3.3.5 \(\int \coth ^{-1}(c+d \tanh (a+b x)) \, dx\) [205]

3.3.5.1 Optimal result

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

output

x*arccoth(c+d*tanh(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
 

3.3.5.2 Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \coth ^{-1}(c+d \tanh (a+b x)) \, dx=x \coth ^{-1}(c+d \tanh (a+b x))+\frac {2 b x \left (\log \left (1+\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )-\log \left (1+\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )-\operatorname {PolyLog}\left (2,-\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )}{4 b} \]

input

Integrate[ArcCoth[c + d*Tanh[a + b*x]],x]
 

output

x*ArcCoth[c + d*Tanh[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)
 

3.3.5.3 Rubi [A] (verified)

Time = 0.63 (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 = {6790, 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.

\(\displaystyle \int \coth ^{-1}(d \tanh (a+b x)+c) \, dx\)

\(\Big \downarrow \) 6790

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

input

Int[ArcCoth[c + d*Tanh[a + b*x]],x]
 

output

x*ArcCoth[c + d*Tanh[a + b*x]] + b*(1 - c - d)*((x*Log[1 + ((1 - c - d)*E^ 
(2*a + 2*b*x))/(1 - c + d)])/(2*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)*((x*L 
og[1 + ((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d)])/(2*b*(1 + c + d)) + Pol 
yLog[2, -(((1 + c + d)*E^(2*a + 2*b*x))/(1 + c - d))]/(4*b^2*(1 + c + d)))
 

3.3.5.3.1 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 6790

Int[ArcCoth[(c_.) + (d_.)*Tanh[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*Ar 
cCoth[c + d*Tanh[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]
 

3.3.5.4 Maple [B] (verified)

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

Time = 2.75 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.32

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccoth}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arccoth}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )-d \right )}{2}-\frac {d^{2} \left (\frac {-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}}{d}-\frac {-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
default	\(\frac {-\frac {\operatorname {arccoth}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arccoth}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )-d \right )}{2}-\frac {d^{2} \left (\frac {-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}}{d}-\frac {-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
risch	\(\text {Expression too large to display}\)	\(3011\)

input

int(arccoth(c+d*tanh(b*x+a)),x,method=_RETURNVERBOSE)
 

output

1/b/d*(-1/2*arccoth(c+d*tanh(b*x+a))*d*ln(-d*tanh(b*x+a)+d)+1/2*arccoth(c+ 
d*tanh(b*x+a))*d*ln(-d*tanh(b*x+a)-d)-1/2*d^2*(1/d*(-1/2*dilog((-d*tanh(b* 
x+a)-c-1)/(-1-c-d))-1/2*ln(-d*tanh(b*x+a)+d)*ln((-d*tanh(b*x+a)-c-1)/(-1-c 
-d))+1/2*dilog((-d*tanh(b*x+a)-c+1)/(1-c-d))+1/2*ln(-d*tanh(b*x+a)+d)*ln(( 
-d*tanh(b*x+a)-c+1)/(1-c-d)))-1/d*(-1/2*dilog((-d*tanh(b*x+a)-c-1)/(-1-c+d 
))-1/2*ln(-d*tanh(b*x+a)-d)*ln((-d*tanh(b*x+a)-c-1)/(-1-c+d))+1/2*dilog((- 
d*tanh(b*x+a)-c+1)/(1-c+d))+1/2*ln(-d*tanh(b*x+a)-d)*ln((-d*tanh(b*x+a)-c+ 
1)/(1-c+d)))))
 

3.3.5.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 551 vs. \(2 (128) = 256\).

Time = 0.28 (sec) , antiderivative size = 551, normalized size of antiderivative = 3.67 \[ \int \coth ^{-1}(c+d \tanh (a+b x)) \, dx=\frac {b x \log \left (\frac {{\left (c + 1\right )} \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}{{\left (c - 1\right )} \cosh \left (b x + a\right ) + d \sinh \left (b x + a\right )}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d + 1\right )} \sqrt {-\frac {c + d + 1}{c - d + 1}}\right ) + a \log \left (2 \, {\left (c + d + 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d + 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d + 1\right )} \sqrt {-\frac {c + d + 1}{c - d + 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) + 2 \, {\left (c - d - 1\right )} \sqrt {-\frac {c + d - 1}{c - d - 1}}\right ) - a \log \left (2 \, {\left (c + d - 1\right )} \cosh \left (b x + a\right ) + 2 \, {\left (c + d - 1\right )} \sinh \left (b x + a\right ) - 2 \, {\left (c - d - 1\right )} \sqrt {-\frac {c + d - 1}{c - d - 1}}\right ) - {\left (b x + a\right )} \log \left (\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\left (b x + a\right )} \log \left (-\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) + {\left (b x + a\right )} \log \left (-\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )} + 1\right ) - {\rm Li}_2\left (\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) - {\rm Li}_2\left (-\sqrt {-\frac {c + d + 1}{c - d + 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right ) + {\rm Li}_2\left (-\sqrt {-\frac {c + d - 1}{c - d - 1}} {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}\right )}{2 \, b} \]

input

integrate(arccoth(c+d*tanh(b*x+a)),x, algorithm="fricas")
 

output

1/2*(b*x*log(((c + 1)*cosh(b*x + a) + d*sinh(b*x + a))/((c - 1)*cosh(b*x + 
 a) + d*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* 
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))) - (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(s 
qrt(-(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) - dilog(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(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))))/b
 

3.3.5.6 Sympy [F]

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

input

integrate(acoth(c+d*tanh(b*x+a)),x)
 

output

Integral(acoth(c + d*tanh(a + b*x)), x)
 

3.3.5.7 Maxima [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.95 \[ \int \coth ^{-1}(c+d \tanh (a+b x)) \, dx=-\frac {1}{4} \, b d {\left (\frac {2 \, b x \log \left (\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (c + d + 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d + 1}\right )}{b^{2} d} - \frac {2 \, b x \log \left (\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1} + 1\right ) + {\rm Li}_2\left (-\frac {{\left (c + d - 1\right )} e^{\left (2 \, b x + 2 \, a\right )}}{c - d - 1}\right )}{b^{2} d}\right )} + x \operatorname {arcoth}\left (d \tanh \left (b x + a\right ) + c\right ) \]

input

integrate(arccoth(c+d*tanh(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*arccoth(d*tanh(b*x + a) + c)
 

3.3.5.8 Giac [F]

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

input

integrate(arccoth(c+d*tanh(b*x+a)),x, algorithm="giac")
 

output

integrate(arccoth(d*tanh(b*x + a) + c), x)
 

3.3.5.9 Mupad [F(-1)]

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

input

int(acoth(c + d*tanh(a + b*x)),x)
 

output

int(acoth(c + d*tanh(a + b*x)), x)