\(\int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx\) [108]

Optimal result

Integrand size = 13, antiderivative size = 97 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=-\frac {1}{2 (b x-\text {arctanh}(\tanh (a+b x))) \text {arctanh}(\tanh (a+b x))^2}+\frac {1}{(b x-\text {arctanh}(\tanh (a+b x)))^2 \text {arctanh}(\tanh (a+b x))}-\frac {\log (x)}{(b x-\text {arctanh}(\tanh (a+b x)))^3}+\frac {\log (\text {arctanh}(\tanh (a+b x)))}{(b x-\text {arctanh}(\tanh (a+b x)))^3} \] Output:

-1/2/(b*x-arctanh(tanh(b*x+a)))/arctanh(tanh(b*x+a))^2+1/(b*x-arctanh(tanh 
(b*x+a)))^2/arctanh(tanh(b*x+a))-ln(x)/(b*x-arctanh(tanh(b*x+a)))^3+ln(arc 
tanh(tanh(b*x+a)))/(b*x-arctanh(tanh(b*x+a)))^3
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.76 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=\frac {b^2 x^2-4 b x \text {arctanh}(\tanh (a+b x))+\text {arctanh}(\tanh (a+b x))^2 (3+2 \log (b x)-2 \log (\text {arctanh}(\tanh (a+b x))))}{2 \text {arctanh}(\tanh (a+b x))^2 (-b x+\text {arctanh}(\tanh (a+b x)))^3} \] Input:

Integrate[1/(x*ArcTanh[Tanh[a + b*x]]^3),x]
 

Output:

(b^2*x^2 - 4*b*x*ArcTanh[Tanh[a + b*x]] + ArcTanh[Tanh[a + b*x]]^2*(3 + 2* 
Log[b*x] - 2*Log[ArcTanh[Tanh[a + b*x]]]))/(2*ArcTanh[Tanh[a + b*x]]^2*(-( 
b*x) + ArcTanh[Tanh[a + b*x]])^3)
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.38, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2594, 2594, 2591, 14, 2588, 14}

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.

\(\displaystyle \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx\)

\(\Big \downarrow \) 2594

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

\(\Big \downarrow \) 2594

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

\(\Big \downarrow \) 2591

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

\(\Big \downarrow \) 14

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

\(\Big \downarrow \) 2588

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

\(\Big \downarrow \) 14

\(\displaystyle -\frac {1}{2 (b x-\text {arctanh}(\tanh (a+b x))) \text {arctanh}(\tanh (a+b x))^2}-\frac {-\frac {1}{(b x-\text {arctanh}(\tanh (a+b x))) \text {arctanh}(\tanh (a+b x))}-\frac {\frac {\log (\text {arctanh}(\tanh (a+b x)))}{b x-\text {arctanh}(\tanh (a+b x))}-\frac {\log (x)}{b x-\text {arctanh}(\tanh (a+b x))}}{b x-\text {arctanh}(\tanh (a+b x))}}{b x-\text {arctanh}(\tanh (a+b x))}\)

Input:

Int[1/(x*ArcTanh[Tanh[a + b*x]]^3),x]
 

Output:

-1/2*1/((b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]]^2) - (-(1/(( 
b*x - ArcTanh[Tanh[a + b*x]])*ArcTanh[Tanh[a + b*x]])) - (-(Log[x]/(b*x - 
ArcTanh[Tanh[a + b*x]])) + Log[ArcTanh[Tanh[a + b*x]]]/(b*x - ArcTanh[Tanh 
[a + b*x]]))/(b*x - ArcTanh[Tanh[a + b*x]]))/(b*x - ArcTanh[Tanh[a + b*x]] 
)
 

Defintions of rubi rules used

rule 14

Int[(a_.)/(x_), x_Symbol] :> Simp[a*Log[x], x] /; FreeQ[a, x]
 

rule 2588

Int[(u_)^(m_.), x_Symbol] :> With[{c = Simplify[D[u, x]]}, Simp[1/c   Subst 
[Int[x^m, x], x, u], x]] /; FreeQ[m, x] && PiecewiseLinearQ[u, x]
 

rule 2591

Int[1/((u_)*(v_)), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D 
[v, x]]}, Simp[b/(b*u - a*v)   Int[1/v, x], x] - Simp[a/(b*u - a*v)   Int[1 
/u, x], x] /; NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x]
 

rule 2594

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[ 
D[v, x]]}, Simp[v^(n + 1)/((n + 1)*(b*u - a*v)), x] - Simp[a*((n + 1)/((n + 
 1)*(b*u - a*v)))   Int[v^(n + 1)/u, x], x] /; NeQ[b*u - a*v, 0]] /; Piecew 
iseLinearQ[u, v, x] && LtQ[n, -1]
 

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.95

\[\frac {\ln \left (x \right )}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}-\frac {\ln \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )\right )}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{3}}+\frac {1}{\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right )^{2} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )}+\frac {1}{2 \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{2}}\]

Input:

int(1/x/arctanh(tanh(b*x+a))^3,x)
 

Output:

