3.171 \(\int (c+d x)^4 \cos (a+b x) \cot ^2(a+b x) \, dx\)

Optimal. Leaf size=299 \[ -\frac{24 d^3 (c+d x) \text{PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac{24 d^3 (c+d x) \text{PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}+\frac{12 i d^2 (c+d x)^2 \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac{24 i d^4 \text{PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^5}+\frac{24 i d^4 \text{PolyLog}\left (4,e^{i (a+b x)}\right )}{b^5}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{24 d^4 \sin (a+b x)}{b^5}-\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{(c+d x)^4 \csc (a+b x)}{b} \]

[Out]

(-8*d*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b^2 + (24*d^3*(c + d*x)*Cos[a + b*x])/b^4 - (4*d*(c + d*x)^3*Cos[a
+ b*x])/b^2 - ((c + d*x)^4*Csc[a + b*x])/b + ((12*I)*d^2*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - ((12
*I)*d^2*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (24*d^3*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (
24*d^3*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^4 - ((24*I)*d^4*PolyLog[4, -E^(I*(a + b*x))])/b^5 + ((24*I)*d^
4*PolyLog[4, E^(I*(a + b*x))])/b^5 - (24*d^4*Sin[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Sin[a + b*x])/b^3 - ((c +
d*x)^4*Sin[a + b*x])/b

________________________________________________________________________________________

Rubi [A]  time = 0.292642, antiderivative size = 299, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 9, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.409, Rules used = {4408, 3296, 2637, 4410, 4183, 2531, 6609, 2282, 6589} \[ -\frac{24 d^3 (c+d x) \text{PolyLog}\left (3,-e^{i (a+b x)}\right )}{b^4}+\frac{24 d^3 (c+d x) \text{PolyLog}\left (3,e^{i (a+b x)}\right )}{b^4}+\frac{12 i d^2 (c+d x)^2 \text{PolyLog}\left (2,-e^{i (a+b x)}\right )}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{PolyLog}\left (2,e^{i (a+b x)}\right )}{b^3}-\frac{24 i d^4 \text{PolyLog}\left (4,-e^{i (a+b x)}\right )}{b^5}+\frac{24 i d^4 \text{PolyLog}\left (4,e^{i (a+b x)}\right )}{b^5}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}+\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{24 d^4 \sin (a+b x)}{b^5}-\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{(c+d x)^4 \csc (a+b x)}{b} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-8*d*(c + d*x)^3*ArcTanh[E^(I*(a + b*x))])/b^2 + (24*d^3*(c + d*x)*Cos[a + b*x])/b^4 - (4*d*(c + d*x)^3*Cos[a
+ b*x])/b^2 - ((c + d*x)^4*Csc[a + b*x])/b + ((12*I)*d^2*(c + d*x)^2*PolyLog[2, -E^(I*(a + b*x))])/b^3 - ((12
*I)*d^2*(c + d*x)^2*PolyLog[2, E^(I*(a + b*x))])/b^3 - (24*d^3*(c + d*x)*PolyLog[3, -E^(I*(a + b*x))])/b^4 + (
24*d^3*(c + d*x)*PolyLog[3, E^(I*(a + b*x))])/b^4 - ((24*I)*d^4*PolyLog[4, -E^(I*(a + b*x))])/b^5 + ((24*I)*d^
4*PolyLog[4, E^(I*(a + b*x))])/b^5 - (24*d^4*Sin[a + b*x])/b^5 + (12*d^2*(c + d*x)^2*Sin[a + b*x])/b^3 - ((c +
d*x)^4*Sin[a + b*x])/b

Rule 4408

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

Rule 3296

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 4410

Int[Cot[(a_.) + (b_.)*(x_)]^(p_.)*Csc[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> -Simp
[((c + d*x)^m*Csc[a + b*x]^n)/(b*n), x] + Dist[(d*m)/(b*n), Int[(c + d*x)^(m - 1)*Csc[a + b*x]^n, x], x] /; Fr
eeQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4183

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 2531

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 6609

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 2282

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 6589

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]

