3.261 \(\int x^2 \coth ^{-1}(1-i d-d \cot (a+b x)) \, dx\)

Optimal. Leaf size=169 \[ -\frac{x \text{PolyLog}\left (3,(1-i d) e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{i \text{PolyLog}\left (4,(1-i d) e^{2 i a+2 i b x}\right )}{8 b^3}+\frac{i x^2 \text{PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{1}{3} x^3 \coth ^{-1}(d (-\cot (a+b x))-i d+1)+\frac{1}{12} i b x^4 \]

[Out]

(I/12)*b*x^4 + (x^3*ArcCoth[1 - I*d - d*Cot[a + b*x]])/3 - (x^3*Log[1 - (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/6
+ ((I/4)*x^2*PolyLog[2, (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/b - (x*PolyLog[3, (1 - I*d)*E^((2*I)*a + (2*I)*b*x
)])/(4*b^2) - ((I/8)*PolyLog[4, (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/b^3

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Rubi [A]  time = 0.294771, antiderivative size = 169, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6266, 2184, 2190, 2531, 6609, 2282, 6589} \[ -\frac{x \text{PolyLog}\left (3,(1-i d) e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{i \text{PolyLog}\left (4,(1-i d) e^{2 i a+2 i b x}\right )}{8 b^3}+\frac{i x^2 \text{PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{1}{3} x^3 \coth ^{-1}(d (-\cot (a+b x))-i d+1)+\frac{1}{12} i b x^4 \]

Antiderivative was successfully verified.

[In]

Int[x^2*ArcCoth[1 - I*d - d*Cot[a + b*x]],x]

[Out]

(I/12)*b*x^4 + (x^3*ArcCoth[1 - I*d - d*Cot[a + b*x]])/3 - (x^3*Log[1 - (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/6
+ ((I/4)*x^2*PolyLog[2, (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/b - (x*PolyLog[3, (1 - I*d)*E^((2*I)*a + (2*I)*b*x
)])/(4*b^2) - ((I/8)*PolyLog[4, (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/b^3

Rule 6266

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

Rule 2184

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

Rule 2190

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*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), 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 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 x^2 \coth ^{-1}(1-i d-d \cot (a+b x)) \, dx &=\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))+\frac{1}{3} (i b) \int \frac{x^3}{1+(-1+i d) e^{2 i a+2 i b x}} \, dx\\ &=\frac{1}{12} i b x^4+\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))+\frac{1}{3} (b (i+d)) \int \frac{e^{2 i a+2 i b x} x^3}{1+(-1+i d) e^{2 i a+2 i b x}} \, dx\\ &=\frac{1}{12} i b x^4+\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{1}{2} \int x^2 \log \left (1+(-1+i d) e^{2 i a+2 i b x}\right ) \, dx\\ &=\frac{1}{12} i b x^4+\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{i x^2 \text{Li}_2\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{i \int x \text{Li}_2\left (-(-1+i d) e^{2 i a+2 i b x}\right ) \, dx}{2 b}\\ &=\frac{1}{12} i b x^4+\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{i x^2 \text{Li}_2\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{x \text{Li}_3\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b^2}+\frac{\int \text{Li}_3\left ((1-i d) e^{2 i a+2 i b x}\right ) \, dx}{4 b^2}\\ &=\frac{1}{12} i b x^4+\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{i x^2 \text{Li}_2\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{x \text{Li}_3\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{i \operatorname{Subst}\left (\int \frac{\text{Li}_3((1-i d) x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^3}\\ &=\frac{1}{12} i b x^4+\frac{1}{3} x^3 \coth ^{-1}(1-i d-d \cot (a+b x))-\frac{1}{6} x^3 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac{i x^2 \text{Li}_2\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac{x \text{Li}_3\left ((1-i d) e^{2 i a+2 i b x}\right )}{4 b^2}-\frac{i \text{Li}_4\left ((1-i d) e^{2 i a+2 i b x}\right )}{8 b^3}\\ \end{align*}

