\(\int \text {arctanh}(c+d \coth (a+b x)) \, dx\) [300]

Optimal result

Integrand size = 11, antiderivative size = 150 \[ \int \text {arctanh}(c+d \coth (a+b x)) \, dx=x \text {arctanh}(c+d \coth (a+b x))+\frac {1}{2} x \log \left (1-\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )-\frac {1}{2} x \log \left (1-\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )+\frac {\operatorname {PolyLog}\left (2,\frac {(1-c-d) e^{2 a+2 b x}}{1-c+d}\right )}{4 b}-\frac {\operatorname {PolyLog}\left (2,\frac {(1+c+d) e^{2 a+2 b x}}{1+c-d}\right )}{4 b} \] Output:

x*arctanh(c+d*coth(b*x+a))+1/2*x*ln(1-(1-c-d)*exp(2*b*x+2*a)/(1-c+d))-1/2* 
x*ln(1-(1+c+d)*exp(2*b*x+2*a)/(1+c-d))+1/4*polylog(2,(1-c-d)*exp(2*b*x+2*a 
)/(1-c+d))/b-1/4*polylog(2,(1+c+d)*exp(2*b*x+2*a)/(1+c-d))/b
 

Mathematica [A] (verified)

Time = 3.11 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \text {arctanh}(c+d \coth (a+b x)) \, dx=x \text {arctanh}(c+d \coth (a+b x))-\frac {-2 b x \left (\log \left (1-\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )-\log \left (1-\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )\right )-\operatorname {PolyLog}\left (2,\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )+\operatorname {PolyLog}\left (2,\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )}{4 b} \] Input:

Integrate[ArcTanh[c + d*Coth[a + b*x]],x]
 

Output:

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

Rubi [A] (verified)

Time = 0.65 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.38, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.364, Rules used = {6791, 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.

\(\displaystyle \int \text {arctanh}(d \coth (a+b x)+c) \, dx\)

\(\Big \downarrow \) 6791

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

Int[ArcTanh[c + d*Coth[a + b*x]],x]
 

Output:

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

Defintions of rubi rules used

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 6791

Int[ArcTanh[(c_.) + Coth[(a_.) + (b_.)*(x_)]*(d_.)], x_Symbol] :> Simp[x*Ar 
cTanh[c + d*Coth[a + b*x]], x] + (-Simp[b*(1 - c - d)   Int[x*(E^(2*a + 2*b 
*x)/(1 - c + d - (1 - c - d)*E^(2*a + 2*b*x))), x], x] + Simp[b*(1 + c + d) 
   Int[x*(E^(2*a + 2*b*x)/(1 + c - d - (1 + c + d)*E^(2*a + 2*b*x))), x], x 
]) /; FreeQ[{a, b, c, d}, x] && NeQ[(c - d)^2, 1]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(347\) vs. \(2(138)=276\).

Time = 3.96 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.32

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arctanh}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )-d \right )}{2}+\frac {d^{2} \left (\frac {\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}}{d}-\frac {\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arctanh}\left (c +d \coth \left (b x +a \right )\right ) d \ln \left (-d \coth \left (b x +a \right )-d \right )}{2}+\frac {d^{2} \left (\frac {\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c +d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )-d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}}{d}-\frac {\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c +1}{1-c -d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \coth \left (b x +a \right )+d \right ) \ln \left (\frac {-d \coth \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
risch	\(\text {Expression too large to display}\)	\(3019\)

Input:

int(arctanh(c+d*coth(b*x+a)),x,method=_RETURNVERBOSE)
 

Output:

