\(\int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx\) [258]

Optimal result

Integrand size = 18, antiderivative size = 132 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (3,(1+i d) e^{2 i a+2 i b x}\right )}{8 b^2} \]

[Out]

1/6*I*b*x^3+1/2*x^2*arccoth(1+I*d+d*cot(b*x+a))-1/4*x^2*ln(1-(1+I*d)*exp(2*I*a+2*I*b*x))+1/4*I*x*polylog(2,(1+
I*d)*exp(2*I*a+2*I*b*x))/b-1/8*polylog(3,(1+I*d)*exp(2*I*a+2*I*b*x))/b^2

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6401, 2215, 2221, 2611, 2320, 6724} \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=-\frac {\operatorname {PolyLog}\left (3,(i d+1) e^{2 i a+2 i b x}\right )}{8 b^2}+\frac {i x \operatorname {PolyLog}\left (2,(i d+1) e^{2 i a+2 i b x}\right )}{4 b}-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {1}{2} x^2 \coth ^{-1}(d \cot (a+b x)+i d+1)+\frac {1}{6} i b x^3 \]

[In]

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

[Out]

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

Rule 2215

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 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 6401

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 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]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))+\frac {1}{2} (i b) \int \frac {x^2}{1+(-1-i d) e^{2 i a+2 i b x}} \, dx \\ & = \frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))+\frac {1}{2} (b (i-d)) \int \frac {e^{2 i a+2 i b x} x^2}{1+(-1-i d) e^{2 i a+2 i b x}} \, dx \\ & = \frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {1}{2} \int x \log \left (1+(-1-i d) e^{2 i a+2 i b x}\right ) \, dx \\ & = \frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {i \int \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right ) \, dx}{4 b} \\ & = \frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {\text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,(1+i d) x)}{x} \, dx,x,e^{2 i a+2 i b x}\right )}{8 b^2} \\ & = \frac {1}{6} i b x^3+\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1+i d) e^{2 i a+2 i b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,(1+i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (3,(1+i d) e^{2 i a+2 i b x}\right )}{8 b^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.90 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {1}{2} x^2 \coth ^{-1}(1+i d+d \cot (a+b x))-\frac {2 b^2 x^2 \log \left (1+\frac {i e^{-2 i (a+b x)}}{-i+d}\right )+2 i b x \operatorname {PolyLog}\left (2,-\frac {i e^{-2 i (a+b x)}}{-i+d}\right )+\operatorname {PolyLog}\left (3,-\frac {i e^{-2 i (a+b x)}}{-i+d}\right )}{8 b^2} \]

[In]

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

[Out]

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

Maple [C] (warning: unable to verify)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 2.09 (sec) , antiderivative size = 2285, normalized size of antiderivative = 17.31

method	result	size
risch	\(\text {Expression too large to display}\)	\(2285\)

[In]

int(x*arccoth(1+I*d+d*cot(b*x+a)),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.18 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {4 i \, b^{3} x^{3} + 6 \, b^{2} x^{2} \log \left (\frac {{\left ({\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i\right )} e^{\left (-2 i \, b x - 2 i \, a\right )}}{d}\right ) + 4 i \, a^{3} + 6 i \, b x {\rm Li}_2\left (-{\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, a^{2} \log \left (\frac {{\left (d - i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + i}{d - i}\right ) - 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left ({\left (-i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \, {\rm polylog}\left (3, {\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\int x \operatorname {acoth}{\left (d \cot {\left (a + b x \right )} + i d + 1 \right )}\, dx \]

[In]

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

[Out]

Integral(x*acoth(d*cot(a + b*x) + I*d + 1), 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. 249 vs. \(2 (94) = 188\).

Time = 0.23 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.89 \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\frac {\frac {12 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right )}{b} - \frac {-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 6 i \, b x {\rm Li}_2\left ({\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), d \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} + 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\rm Li}_{3}({\left (i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )})}{b}}{24 \, b} \]

[In]

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

[Out]

1/24*(12*((b*x + a)^2 - 2*(b*x + a)*a)*arccoth(d*cot(b*x + a) + I*d + 1)/b - (-4*I*(b*x + a)^3 + 12*I*(b*x + a
)^2*a - 6*I*b*x*dilog((I*d + 1)*e^(2*I*b*x + 2*I*a)) - 6*(I*(b*x + a)^2 - 2*I*(b*x + a)*a)*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) + 3*((b*x + a)^2 - 2*(b*x + a)*a)*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*polylog(3, (I*d + 1)*e^(2*I*b*x + 2*I*a)))/b)/b

Giac [F]

\[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\int { x \operatorname {arcoth}\left (d \cot \left (b x + a\right ) + i \, d + 1\right ) \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int x \coth ^{-1}(1+i d+d \cot (a+b x)) \, dx=\int x\,\mathrm {acoth}\left (d\,\mathrm {cot}\left (a+b\,x\right )+1+d\,1{}\mathrm {i}\right ) \,d x \]

[In]

int(x*acoth(d*1i + d*cot(a + b*x) + 1),x)

[Out]

int(x*acoth(d*1i + d*cot(a + b*x) + 1), x)