Mathematica [A]  time = 0.196546, size = 155, normalized size = 0.92 \[ \frac{1}{3} x^3 \coth ^{-1}(d (-\cot (a+b x))-i d+1)-\frac{6 i b^2 x^2 \text{PolyLog}\left (2,\frac{i e^{-2 i (a+b x)}}{d+i}\right )+6 b x \text{PolyLog}\left (3,\frac{i e^{-2 i (a+b x)}}{d+i}\right )-3 i \text{PolyLog}\left (4,\frac{i e^{-2 i (a+b x)}}{d+i}\right )+4 b^3 x^3 \log \left (1+\frac{e^{-2 i (a+b x)}}{-1+i d}\right )}{24 b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2*ArcCoth[1 - I*d - d*Cot[a + b*x]],x]

[Out]

(x^3*ArcCoth[1 - I*d - d*Cot[a + b*x]])/3 - (4*b^3*x^3*Log[1 + 1/((-1 + I*d)*E^((2*I)*(a + b*x)))] + (6*I)*b^2
*x^2*PolyLog[2, I/((I + d)*E^((2*I)*(a + b*x)))] + 6*b*x*PolyLog[3, I/((I + d)*E^((2*I)*(a + b*x)))] - (3*I)*P
olyLog[4, I/((I + d)*E^((2*I)*(a + b*x)))])/(24*b^3)

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Maple [C]  time = 29.139, size = 2339, normalized size = 13.8 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*arccoth(1-I*d-d*cot(b*x+a)),x)

[Out]

