I*sin(b*x + a)) + 8*
(b^3*d^4*x^3 + 3*b^3*c*d^3*x^2 + 3*b^3*c^2*d^2*x + b^3*c^3*d)*sin(b*x + a))/(b^5*cos(b*x + a)^2 - b^5)

Sympy [F]

\[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=\int \left (c + d x\right )^{4} \cot ^{2}{\left (a + b x \right )} \csc {\left (a + b x \right )}\, dx \]

[In]

integrate((d*x+c)**4*cot(b*x+a)**2*csc(b*x+a),x)

[Out]

Integral((c + d*x)**4*cot(a + b*x)**2*csc(a + b*x), x)

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. 7004 vs. \(2 (370) = 740\).

Time = 3.26 (sec) , antiderivative size = 7004, normalized size of antiderivative = 16.84 \[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=\text {Too large to display} \]

[In]

integrate((d*x+c)^4*cot(b*x+a)^2*csc(b*x+a),x, algorithm="maxima")

[Out]

1/4*(c^4*(2*cos(b*x + a)/(cos(b*x + a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1)) - 4*a*c^3*d*(2*
cos(b*x + a)/(cos(b*x + a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1))/b + 6*a^2*c^2*d^2*(2*cos(b*
x + a)/(cos(b*x + a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1))/b^2 - 4*a^3*c*d^3*(2*cos(b*x + a)
/(cos(b*x + a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1))/b^3 + a^4*d^4*(2*cos(b*x + a)/(cos(b*x
+ a)^2 - 1) + log(cos(b*x + a) + 1) - log(cos(b*x + a) - 1))/b^4 + 4*(2*((b*x + a)^4*d^4 - 12*b^2*c^2*d^2 + 24
*a*b*c*d^3 - 12*a^2*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 - 2)*d^4)*(b*x
+ a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*d^4)*(b*x + a) + ((b*x + a)^4*d^4
- 12*b^2*c^2*d^2 + 24*a*b*c*d^3 - 12*a^2*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 - 2)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*d^4)*(b*x +
a))*cos(4*b*x + 4*a) - 2*((b*x + a)^4*d^4 - 12*b^2*c^2*d^2 + 24*a*b*c*d^3 - 12*a^2*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 - 2)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(
a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*d^4)*(b*x + a))*cos(2*b*x + 2*a) - (-I*(b*x + a)^4*d^4 + 12*I*b^2*c^2*d^2 - 24*
I*a*b*c*d^3 + 12*I*a^2*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 + 2*I)*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 + 2*I)*b*c*d^3 + (I*a^3 - 6*I*a)*d
^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*(I*(b*x + a)^4*d^4 - 12*I*b^2*c^2*d^2 + 24*I*a*b*c*d^3 - 12*I*a^2*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 - 2*I)*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 - 2*I)*b*c*d^3 + (-I*a^3 + 6*I*a)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*ar
ctan2(sin(b*x + a), cos(b*x + a) + 1) + 24*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4 + (b^2*c^2*d^2 - 2*a*b*c*d^3 +
a^2*d^4)*cos(4*b*x + 4*a) - 2*(b^2*c^2*d^2 - 2*a*b*c*d^3 + a^2*d^4)*cos(2*b*x + 2*a) - (-I*b^2*c^2*d^2 + 2*I*
a*b*c*d^3 - I*a^2*d^4)*sin(4*b*x + 4*a) - 2*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*a^2*d^4)*sin(2*b*x + 2*a))*arct
an2(sin(b*x + a), cos(b*x + a) - 1) + 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 - 2)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*
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 - 2)
*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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 - 2)*d^
4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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 + 2*I)*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 + 2*I)*b*c*d^3 + (I*a^3 - 6*I*a)
*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 - 2*I)*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 - 2*I)*b*
c*d^3 + (-I*a^3 + 6*I*a)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*arctan2(sin(b*x + a), -cos(b*x + a) + 1) - 4*(I*(b*
