\(\int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx\) [343]

Optimal result

Integrand size = 19, antiderivative size = 133 \[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\frac {1}{6} i b x^3+\frac {1}{2} x^2 \text {arctanh}(1-i d-d \cot (a+b x))-\frac {1}{4} x^2 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac {i x \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (3,(1-i d) e^{2 i a+2 i b x}\right )}{8 b^2} \] Output:

1/6*I*b*x^3-1/2*x^2*arctanh(-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*polylo 
g(3,(1-I*d)*exp(2*I*a+2*I*b*x))/b^2
 

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.89 \[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\frac {1}{2} x^2 \text {arctanh}(1-i d-d \cot (a+b x))-\frac {2 b^2 x^2 \log \left (1+\frac {e^{-2 i (a+b x)}}{-1+i d}\right )+2 i b x \operatorname {PolyLog}\left (2,\frac {i e^{-2 i (a+b x)}}{i+d}\right )+\operatorname {PolyLog}\left (3,\frac {i e^{-2 i (a+b x)}}{i+d}\right )}{8 b^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.71 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.27, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {6819, 2615, 2620, 3011, 2720, 7143}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int x \text {arctanh}(d (-\cot (a+b x))-i d+1) \, dx\)

\(\Big \downarrow \) 6819

\(\displaystyle \frac {1}{2} i b \int \frac {x^2}{1-(1-i d) e^{2 i a+2 i b x}}dx+\frac {1}{2} x^2 \text {arctanh}(d (-\cot (a+b x))-i d+1)\)

\(\Big \downarrow \) 2615

\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1-i d) \int \frac {e^{2 i a+2 i b x} x^2}{1-(1-i d) e^{2 i a+2 i b x}}dx\right )+\frac {1}{2} x^2 \text {arctanh}(d (-\cot (a+b x))-i d+1)\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1-i d) \left (\frac {\int x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )dx}{b (d+i)}-\frac {x^2 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )}{2 b (d+i)}\right )\right )+\frac {1}{2} x^2 \text {arctanh}(d (-\cot (a+b x))-i d+1)\)

\(\Big \downarrow \) 3011

\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1-i d) \left (\frac {\frac {i x \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{2 b}-\frac {i \int \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )dx}{2 b}}{b (d+i)}-\frac {x^2 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )}{2 b (d+i)}\right )\right )+\frac {1}{2} x^2 \text {arctanh}(d (-\cot (a+b x))-i d+1)\)

\(\Big \downarrow \) 2720

\(\displaystyle \frac {1}{2} i b \left (\frac {x^3}{3}+(1-i d) \left (\frac {\frac {i x \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{2 b}-\frac {\int e^{-2 i a-2 i b x} \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )de^{2 i a+2 i b x}}{4 b^2}}{b (d+i)}-\frac {x^2 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )}{2 b (d+i)}\right )\right )+\frac {1}{2} x^2 \text {arctanh}(d (-\cot (a+b x))-i d+1)\)

\(\Big \downarrow \) 7143

\(\displaystyle \frac {1}{2} x^2 \text {arctanh}(d (-\cot (a+b x))-i d+1)+\frac {1}{2} i b \left (\frac {x^3}{3}+(1-i d) \left (\frac {\frac {i x \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{2 b}-\frac {\operatorname {PolyLog}\left (3,(1-i d) e^{2 i a+2 i b x}\right )}{4 b^2}}{b (d+i)}-\frac {x^2 \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )}{2 b (d+i)}\right )\right )\)

Input:

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

Output:

(x^2*ArcTanh[1 - I*d - d*Cot[a + b*x]])/2 + (I/2)*b*(x^3/3 + (1 - I*d)*(-1 
/2*(x^2*Log[1 - (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/(b*(I + d)) + (((I/2)* 
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)]/(4*b^2))/(b*(I + d))))
 

Defintions of rubi rules used

rule 2615

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] - Simp[ 
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 2620

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] - Si 
mp[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 2720

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] 
   Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct 
ionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ 
[{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 3011

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] + Simp[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 6819

Int[ArcTanh[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)]*((e_.) + (f_.)*(x_))^(m_ 
.), x_Symbol] :> Simp[(e + f*x)^(m + 1)*(ArcTanh[c + d*Cot[a + b*x]]/(f*(m 
+ 1))), x] + Simp[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 7143

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S 
ymbol] :> 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]
 

Maple [C] (warning: unable to verify)

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

Time = 3.55 (sec) , antiderivative size = 2187, normalized size of antiderivative = 16.44

method	result	size
risch	\(\text {Expression too large to display}\)	\(2187\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.18 \[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\frac {4 i \, b^{3} x^{3} - 6 \, b^{2} x^{2} \log \left (-\frac {d e^{\left (2 i \, b x + 2 i \, a\right )}}{{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) + 4 i \, a^{3} + 6 i \, b x {\rm Li}_2\left (-{\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, a^{2} \log \left (\frac {{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}{d + i}\right ) - 6 \, {\left (b^{2} x^{2} - a^{2}\right )} \log \left ({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) - 3 \, {\rm polylog}\left (3, {\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{24 \, b^{2}} \] Input:

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

Output:

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

\[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=- \int x \operatorname {atanh}{\left (d \cot {\left (a + b x \right )} + i d - 1 \right )}\, dx \] Input:

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

Output:

-Integral(x*atanh(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. 250 vs. \(2 (94) = 188\).

Time = 0.06 (sec) , antiderivative size = 250, normalized size of antiderivative = 1.88 \[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=-\frac {\frac {12 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \operatorname {artanh}\left (d \cot \left (b x + a\right ) + i \, d - 1\right )}{b} + \frac {-4 i \, {\left (b x + a\right )}^{3} + 12 i \, {\left (b x + a\right )}^{2} a - 6 i \, b x {\rm Li}_2\left ({\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right ) - 6 \, {\left (i \, {\left (b x + a\right )}^{2} - 2 i \, {\left (b x + a\right )} a\right )} \arctan \left (-d \cos \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right ), -d \sin \left (2 \, b x + 2 \, a\right ) - \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\left ({\left (b x + a\right )}^{2} - 2 \, {\left (b x + a\right )} a\right )} \log \left ({\left (d^{2} + 1\right )} \cos \left (2 \, b x + 2 \, a\right )^{2} + {\left (d^{2} + 1\right )} \sin \left (2 \, b x + 2 \, a\right )^{2} - 2 \, d \sin \left (2 \, b x + 2 \, a\right ) - 2 \, \cos \left (2 \, b x + 2 \, a\right ) + 1\right ) + 3 \, {\rm Li}_{3}({\left (-i \, d + 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )})}{b}}{24 \, b} \] Input:

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

Output:

-1/24*(12*((b*x + a)^2 - 2*(b*x + a)*a)*arctanh(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]

\[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\int { -x \operatorname {artanh}\left (d \cot \left (b x + a\right ) + i \, d - 1\right ) \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

Timed out. \[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\int -x\,\mathrm {atanh}\left (d\,\mathrm {cot}\left (a+b\,x\right )-1+d\,1{}\mathrm {i}\right ) \,d x \] Input:

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

Output:

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

Reduce [F]

\[ \int x \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=-\left (\int \mathit {atanh} \left (\cot \left (b x +a \right ) d +d i -1\right ) x d x \right ) \] Input:

int(-x*atanh(-1+I*d+d*cot(b*x+a)),x)
 

Output:

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