-1/6*x^3*ln(d)+1/12*I*b*x^4+1/4*I/b*d/(I+d)*polylog(2,-I*(I+d)*exp(2*I*(b*x+a)))*x^2+1/12*I*x^3*Pi*csgn((I*exp
(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))^2-1/3*x^3*ln(exp(I*(b*x+a)))-1/12*I*x^3*Pi*csgn(I*(I
*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))^3-1/12*I*x^3*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*ex
p(2*I*(b*x+a)))*csgn(d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2-1/12*I*x^3*Pi*csgn((I*exp(2*I*(b*x+a))+exp(2*I
*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))^3+1/12*I*x^3*Pi*csgn(d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^3-1/2/b^3*a
^2/(I+d)*dilog(1+I*exp(I*(b*x+a))*(I*(I+d))^(1/2))-1/2/b^3*a^2/(I+d)*dilog(1-I*exp(I*(b*x+a))*(I*(I+d))^(1/2))
-1/4/b/(I+d)*polylog(2,-I*(I+d)*exp(2*I*(b*x+a)))*x^2+1/4/b^3/(I+d)*polylog(2,-I*(I+d)*exp(2*I*(b*x+a)))*a^2-1
/6*d/(I+d)*ln(1+I*(I+d)*exp(2*I*(b*x+a)))*x^3+1/6*x^3*ln(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)-1/4*I/b^3*d/
(I+d)*polylog(2,-I*(I+d)*exp(2*I*(b*x+a)))*a^2+1/12*I*x^3*Pi*csgn(I*exp(I*(b*x+a)))^2*csgn(I*exp(2*I*(b*x+a)))
-1/6*I*x^3*Pi*csgn(I*exp(I*(b*x+a)))*csgn(I*exp(2*I*(b*x+a)))^2-1/12*I*x^3*Pi*csgn(d/(exp(2*I*(b*x+a))-1)*exp(
2*I*(b*x+a)))^2+1/2*I/b^3*a^2*d/(I+d)*dilog(1+I*exp(I*(b*x+a))*(I*(I+d))^(1/2))-1/2*I/b^3*a^3/(I+d)*ln(1-I*exp
(I*(b*x+a))*(I*(I+d))^(1/2))-1/8*I/b^3*d/(I+d)*polylog(4,-I*(I+d)*exp(2*I*(b*x+a)))+1/12*I*x^3*Pi*csgn(I*exp(2
*I*(b*x+a)))*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))+1/12*I*x^3*Pi*csgn(I*d
)*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))+1/6*I/b^3*a^3/
(I+d)*ln(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)+1/3*I/b^3/(I+d)*ln(1+I*(I+d)*exp(2*I*(b*x+a)))*a^3-1/4*I/b^2
/(I+d)*polylog(3,-I*(I+d)*exp(2*I*(b*x+a)))*x-1/2*I/b^3*a^3/(I+d)*ln(1+I*exp(I*(b*x+a))*(I*(I+d))^(1/2))+1/2*I
/b^3*a^2*d/(I+d)*dilog(1-I*exp(I*(b*x+a))*(I*(I+d))^(1/2))-1/12*I*x^3*Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b
*x+a))*d-I)/(exp(2*I*(b*x+a))-1))*csgn((I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))+1/12*I*
x^3*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))*csgn(d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))+1/12*I*x^
3*Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I))*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2
*I*(b*x+a))-1))^2-1/12*I*x^3*Pi*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))*csgn(I*d/(exp(2*I*(b*x+a))-1)*ex
p(2*I*(b*x+a)))^2+1/8/b^3/(I+d)*polylog(4,-I*(I+d)*exp(2*I*(b*x+a)))-1/12*I*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))*cs
gn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))^2-1/12*I*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*exp(2*I*(b*x+a
))/(exp(2*I*(b*x+a))-1))^2-1/12*I*x^3*Pi*csgn(I*d)*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x+a)))^2+1/12*I*x^
3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))^2-1/1
2*I*x^3*Pi*csgn(I/(exp(2*I*(b*x+a))-1))*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I))*csgn(I*(I*exp(2*I*(b
*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))-1/6*I/(I+d)*ln(1+I*(I+d)*exp(2*I*(b*x+a)))*x^3+1/2*I/b^2/(I
+d)*ln(1+I*(I+d)*exp(2*I*(b*x+a)))*x*a^2+1/12*I*x^3*Pi*csgn(I*(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2
*I*(b*x+a))-1))*csgn((I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)/(exp(2*I*(b*x+a))-1))^2-1/2/b^3*a^3*d/(I+d)*ln(
1+I*exp(I*(b*x+a))*(I*(I+d))^(1/2))-1/2/b^3*a^3*d/(I+d)*ln(1-I*exp(I*(b*x+a))*(I*(I+d))^(1/2))+1/6/b^3*a^3*d/(
I+d)*ln(I*exp(2*I*(b*x+a))+exp(2*I*(b*x+a))*d-I)+1/3/b^3*d/(I+d)*ln(1+I*(I+d)*exp(2*I*(b*x+a)))*a^3-1/4/b^2*d/
(I+d)*polylog(3,-I*(I+d)*exp(2*I*(b*x+a)))*x+1/12*I*x^3*Pi*csgn(I*exp(2*I*(b*x+a)))^3-1/2*I/b^2*a^2/(I+d)*ln(1
-I*exp(I*(b*x+a))*(I*(I+d))^(1/2))*x-1/2/b^2*a^2*d/(I+d)*ln(1-I*exp(I*(b*x+a))*(I*(I+d))^(1/2))*x-1/2/b^2*a^2*
d/(I+d)*ln(1+I*exp(I*(b*x+a))*(I*(I+d))^(1/2))*x+1/2/b^2*d/(I+d)*ln(1+I*(I+d)*exp(2*I*(b*x+a)))*x*a^2+1/12*I*x
^3*Pi*csgn(I*exp(2*I*(b*x+a))/(exp(2*I*(b*x+a))-1))^3+1/12*I*x^3*Pi*csgn(I*d/(exp(2*I*(b*x+a))-1)*exp(2*I*(b*x
+a)))^3-1/2*I/b^2*a^2/(I+d)*ln(1+I*exp(I*(b*x+a))*(I*(I+d))^(1/2))*x

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Maxima [B]  time = 1.37798, size = 463, normalized size = 2.74 \begin{align*} -\frac{\frac{12 \,{\left ({\left (b x + a\right )}^{3} - 3 \,{\left (b x + a\right )}^{2} a + 3 \,{\left (b x + a\right )} a^{2}\right )} \operatorname{arcoth}\left (d \cot \left (b x + a\right ) + i \, d - 1\right )}{b^{2}} + \frac{-3 i \,{\left (b x + a\right )}^{4} + 12 i \,{\left (b x + a\right )}^{3} a - 18 i \,{\left (b x + a\right )}^{2} a^{2} +{\left (-8 i \,{\left (b x + a\right )}^{3} + 18 i \,{\left (b x + a\right )}^{2} a - 18 i \,{\left (b x + a\right )} a^{2}\right )} \arctan \left (-d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), -d \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) +{\left (-12 i \,{\left (b x + a\right )}^{2} + 18 i \,{\left (b x + a\right )} a - 9 i \, a^{2}\right )}{\rm Li}_2\left ({\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) +{\left (4 \,{\left (b x + a\right )}^{3} - 9 \,{\left (b x + a\right )}^{2} a + 9 \,{\left (b x + a\right )} a^{2}\right )} \log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} +{\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \,{\left (4 \, b x + a\right )}{\rm Li}_{3}({\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}) + 6 i \,{\rm Li}_{4}({\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )})}{b^{2}}}{36 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccoth(1-I*d-d*cot(b*x+a)),x, algorithm="maxima")