x + a)^4*d^4 + 4*b^3*c^3*d - 12*a*b^2*c^2*d^2 + 12*a^2*b*c*d^3 - 4*a^3*d^4 + 4*(I*b*c*d^3 + (-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 - 2*a)*d^4)*(b*x + a)^2 + 4*(I*b^3*c^3*d + 3*(-I*
a + 1)*b^2*c^2*d^2 + 3*(I*a^2 - 2*a)*b*c*d^3 + (-I*a^3 + 3*a^2)*d^4)*(b*x + a))*cos(3*b*x + 3*a) - 4*(I*(b*x +
a)^4*d^4 - 4*b^3*c^3*d + 12*a*b^2*c^2*d^2 - 12*a^2*b*c*d^3 + 4*a^3*d^4 + 4*(I*b*c*d^3 + (-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 + 2*a)*d^4)*(b*x + a)^2 + 4*(I*b^3*c^3*d + 3*(-I*a -
1)*b^2*c^2*d^2 + 3*(I*a^2 + 2*a)*b*c*d^3 + (-I*a^3 - 3*a^2)*d^4)*(b*x + a))*cos(b*x + a) - 8*(b^3*c^3*d - 3*a
*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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 - 2)*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2
- 2)*b*c*d^3 - (a^3 - 6*a)*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 - 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 + (b*x + a)^3*d^4 + 3*(a^2 - 2)*b*c*d^3 - (a^
3 - 6*a)*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 - 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 + I*(b*x + a)^3*d^4 + 3*(I*a^2 - 2*I)*b*c*d^3 + (-I*a^3 + 6*I*
a)*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 - 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 - I*(b*x + a)^3*d^4 + 3*(-I*a^2 + 2*I)*b*c*d^3 + (I*a
^3 - 6*I*a)*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 + 2*I)*d^
4)*(b*x + a))*sin(2*b*x + 2*a))*dilog(-e^(I*b*x + I*a)) + 8*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3
*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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 -
2)*d^4)*(b*x + a) + (b^3*c^3*d - 3*a*b^2*c^2*d^2 + (b*x + a)^3*d^4 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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 - 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 + (b*x + a)^3*d^4 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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 - 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 - I*(b*x + a)^3*d^4 + 3*(-I*a^2 + 2*I)*b*c*d^3 + (I*a^3 - 6*I*a)*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 + 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 + I*(b*x + a)^3*d^4 + 3*(I*a^2 - 2*I)*b*c*d^3 + (-I*a^3 + 6*I*a)*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 - 2*I)*d^4)*(b*x + a))*sin(2*b*x + 2*a))*di
log(e^(I*b*x + I*a)) + (-I*(b*x + a)^4*d^4 + 12*I*b^2*c^2*d^2 - 24*I*a*b*c*d^3 + 12*I*a^2*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 - 2*I)*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 - 2*I)*b*c*d^3 + (-I*a^3 + 6*I*a)*d^4)*(b*x + a) + (-I*(b*x + a)^4*d^4 + 12*I*b^2*
c^2*d^2 - 24*I*a*b*c*d^3 + 12*I*a^2*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 - 2*I)*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 - 2*I)*b*c*d^3 + (-I*a^3 +
6*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 2*(-I*(b*x + a)^4*d^4 + 12*I*b^2*c^2*d^2 - 24*I*a*b*c*d^3 + 12*I*a^
2*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 + 2*I)*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 + 2*I)*b*c*d^3 + (I*a^3 - 6*I*a)*d^4)*(b*x + a))*cos(2*
b*x + 2*a) + ((b*x + a)^4*d^4 - 12*b^2*c^2*d^2 + 24*a*b*c*d^3 - 12*a^2*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 - 2)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*
d^3 - (a^3 - 6*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) - 2*((b*x + a)^4*d^4 - 12*b^2*c^2*d^2 + 24*a*b*c*d^3 - 12*a
^2*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 - 2)*d^4)*(b*x + a)^2 + 4*(b^3*
