\(\int \text {arctanh}(c+d \tanh (a+b x)) \, dx\) [286]

Optimal result

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

x*arctanh(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
 

Mathematica [A] (verified)

Time = 3.34 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.87 \[ \int \text {arctanh}(c+d \tanh (a+b x)) \, dx=x \text {arctanh}(c+d \tanh (a+b x))+\frac {2 b x \left (\log \left (1+\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )-\log \left (1+\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )\right )+\operatorname {PolyLog}\left (2,-\frac {(-1+c+d) e^{2 (a+b x)}}{-1+c-d}\right )-\operatorname {PolyLog}\left (2,-\frac {(1+c+d) e^{2 (a+b x)}}{1+c-d}\right )}{4 b} \] Input:

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

Output:

x*ArcTanh[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)
 

Rubi [A] (verified)

Time = 1.06 (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 = {6789, 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.

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

\(\Big \downarrow \) 6789

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

\(\Big \downarrow \) 2620

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

\(\Big \downarrow \) 2715

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

\(\Big \downarrow \) 2838

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

Input:

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

Output:

x*ArcTanh[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)))
 

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 6789

Int[ArcTanh[(c_.) + (d_.)*Tanh[(a_.) + (b_.)*(x_)]], x_Symbol] :> Simp[x*Ar 
cTanh[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]
 

Maple [B] (verified)

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

Time = 3.83 (sec) , antiderivative size = 348, normalized size of antiderivative = 2.32

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arctanh}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arctanh}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )-d \right )}{2}+\frac {d^{2} \left (\frac {\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}}{d}-\frac {-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
default	\(\frac {-\frac {\operatorname {arctanh}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )+d \right )}{2}+\frac {\operatorname {arctanh}\left (c +d \tanh \left (b x +a \right )\right ) d \ln \left (-d \tanh \left (b x +a \right )-d \right )}{2}+\frac {d^{2} \left (\frac {\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c +d}\right )}{2}-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )-d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c +d}\right )}{2}}{d}-\frac {-\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}-\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c -1}{-1-c -d}\right )}{2}+\frac {\operatorname {dilog}\left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}+\frac {\ln \left (-d \tanh \left (b x +a \right )+d \right ) \ln \left (\frac {-d \tanh \left (b x +a \right )-c +1}{1-c -d}\right )}{2}}{d}\right )}{2}}{b d}\)	\(348\)
risch	\(\text {Expression too large to display}\)	\(3079\)

Input:

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

Output:

1/b/d*(-1/2*arctanh(c+d*tanh(b*x+a))*d*ln(-d*tanh(b*x+a)+d)+1/2*arctanh(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)))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 552 vs. \(2 (128) = 256\).

Time = 0.12 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.68 \[ \int \text {arctanh}(c+d \tanh (a+b x)) \, dx =\text {Too large to display} \] Input:

integrate(arctanh(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)*s 
qrt(-(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( 
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) - 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
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

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

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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