Rubi steps

\begin{align*} \int (c+d x)^4 \cos (a+b x) \cot ^2(a+b x) \, dx &=-\int (c+d x)^4 \cos (a+b x) \, dx+\int (c+d x)^4 \cot (a+b x) \csc (a+b x) \, dx\\ &=-\frac{(c+d x)^4 \csc (a+b x)}{b}-\frac{(c+d x)^4 \sin (a+b x)}{b}+\frac{(4 d) \int (c+d x)^3 \csc (a+b x) \, dx}{b}+\frac{(4 d) \int (c+d x)^3 \sin (a+b x) \, dx}{b}\\ &=-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{(c+d x)^4 \csc (a+b x)}{b}-\frac{(c+d x)^4 \sin (a+b x)}{b}+\frac{\left (12 d^2\right ) \int (c+d x)^2 \cos (a+b x) \, dx}{b^2}-\frac{\left (12 d^2\right ) \int (c+d x)^2 \log \left (1-e^{i (a+b x)}\right ) \, dx}{b^2}+\frac{\left (12 d^2\right ) \int (c+d x)^2 \log \left (1+e^{i (a+b x)}\right ) \, dx}{b^2}\\ &=-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{(c+d x)^4 \csc (a+b x)}{b}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{b^3}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{\left (24 i d^3\right ) \int (c+d x) \text{Li}_2\left (-e^{i (a+b x)}\right ) \, dx}{b^3}+\frac{\left (24 i d^3\right ) \int (c+d x) \text{Li}_2\left (e^{i (a+b x)}\right ) \, dx}{b^3}-\frac{\left (24 d^3\right ) \int (c+d x) \sin (a+b x) \, dx}{b^3}\\ &=-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{(c+d x)^4 \csc (a+b x)}{b}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{b^3}-\frac{24 d^3 (c+d x) \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac{24 d^3 (c+d x) \text{Li}_3\left (e^{i (a+b x)}\right )}{b^4}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{\left (24 d^4\right ) \int \cos (a+b x) \, dx}{b^4}+\frac{\left (24 d^4\right ) \int \text{Li}_3\left (-e^{i (a+b x)}\right ) \, dx}{b^4}-\frac{\left (24 d^4\right ) \int \text{Li}_3\left (e^{i (a+b x)}\right ) \, dx}{b^4}\\ &=-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{(c+d x)^4 \csc (a+b x)}{b}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{b^3}-\frac{24 d^3 (c+d x) \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac{24 d^3 (c+d x) \text{Li}_3\left (e^{i (a+b x)}\right )}{b^4}-\frac{24 d^4 \sin (a+b x)}{b^5}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{(c+d x)^4 \sin (a+b x)}{b}-\frac{\left (24 i d^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}+\frac{\left (24 i d^4\right ) \operatorname{Subst}\left (\int \frac{\text{Li}_3(x)}{x} \, dx,x,e^{i (a+b x)}\right )}{b^5}\\ &=-\frac{8 d (c+d x)^3 \tanh ^{-1}\left (e^{i (a+b x)}\right )}{b^2}+\frac{24 d^3 (c+d x) \cos (a+b x)}{b^4}-\frac{4 d (c+d x)^3 \cos (a+b x)}{b^2}-\frac{(c+d x)^4 \csc (a+b x)}{b}+\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (-e^{i (a+b x)}\right )}{b^3}-\frac{12 i d^2 (c+d x)^2 \text{Li}_2\left (e^{i (a+b x)}\right )}{b^3}-\frac{24 d^3 (c+d x) \text{Li}_3\left (-e^{i (a+b x)}\right )}{b^4}+\frac{24 d^3 (c+d x) \text{Li}_3\left (e^{i (a+b x)}\right )}{b^4}-\frac{24 i d^4 \text{Li}_4\left (-e^{i (a+b x)}\right )}{b^5}+\frac{24 i d^4 \text{Li}_4\left (e^{i (a+b x)}\right )}{b^5}-\frac{24 d^4 \sin (a+b x)}{b^5}+\frac{12 d^2 (c+d x)^2 \sin (a+b x)}{b^3}-\frac{(c+d x)^4 \sin (a+b x)}{b}\\ \end{align*}