c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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) + (I*(b*x + a)^4*d^4 - 12*I*b^2*c^2*d^2 + 24*I*a*b*c*d^3 - 12*I*a^2*
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 + 2*I)*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 + 2*I)*b*c*d^3 + (I*a^3 - 6*I*a)*d^4)*(b*x + a) + (I*(b*x
+ a)^4*d^4 - 12*I*b^2*c^2*d^2 + 24*I*a*b*c*d^3 - 12*I*a^2*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 + 2*I)*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 + 2*I)*b*c*d^3 + (I*a^3 - 6*I*a)*d^4)*(b*x + a))*cos(4*b*x + 4*a) - 2*(I*(b*x + a)^4*d^4 - 12*I*b^2*c^2*d^2
+ 24*I*a*b*c*d^3 - 12*I*a^2*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 - 2*I)*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 - 2*I)*b*c*d^3 + (-I*a^3 + 6*I*a)*
d^4)*(b*x + a))*cos(2*b*x + 2*a) - ((b*x + a)^4*d^4 - 12*b^2*c^2*d^2 + 24*a*b*c*d^3 - 12*a^2*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 - 2)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*
d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*d^4)*(b*x + a))*sin(4*b*x + 4*a) + 2*((b*x + a)^4*d^4 - 12*b^2*c^2*d^2
+ 24*a*b*c*d^3 - 12*a^2*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 - 2)*d^4)
*(b*x + a)^2 + 4*(b^3*c^3*d - 3*a*b^2*c^2*d^2 + 3*(a^2 - 2)*b*c*d^3 - (a^3 - 6*a)*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) - 48*(-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)) - 48*(I*d^4*c
os(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)) + 48*(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)) - 48*(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)) - 24*(I*b^2*c^2*d^2 - 2*I*a*b*c*d^3 + I*(b*x + a)
^2*d^4 + (I*a^2 - 2*I)*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 - 2*I)*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 + 2*I)*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 - 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 - 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^2*c^2*d^2 + 2*I*a*b*c*d^3 - I*(b*x + a)^2*d^4 + (-I*a^2 + 2*I)*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 + 2*I)*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 - 2*I)*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 - 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 - 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))
+ 4*((b*x + a)^4*d^4 - 4*I*b^3*c^3*d + 12*I*a*b^2*c^2*d^2 - 12*I*a^2*b*c*d^3 + 4*I*a^3*d^4 + 4*(b*c*d^3 - (a
+ I)*d^4)*(b*x + a)^3 + 6*(b^2*c^2*d^2 - 2*(a + I)*b*c*d^3 + (a^2 + 2*I*a)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3
*(a + I)*b^2*c^2*d^2 + 3*(a^2 + 2*I*a)*b*c*d^3 - (a^3 + 3*I*a^2)*d^4)*(b*x + a))*sin(3*b*x + 3*a) + 4*((b*x +
a)^4*d^4 + 4*I*b^3*c^3*d - 12*I*a*b^2*c^2*d^2 + 12*I*a^2*b*c*d^3 - 4*I*a^3*d^4 + 4*(b*c*d^3 - (a - I)*d^4)*(b*
x + a)^3 + 6*(b^2*c^2*d^2 - 2*(a - I)*b*c*d^3 + (a^2 - 2*I*a)*d^4)*(b*x + a)^2 + 4*(b^3*c^3*d - 3*(a - I)*b^2*
c^2*d^2 + 3*(a^2 - 2*I*a)*b*c*d^3 - (a^3 - 3*I*a^2)*d^4)*(b*x + a))*sin(b*x + a))/(-4*I*b^4*cos(4*b*x + 4*a) +
8*I*b^4*cos(2*b*x + 2*a) + 4*b^4*sin(4*b*x + 4*a) - 8*b^4*sin(2*b*x + 2*a) - 4*I*b^4))/b

Giac [F]

\[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=\int { {\left (d x + c\right )}^{4} \cot \left (b x + a\right )^{2} \csc \left (b x + a\right ) \,d x } \]

[In]

integrate((d*x+c)^4*cot(b*x+a)^2*csc(b*x+a),x, algorithm="giac")

[Out]

integrate((d*x + c)^4*cot(b*x + a)^2*csc(b*x + a), x)

Mupad [F(-1)]

Timed out. \[ \int (c+d x)^4 \cot ^2(a+b x) \csc (a+b x) \, dx=\text {Hanged} \]

[In]

int((cot(a + b*x)^2*(c + d*x)^4)/sin(a + b*x),x)

[Out]

\text{Hanged}