\(\int \text {arctanh}(1-i d-d \cot (a+b x)) \, dx\) [344]

Optimal result

Integrand size = 17, antiderivative size = 94 \[ \int \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\frac {1}{2} i b x^2+x \text {arctanh}(1-i d-d \cot (a+b x))-\frac {1}{2} x \log \left (1-(1-i d) e^{2 i a+2 i b x}\right )+\frac {i \operatorname {PolyLog}\left (2,(1-i d) e^{2 i a+2 i b x}\right )}{4 b} \] Output:

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

Mathematica [B] (warning: unable to verify)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(605\) vs. \(2(94)=188\).

Time = 8.98 (sec) , antiderivative size = 605, normalized size of antiderivative = 6.44 \[ \int \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=x \text {arctanh}(1-i d-d \cot (a+b x))+\frac {x \csc ^2(a+b x) \left (2 b x \log (2 \cos (b x) (\cos (b x)-i \sin (b x)))+i \log \left (\frac {\sec (b x) (\cos (a)-i \sin (a)) (d \cos (a+b x)+i (2 i+d) \sin (a+b x))}{2 (i+d)}\right ) \log (1-i \tan (b x))-i \log \left (\frac {i \sec (b x) (d \cos (a+b x)+i (2 i+d) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right ) \log (1+i \tan (b x))+i \operatorname {PolyLog}(2,-\cos (2 b x)+i \sin (2 b x))-i \operatorname {PolyLog}\left (2,\frac {1}{2} \sec (b x) ((2-i d) \cos (a)+d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right )+i \operatorname {PolyLog}\left (2,\frac {(\cos (a)-i \sin (a)) ((2-i d) \cos (a)+d \sin (a)) (i+\tan (b x))}{2 (i+d)}\right )\right ) (\cos (b x)-i \sin (b x)) (\cos (b x)+i \sin (b x))}{(i+\cot (a+b x)) (-2+i d+d \cot (a+b x)) \left (-\frac {\log (1-i \tan (b x)) \sec (b x) ((2 i+d) \cos (a)+i d \sin (a))}{d \cos (a+b x)+i (2 i+d) \sin (a+b x)}+\frac {\log (1+i \tan (b x)) \sec (b x) ((2 i+d) \cos (a)+i d \sin (a))}{d \cos (a+b x)+i (2 i+d) \sin (a+b x)}+\frac {\log \left (\frac {i \sec (b x) (d \cos (a+b x)+i (2 i+d) \sin (a+b x))}{2 \cos (a)-2 i \sin (a)}\right ) \sec ^2(b x)}{1+i \tan (b x)}-2 b x (i+\tan (b x))+i \log \left (1-\frac {1}{2} \sec (b x) ((2-i d) \cos (a)+d \sin (a)) (\cos (a+b x)+i \sin (a+b x))\right ) (i+\tan (b x))\right )} \] Input:

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

Output:

x*ArcTanh[1 - I*d - d*Cot[a + b*x]] + (x*Csc[a + b*x]^2*(2*b*x*Log[2*Cos[b 
*x]*(Cos[b*x] - I*Sin[b*x])] + I*Log[(Sec[b*x]*(Cos[a] - I*Sin[a])*(d*Cos[ 
a + b*x] + I*(2*I + d)*Sin[a + b*x]))/(2*(I + d))]*Log[1 - I*Tan[b*x]] - I 
*Log[(I*Sec[b*x]*(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x]))/(2*Cos[a] - 
(2*I)*Sin[a])]*Log[1 + I*Tan[b*x]] + I*PolyLog[2, -Cos[2*b*x] + I*Sin[2*b* 
x]] - I*PolyLog[2, (Sec[b*x]*((2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + 
 I*Sin[a + b*x]))/2] + I*PolyLog[2, ((Cos[a] - I*Sin[a])*((2 - I*d)*Cos[a] 
 + d*Sin[a])*(I + Tan[b*x]))/(2*(I + d))])*(Cos[b*x] - I*Sin[b*x])*(Cos[b* 
x] + I*Sin[b*x]))/((I + Cot[a + b*x])*(-2 + I*d + d*Cot[a + b*x])*(-((Log[ 
1 - I*Tan[b*x]]*Sec[b*x]*((2*I + d)*Cos[a] + I*d*Sin[a]))/(d*Cos[a + b*x] 
+ I*(2*I + d)*Sin[a + b*x])) + (Log[1 + I*Tan[b*x]]*Sec[b*x]*((2*I + d)*Co 
s[a] + I*d*Sin[a]))/(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x]) + (Log[(I* 
Sec[b*x]*(d*Cos[a + b*x] + I*(2*I + d)*Sin[a + b*x]))/(2*Cos[a] - (2*I)*Si 
n[a])]*Sec[b*x]^2)/(1 + I*Tan[b*x]) - 2*b*x*(I + Tan[b*x]) + I*Log[1 - (Se 
c[b*x]*((2 - I*d)*Cos[a] + d*Sin[a])*(Cos[a + b*x] + I*Sin[a + b*x]))/2]*( 
I + Tan[b*x])))
 

Rubi [A] (verified)

Time = 0.48 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.31, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {6811, 2615, 2620, 2715, 2838}

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.