Mathematica [B]  time = 1.72215, size = 798, normalized size = 2.67 \[ \frac{\csc (a+b x) \left (-3 c^4 b^4-3 d^4 x^4 b^4-12 c d^3 x^3 b^4-18 c^2 d^2 x^2 b^4-12 c^3 d x b^4+c^4 \cos (2 (a+b x)) b^4+d^4 x^4 \cos (2 (a+b x)) b^4+4 c d^3 x^3 \cos (2 (a+b x)) b^4+6 c^2 d^2 x^2 \cos (2 (a+b x)) b^4+4 c^3 d x \cos (2 (a+b x)) b^4-16 d^4 x^3 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-48 c d^3 x^2 \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-16 c^3 d \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-48 c^2 d^2 x \tanh ^{-1}(\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^3-4 d^4 x^3 \sin (2 (a+b x)) b^3-12 c d^3 x^2 \sin (2 (a+b x)) b^3-4 c^3 d \sin (2 (a+b x)) b^3-12 c^2 d^2 x \sin (2 (a+b x)) b^3+12 c^2 d^2 b^2+12 d^4 x^2 b^2+24 c d^3 x b^2-12 c^2 d^2 \cos (2 (a+b x)) b^2-12 d^4 x^2 \cos (2 (a+b x)) b^2-24 c d^3 x \cos (2 (a+b x)) b^2+24 i d^2 (c+d x)^2 \text{PolyLog}(2,-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b^2-24 i d^2 (c+d x)^2 \text{PolyLog}(2,\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b^2-48 c d^3 \text{PolyLog}(3,-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b-48 d^4 x \text{PolyLog}(3,-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x) b+48 c d^3 \text{PolyLog}(3,\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b+48 d^4 x \text{PolyLog}(3,\cos (a+b x)+i \sin (a+b x)) \sin (a+b x) b+24 c d^3 \sin (2 (a+b x)) b+24 d^4 x \sin (2 (a+b x)) b-24 d^4+24 d^4 \cos (2 (a+b x))-48 i d^4 \text{PolyLog}(4,-\cos (a+b x)-i \sin (a+b x)) \sin (a+b x)+48 i d^4 \text{PolyLog}(4,\cos (a+b x)+i \sin (a+b x)) \sin (a+b x)\right )}{2 b^5} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

(Csc[a + b*x]*(-3*b^4*c^4 + 12*b^2*c^2*d^2 - 24*d^4 - 12*b^4*c^3*d*x + 24*b^2*c*d^3*x - 18*b^4*c^2*d^2*x^2 + 1
2*b^2*d^4*x^2 - 12*b^4*c*d^3*x^3 - 3*b^4*d^4*x^4 + b^4*c^4*Cos[2*(a + b*x)] - 12*b^2*c^2*d^2*Cos[2*(a + b*x)]
+ 24*d^4*Cos[2*(a + b*x)] + 4*b^4*c^3*d*x*Cos[2*(a + b*x)] - 24*b^2*c*d^3*x*Cos[2*(a + b*x)] + 6*b^4*c^2*d^2*x
^2*Cos[2*(a + b*x)] - 12*b^2*d^4*x^2*Cos[2*(a + b*x)] + 4*b^4*c*d^3*x^3*Cos[2*(a + b*x)] + b^4*d^4*x^4*Cos[2*(
a + b*x)] - 16*b^3*c^3*d*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b^3*c^2*d^2*x*ArcTanh[Cos[a
+ b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 48*b^3*c*d^3*x^2*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] -
16*b^3*d^4*x^3*ArcTanh[Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] + (24*I)*b^2*d^2*(c + d*x)^2*PolyLog[2, -C
os[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] - (24*I)*b^2*d^2*(c + d*x)^2*PolyLog[2, Cos[a + b*x] + I*Sin[a + b*
x]]*Sin[a + b*x] - 48*b*c*d^3*PolyLog[3, -Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] - 48*b*d^4*x*PolyLog[3,
-Cos[a + b*x] - I*Sin[a + b*x]]*Sin[a + b*x] + 48*b*c*d^3*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*
x] + 48*b*d^4*x*PolyLog[3, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - (48*I)*d^4*PolyLog[4, -Cos[a + b*x] -
I*Sin[a + b*x]]*Sin[a + b*x] + (48*I)*d^4*PolyLog[4, Cos[a + b*x] + I*Sin[a + b*x]]*Sin[a + b*x] - 4*b^3*c^3*
d*Sin[2*(a + b*x)] + 24*b*c*d^3*Sin[2*(a + b*x)] - 12*b^3*c^2*d^2*x*Sin[2*(a + b*x)] + 24*b*d^4*x*Sin[2*(a + b
*x)] - 12*b^3*c*d^3*x^2*Sin[2*(a + b*x)] - 4*b^3*d^4*x^3*Sin[2*(a + b*x)]))/(2*b^5)