1/b/d*(-1/2*arctanh(c+d*coth(b*x+a))*d*ln(-d*coth(b*x+a)+d)+1/2*arctanh(c+ 
d*coth(b*x+a))*d*ln(-d*coth(b*x+a)-d)+1/2*d^2*(1/d*(1/2*dilog((-d*coth(b*x 
+a)-c+1)/(1-c+d))+1/2*ln(-d*coth(b*x+a)-d)*ln((-d*coth(b*x+a)-c+1)/(1-c+d) 
)-1/2*dilog((-d*coth(b*x+a)-c-1)/(-1-c+d))-1/2*ln(-d*coth(b*x+a)-d)*ln((-d 
*coth(b*x+a)-c-1)/(-1-c+d)))-1/d*(1/2*dilog((-d*coth(b*x+a)-c+1)/(1-c-d))+ 
1/2*ln(-d*coth(b*x+a)+d)*ln((-d*coth(b*x+a)-c+1)/(1-c-d))-1/2*dilog((-d*co 
th(b*x+a)-c-1)/(-1-c-d))-1/2*ln(-d*coth(b*x+a)+d)*ln((-d*coth(b*x+a)-c-1)/ 
(-1-c-d)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 540 vs. \(2 (128) = 256\).

Time = 0.12 (sec) , antiderivative size = 540, normalized size of antiderivative = 3.60 \[ \int \text {arctanh}(c+d \coth (a+b x)) \, dx =\text {Too large to display} \] Input:

integrate(arctanh(c+d*coth(b*x+a)),x, algorithm="fricas")
 

Output:

1/2*(b*x*log(-(d*cosh(b*x + a) + (c + 1)*sinh(b*x + a))/(d*cosh(b*x + a) + 
 (c - 1)*sinh(b*x + a))) + a*log(2*(c + d + 1)*cosh(b*x + a) + 2*(c + d + 
1)*sinh(b*x + a) + 2*(c - d + 1)*sqrt((c + d + 1)/(c - d + 1))) + a*log(2* 
(c + d + 1)*cosh(b*x + a) + 2*(c + d + 1)*sinh(b*x + a) - 2*(c - d + 1)*sq 
rt((c + d + 1)/(c - d + 1))) - a*log(2*(c + d - 1)*cosh(b*x + a) + 2*(c + 
d - 1)*sinh(b*x + a) + 2*(c - d - 1)*sqrt((c + d - 1)/(c - d - 1))) - a*lo 
g(2*(c + d - 1)*cosh(b*x + a) + 2*(c + d - 1)*sinh(b*x + a) - 2*(c - d - 1 
)*sqrt((c + d - 1)/(c - d - 1))) - (b*x + a)*log(sqrt((c + d + 1)/(c - d + 
 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) - (b*x + a)*log(-sqrt((c + d + 1 
)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (b*x + a)*log(sqrt(( 
c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) + (b*x + a)*l 
og(-sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a)) + 1) - d 
ilog(sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a))) - dilo 
g(-sqrt((c + d + 1)/(c - d + 1))*(cosh(b*x + a) + sinh(b*x + a))) + dilog( 
sqrt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))) + dilog(-sq 
rt((c + d - 1)/(c - d - 1))*(cosh(b*x + a) + sinh(b*x + a))))/b
 

Sympy [F]

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

integrate(atanh(c+d*coth(b*x+a)),x)
 

Output:

Integral(atanh(c + d*coth(a + b*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.30 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \text {arctanh}(c+d \coth (a+b x)) \, dx=-\frac {1}{4} \, b d {\left (\frac {2 \, b x \log \left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + {\rm Li}_2\left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right )}{b^{2} d} - \frac {2 \, b x \log \left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + {\rm Li}_2\left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right )}{b^{2} d}\right )} + x \operatorname {artanh}\left (d \coth \left (b x + a\right ) + c\right ) \] Input:

integrate(arctanh(c+d*coth(b*x+a)),x, algorithm="maxima")
 

Output:

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

Giac [F]

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

integrate(arctanh(c+d*coth(b*x+a)),x, algorithm="giac")
 

Output:

integrate(arctanh(d*coth(b*x + a) + c), x)
 

Mupad [F(-1)]

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

int(atanh(c + d*coth(a + b*x)),x)
 

Output:

int(atanh(c + d*coth(a + b*x)), x)
 

Reduce [F]

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

int(atanh(c+d*coth(b*x+a)),x)
 

Output:

int(atanh(coth(a + b*x)*d + c),x)