1/(arctanh(tanh(b*x+a))-b*x)^3*ln(x)-1/(arctanh(tanh(b*x+a))-b*x)^3*ln(arc 
tanh(tanh(b*x+a)))+1/(arctanh(tanh(b*x+a))-b*x)^2/arctanh(tanh(b*x+a))+1/2 
/(arctanh(tanh(b*x+a))-b*x)/arctanh(tanh(b*x+a))^2
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=\frac {2 \, a b x + 3 \, a^{2} - 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) + 2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \log \left (x\right )}{2 \, {\left (a^{3} b^{2} x^{2} + 2 \, a^{4} b x + a^{5}\right )}} \] Input:

integrate(1/x/arctanh(tanh(b*x+a))^3,x, algorithm="fricas")
 

Output:

1/2*(2*a*b*x + 3*a^2 - 2*(b^2*x^2 + 2*a*b*x + a^2)*log(b*x + a) + 2*(b^2*x 
^2 + 2*a*b*x + a^2)*log(x))/(a^3*b^2*x^2 + 2*a^4*b*x + a^5)
 

Sympy [F]

\[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=\int \frac {1}{x \operatorname {atanh}^{3}{\left (\tanh {\left (a + b x \right )} \right )}}\, dx \] Input:

integrate(1/x/atanh(tanh(b*x+a))**3,x)
 

Output:

Integral(1/(x*atanh(tanh(a + b*x))**3), x)
 

Maxima [A] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.53 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=\frac {2 \, b x + 3 \, a}{2 \, {\left (a^{2} b^{2} x^{2} + 2 \, a^{3} b x + a^{4}\right )}} - \frac {\log \left (b x + a\right )}{a^{3}} + \frac {\log \left (x\right )}{a^{3}} \] Input:

integrate(1/x/arctanh(tanh(b*x+a))^3,x, algorithm="maxima")
 

Output:

1/2*(2*b*x + 3*a)/(a^2*b^2*x^2 + 2*a^3*b*x + a^4) - log(b*x + a)/a^3 + log 
(x)/a^3
 

Giac [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.44 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=-\frac {\log \left ({\left | b x + a \right |}\right )}{a^{3}} + \frac {\log \left ({\left | x \right |}\right )}{a^{3}} + \frac {2 \, a b x + 3 \, a^{2}}{2 \, {\left (b x + a\right )}^{2} a^{3}} \] Input:

integrate(1/x/arctanh(tanh(b*x+a))^3,x, algorithm="giac")
 

Output:

-log(abs(b*x + a))/a^3 + log(abs(x))/a^3 + 1/2*(2*a*b*x + 3*a^2)/((b*x + a 
)^2*a^3)
 

Mupad [B] (verification not implemented)

Time = 5.43 (sec) , antiderivative size = 645, normalized size of antiderivative = 6.65 \[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx =\text {Too large to display} \] Input:

int(1/(x*atanh(tanh(a + b*x))^3),x)
 

Output:

-(12*log(1/(exp(2*a)*exp(2*b*x) + 1))^2 - 24*log(1/(exp(2*a)*exp(2*b*x) + 
1))*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) - log((exp(2*a)*e 
xp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))^2*atan((log((exp(2*a)*exp(2*b*x))/(e 
xp(2*a)*exp(2*b*x) + 1))*1i - log(1/(exp(2*a)*exp(2*b*x) + 1))*1i + b*x*2i 
)/(log(1/(exp(2*a)*exp(2*b*x) + 1)) - log((exp(2*a)*exp(2*b*x))/(exp(2*a)* 
exp(2*b*x) + 1)) + 2*b*x))*16i - log(1/(exp(2*a)*exp(2*b*x) + 1))^2*atan(( 
log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))*1i - log(1/(exp(2*a)* 
exp(2*b*x) + 1))*1i + b*x*2i)/(log(1/(exp(2*a)*exp(2*b*x) + 1)) - log((exp 
(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x))*16i + 12*log((exp(2 
*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))^2 + 16*b^2*x^2 + log(1/(exp(2*a 
)*exp(2*b*x) + 1))*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))*at 
an((log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))*1i - log(1/(exp(2 
*a)*exp(2*b*x) + 1))*1i + b*x*2i)/(log(1/(exp(2*a)*exp(2*b*x) + 1)) - log( 
(exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x))*32i + b*x*(32*l 
og(1/(exp(2*a)*exp(2*b*x) + 1)) - 32*log((exp(2*a)*exp(2*b*x))/(exp(2*a)*e 
xp(2*b*x) + 1))))/((log(1/(exp(2*a)*exp(2*b*x) + 1)) - log((exp(2*a)*exp(2 
*b*x))/(exp(2*a)*exp(2*b*x) + 1)))^2*(log(1/(exp(2*a)*exp(2*b*x) + 1)) - l 
og((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^3)
 

Reduce [F]

\[ \int \frac {1}{x \text {arctanh}(\tanh (a+b x))^3} \, dx=\int \frac {1}{\mathit {atanh} \left (\tanh \left (b x +a \right )\right )^{3} x}d x \] Input:

int(1/x/atanh(tanh(b*x+a))^3,x)
 

Output:

int(1/(atanh(tanh(a + b*x))**3*x),x)