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Maple [B]  time = 0.209, size = 1056, normalized size = 3.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*d^4/b^5*ln(exp(I*(b*x+a))+1)*a^3-24*d^3/b^4*c*a^2*arctanh(exp(I*(b*x+a)))+24*d^2/b^3*c^2*a*arctanh(exp(I*(b
*x+a)))-24*I*d^3/b^3*c*polylog(2,exp(I*(b*x+a)))*x+8*d^4/b^5*a^3*arctanh(exp(I*(b*x+a)))-8*d/b^2*c^3*arctanh(e
xp(I*(b*x+a)))+4*d^4/b^2*ln(1-exp(I*(b*x+a)))*x^3+24*d^4/b^4*polylog(3,exp(I*(b*x+a)))*x-24*d^4/b^4*polylog(3,
-exp(I*(b*x+a)))*x-24*d^3/b^4*c*polylog(3,-exp(I*(b*x+a)))+24*d^3/b^4*c*polylog(3,exp(I*(b*x+a)))+12*d^3/b^4*c
*a^2*ln(exp(I*(b*x+a))+1)-12*d^2/b^3*c^2*ln(exp(I*(b*x+a))+1)*a+12*I*d^2/b^3*c^2*polylog(2,-exp(I*(b*x+a)))+12
*I*d^4/b^3*polylog(2,-exp(I*(b*x+a)))*x^2+24*I*d^4*polylog(4,exp(I*(b*x+a)))/b^5-2*I*(d^4*x^4+4*c*d^3*x^3+6*c^
2*d^2*x^2+4*c^3*d*x+c^4)*exp(I*(b*x+a))/b/(exp(2*I*(b*x+a))-1)-1/2*I*(d^4*x^4*b^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-4*I*b^3*d^4*x^3+b^4*c^4-12*b^2*d^4*x^2-12*I*b^3*c*d^3*x^2-24*b^2*c*d^3*x-12*I*b^3*c^2*d^2*
x-12*c^2*d^2*b^2-4*I*b^3*c^3*d+24*I*b*d^4*x+24*d^4+24*I*b*c*d^3)/b^5*exp(-I*(b*x+a))-12*d^3/b^2*c*ln(exp(I*(b*
x+a))+1)*x^2+12*d^3/b^2*c*ln(1-exp(I*(b*x+a)))*x^2-12*d^3/b^4*c*ln(1-exp(I*(b*x+a)))*a^2-12*I*d^2/b^3*c^2*poly
log(2,exp(I*(b*x+a)))-12*I*d^4/b^3*polylog(2,exp(I*(b*x+a)))*x^2+24*I*d^3/b^3*c*polylog(2,-exp(I*(b*x+a)))*x-2
4*I*d^4*polylog(4,-exp(I*(b*x+a)))/b^5+1/2*I*(d^4*x^4*b^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+4*I*
b^3*d^4*x^3+b^4*c^4-12*b^2*d^4*x^2+12*I*b^3*c*d^3*x^2-24*b^2*c*d^3*x+12*I*b^3*c^2*d^2*x-12*c^2*d^2*b^2+4*I*b^3
*c^3*d-24*I*b*d^4*x+24*d^4-24*I*b*c*d^3)/b^5*exp(I*(b*x+a))+4*d^4/b^5*ln(1-exp(I*(b*x+a)))*a^3-4*d^4/b^2*ln(ex
p(I*(b*x+a))+1)*x^3+12*d^2/b^2*c^2*ln(1-exp(I*(b*x+a)))*x+12*d^2/b^3*c^2*ln(1-exp(I*(b*x+a)))*a-12*d^2/b^2*c^2
*ln(exp(I*(b*x+a))+1)*x

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Fricas [C]  time = 0.88748, size = 3009, normalized size = 10.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x + c\right )}^{4} \cos \left (b x + a\right ) \cot \left (b x + a\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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