[Out]

-1/36*(12*((b*x + a)^3 - 3*(b*x + a)^2*a + 3*(b*x + a)*a^2)*arccoth(d*cot(b*x + a) + I*d - 1)/b^2 + (-3*I*(b*x
+ a)^4 + 12*I*(b*x + a)^3*a - 18*I*(b*x + a)^2*a^2 + (-8*I*(b*x + a)^3 + 18*I*(b*x + a)^2*a - 18*I*(b*x + a)*
a^2)*arctan2(-d*cos(2*b*x + 2*a) + sin(2*b*x + 2*a), -d*sin(2*b*x + 2*a) - cos(2*b*x + 2*a) + 1) + (-12*I*(b*x
+ a)^2 + 18*I*(b*x + a)*a - 9*I*a^2)*dilog((-I*d + 1)*e^(2*I*b*x + 2*I*a)) + (4*(b*x + a)^3 - 9*(b*x + a)^2*a
+ 9*(b*x + a)*a^2)*log((d^2 + 1)*cos(2*b*x + 2*a)^2 + (d^2 + 1)*sin(2*b*x + 2*a)^2 - 2*d*sin(2*b*x + 2*a) - 2
*cos(2*b*x + 2*a) + 1) + 3*(4*b*x + a)*polylog(3, (-I*d + 1)*e^(2*I*b*x + 2*I*a)) + 6*I*polylog(4, (-I*d + 1)*
e^(2*I*b*x + 2*I*a)))/b^2)/b

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Fricas [C]  time = 1.68903, size = 504, normalized size = 2.98 \begin{align*} \frac{2 i \, b^{4} x^{4} - 4 \, b^{3} x^{3} \log \left (\frac{d e^{\left (2 i \, b x + 2 i \, a\right )}}{{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) + 6 i \, b^{2} x^{2}{\rm Li}_2\left (-{\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 2 i \, a^{4} + 4 \, a^{3} \log \left (\frac{{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}{d + i}\right ) - 6 \, b x{\rm polylog}\left (3,{\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 4 \,{\left (b^{3} x^{3} + a^{3}\right )} \log \left ({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 i \,{\rm polylog}\left (4,{\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccoth(1-I*d-d*cot(b*x+a)),x, algorithm="fricas")

[Out]

1/24*(2*I*b^4*x^4 - 4*b^3*x^3*log(d*e^(2*I*b*x + 2*I*a)/((d + I)*e^(2*I*b*x + 2*I*a) - I)) + 6*I*b^2*x^2*dilog
(-(I*d - 1)*e^(2*I*b*x + 2*I*a)) - 2*I*a^4 + 4*a^3*log(((d + I)*e^(2*I*b*x + 2*I*a) - I)/(d + I)) - 6*b*x*poly
log(3, (-I*d + 1)*e^(2*I*b*x + 2*I*a)) - 4*(b^3*x^3 + a^3)*log((I*d - 1)*e^(2*I*b*x + 2*I*a) + 1) - 3*I*polylo
g(4, (-I*d + 1)*e^(2*I*b*x + 2*I*a)))/b^3

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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(x**2*acoth(1-I*d-d*cot(b*x+a)),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{2} \operatorname{arcoth}\left (-d \cot \left (b x + a\right ) - i \, d + 1\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*arccoth(1-I*d-d*cot(b*x+a)),x, algorithm="giac")

[Out]

integrate(x^2*arccoth(-d*cot(b*x + a) - I*d + 1), x)