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

\(\Big \downarrow \) 6811

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

\(\Big \downarrow \) 2615

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

x*ArcTanh[1 - I*d - d*Cot[a + b*x]] + I*b*(x^2/2 + (1 - I*d)*(-1/2*(x*Log[ 
1 - (1 - I*d)*E^((2*I)*a + (2*I)*b*x)])/(b*(I + d)) + ((I/4)*PolyLog[2, (1 
 - I*d)*E^((2*I)*a + (2*I)*b*x)])/(b^2*(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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 6811

Int[ArcTanh[(c_.) + Cot[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*Arc 
Tanh[c + d*Cot[a + b*x]], x] + Simp[I*b   Int[x/(c - I*d - c*E^(2*I*a + 2*I 
*b*x)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[(c - I*d)^2, 1]
 

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (77 ) = 154\).

Time = 4.68 (sec) , antiderivative size = 332, normalized size of antiderivative = 3.53

method	result	size
derivativedivides	\(-\frac {-\frac {i \operatorname {arctanh}\left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d +d \cot \left (b x +a \right )\right )}{2}+\frac {i \operatorname {arctanh}\left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d -d \cot \left (b x +a \right )\right )}{2}-\frac {d^{2} \left (-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 i+2 d \right )\right )}{2 i+2 d}\right )}{2}+\frac {\ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 i+2 d \right )\right )}{2 i+2 d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}-\frac {\ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}\right )}{d}+\frac {i \left (-\frac {\ln \left (i d +d \cot \left (b x +a \right )\right )^{2}}{4}+\frac {\left (\ln \left (i d +d \cot \left (b x +a \right )\right )-\ln \left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )\right ) \ln \left (1-\frac {i d}{2}-\frac {d \cot \left (b x +a \right )}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2}\right )}{d}\right )}{2}}{b d}\)	\(332\)
default	\(-\frac {-\frac {i \operatorname {arctanh}\left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d +d \cot \left (b x +a \right )\right )}{2}+\frac {i \operatorname {arctanh}\left (-1+i d +d \cot \left (b x +a \right )\right ) d \ln \left (i d -d \cot \left (b x +a \right )\right )}{2}-\frac {d^{2} \left (-\frac {i \left (\frac {\operatorname {dilog}\left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 i+2 d \right )\right )}{2 i+2 d}\right )}{2}+\frac {\ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (i d -d \cot \left (b x +a \right )-i \left (2 i+2 d \right )\right )}{2 i+2 d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}-\frac {\ln \left (i d -d \cot \left (b x +a \right )\right ) \ln \left (\frac {i \left (-i d -d \cot \left (b x +a \right )\right )}{2 d}\right )}{2}\right )}{d}+\frac {i \left (-\frac {\ln \left (i d +d \cot \left (b x +a \right )\right )^{2}}{4}+\frac {\left (\ln \left (i d +d \cot \left (b x +a \right )\right )-\ln \left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )\right ) \ln \left (1-\frac {i d}{2}-\frac {d \cot \left (b x +a \right )}{2}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {i d}{2}+\frac {d \cot \left (b x +a \right )}{2}\right )}{2}\right )}{d}\right )}{2}}{b d}\)	\(332\)
risch	\(\text {Expression too large to display}\)	\(1623\)

Input:

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

Output:

-1/b/d*(-1/2*I*arctanh(-1+I*d+d*cot(b*x+a))*d*ln(I*d+d*cot(b*x+a))+1/2*I*a 
rctanh(-1+I*d+d*cot(b*x+a))*d*ln(I*d-d*cot(b*x+a))-1/2*d^2*(-I/d*(1/2*dilo 
g(I*(I*d-d*cot(b*x+a)-I*(2*I+2*d))/(2*I+2*d))+1/2*ln(I*d-d*cot(b*x+a))*ln( 
I*(I*d-d*cot(b*x+a)-I*(2*I+2*d))/(2*I+2*d))-1/2*dilog(1/2*I*(-I*d-d*cot(b* 
x+a))/d)-1/2*ln(I*d-d*cot(b*x+a))*ln(1/2*I*(-I*d-d*cot(b*x+a))/d))+I/d*(-1 
/4*ln(I*d+d*cot(b*x+a))^2+1/2*(ln(I*d+d*cot(b*x+a))-ln(1/2*I*d+1/2*d*cot(b 
*x+a)))*ln(1-1/2*I*d-1/2*d*cot(b*x+a))-1/2*dilog(1/2*I*d+1/2*d*cot(b*x+a)) 
)))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.30 \[ \int \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=\frac {2 i \, b^{2} x^{2} - 2 \, b x \log \left (-\frac {d e^{\left (2 i \, b x + 2 i \, a\right )}}{{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}\right ) - 2 i \, a^{2} - 2 \, {\left (b x + a\right )} \log \left ({\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )} + 1\right ) + 2 \, a \log \left (\frac {{\left (d + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )} - i}{d + i}\right ) + i \, {\rm Li}_2\left (-{\left (i \, d - 1\right )} e^{\left (2 i \, b x + 2 i \, a\right )}\right )}{4 \, b} \] Input:

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

Output:

1/4*(2*I*b^2*x^2 - 2*b*x*log(-d*e^(2*I*b*x + 2*I*a)/((d + I)*e^(2*I*b*x + 
2*I*a) - I)) - 2*I*a^2 - 2*(b*x + a)*log((I*d - 1)*e^(2*I*b*x + 2*I*a) + 1 
) + 2*a*log(((d + I)*e^(2*I*b*x + 2*I*a) - I)/(d + I)) + I*dilog(-(I*d - 1 
)*e^(2*I*b*x + 2*I*a)))/b
 

Sympy [F]

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

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

Output:

-Integral(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. 286 vs. \(2 (67) = 134\).

Time = 0.12 (sec) , antiderivative size = 286, normalized size of antiderivative = 3.04 \[ \int \text {arctanh}(1-i d-d \cot (a+b x)) \, dx=-\frac {4 \, {\left (b x + a\right )} d {\left (\frac {\log \left ({\left (i \, d - 2\right )} \tan \left (b x + a\right ) + d\right )}{d} - \frac {\log \left (i \, \tan \left (b x + a\right ) + 1\right )}{d}\right )} + d {\left (-\frac {2 i \, {\left (\log \left ({\left (i \, d - 2\right )} \tan \left (b x + a\right ) + d\right ) \log \left (\frac {{\left (d + 2 i\right )} \tan \left (b x + a\right ) - i \, d}{2 i \, d - 2} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (d + 2 i\right )} \tan \left (b x + a\right ) - i \, d}{2 i \, d - 2}\right )\right )}}{d} - \frac {2 i \, {\left (\log \left (-\frac {1}{2} \, {\left (d + 2 i\right )} \tan \left (b x + a\right ) + \frac {1}{2} i \, d\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) + {\rm Li}_2\left (\frac {1}{2} \, {\left (d + 2 i\right )} \tan \left (b x + a\right ) - \frac {1}{2} i \, d + 1\right )\right )}}{d} + \frac {2 i \, \log \left ({\left (i \, d - 2\right )} \tan \left (b x + a\right ) + d\right ) \log \left (i \, \tan \left (b x + a\right ) + 1\right ) - i \, \log \left (i \, \tan \left (b x + a\right ) + 1\right )^{2}}{d} + \frac {2 i \, {\left (\log \left (i \, \tan \left (b x + a\right ) + 1\right ) \log \left (-\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right ) + {\rm Li}_2\left (\frac {1}{2} i \, \tan \left (b x + a\right ) + \frac {1}{2}\right )\right )}}{d}\right )} + 8 \, {\left (b x + a\right )} \operatorname {artanh}\left (i \, d + \frac {d}{\tan \left (b x + a\right )} - 1\right )}{8 \, b} \] Input:

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

Output:

-1/8*(4*(b*x + a)*d*(log((I*d - 2)*tan(b*x + a) + d)/d - log(I*tan(b*x + a 
) + 1)/d) + d*(-2*I*(log((I*d - 2)*tan(b*x + a) + d)*log(((d + 2*I)*tan(b* 
x + a) - I*d)/(2*I*d - 2) + 1) + dilog(-((d + 2*I)*tan(b*x + a) - I*d)/(2* 
I*d - 2)))/d - 2*I*(log(-1/2*(d + 2*I)*tan(b*x + a) + 1/2*I*d)*log(I*tan(b 
*x + a) + 1) + dilog(1/2*(d + 2*I)*tan(b*x + a) - 1/2*I*d + 1))/d + (2*I*l 
og((I*d - 2)*tan(b*x + a) + d)*log(I*tan(b*x + a) + 1) - I*log(I*tan(b*x + 
 a) + 1)^2)/d + 2*I*(log(I*tan(b*x + a) + 1)*log(-1/2*I*tan(b*x + a) + 1/2 
) + dilog(1/2*I*tan(b*x + a) + 1/2))/d) + 8*(b*x + a)*arctanh(I*d + d/tan( 
b*x + a) - 1))/b
 

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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