\(\int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx\) [73]

Optimal result

Integrand size = 13, antiderivative size = 105 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=-b x (b x-\text {arctanh}(\tanh (a+b x)))^3+\frac {1}{2} (b x-\text {arctanh}(\tanh (a+b x)))^2 \text {arctanh}(\tanh (a+b x))^2-\frac {1}{3} (b x-\text {arctanh}(\tanh (a+b x))) \text {arctanh}(\tanh (a+b x))^3+\frac {1}{4} \text {arctanh}(\tanh (a+b x))^4+(b x-\text {arctanh}(\tanh (a+b x)))^4 \log (x) \] Output:

-b*x*(b*x-arctanh(tanh(b*x+a)))^3+1/2*(b*x-arctanh(tanh(b*x+a)))^2*arctanh 
(tanh(b*x+a))^2-1/3*(b*x-arctanh(tanh(b*x+a)))*arctanh(tanh(b*x+a))^3+1/4* 
arctanh(tanh(b*x+a))^4+(b*x-arctanh(tanh(b*x+a)))^4*ln(x)
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.67 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\frac {1}{4} (a+b x)^4+\frac {1}{2} (a+b x)^2 \left (a^2-4 a (a+b x-\text {arctanh}(\tanh (a+b x)))+6 (a+b x-\text {arctanh}(\tanh (a+b x)))^2\right )+(a+b x) \left (a^3-4 a^2 (a+b x-\text {arctanh}(\tanh (a+b x)))+6 a (a+b x-\text {arctanh}(\tanh (a+b x)))^2-4 (a+b x-\text {arctanh}(\tanh (a+b x)))^3\right )-\frac {1}{3} (a+b x)^3 (3 a+4 b x-4 \text {arctanh}(\tanh (a+b x)))+(-b x+\text {arctanh}(\tanh (a+b x)))^4 \log (b x) \] Input:

Integrate[ArcTanh[Tanh[a + b*x]]^4/x,x]
 

Output:

(a + b*x)^4/4 + ((a + b*x)^2*(a^2 - 4*a*(a + b*x - ArcTanh[Tanh[a + b*x]]) 
 + 6*(a + b*x - ArcTanh[Tanh[a + b*x]])^2))/2 + (a + b*x)*(a^3 - 4*a^2*(a 
+ b*x - ArcTanh[Tanh[a + b*x]]) + 6*a*(a + b*x - ArcTanh[Tanh[a + b*x]])^2 
 - 4*(a + b*x - ArcTanh[Tanh[a + b*x]])^3) - ((a + b*x)^3*(3*a + 4*b*x - 4 
*ArcTanh[Tanh[a + b*x]]))/3 + (-(b*x) + ArcTanh[Tanh[a + b*x]])^4*Log[b*x]
 

Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 108, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2590, 2590, 2590, 2589, 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.

\(\displaystyle \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {1}{4} \text {arctanh}(\tanh (a+b x))^4-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\text {arctanh}(\tanh (a+b x))^3}{x}dx\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {1}{4} \text {arctanh}(\tanh (a+b x))^4-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\text {arctanh}(\tanh (a+b x))^2}{x}dx\right )\)

\(\Big \downarrow \) 2590

\(\displaystyle \frac {1}{4} \text {arctanh}(\tanh (a+b x))^4-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{2} \text {arctanh}(\tanh (a+b x))^2-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {\text {arctanh}(\tanh (a+b x))}{x}dx\right )\right )\)

\(\Big \downarrow \) 2589

\(\displaystyle \frac {1}{4} \text {arctanh}(\tanh (a+b x))^4-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{2} \text {arctanh}(\tanh (a+b x))^2-(b x-\text {arctanh}(\tanh (a+b x))) \left (b x-(b x-\text {arctanh}(\tanh (a+b x))) \int \frac {1}{x}dx\right )\right )\right )\)

\(\Big \downarrow \) 14

\(\displaystyle \frac {1}{4} \text {arctanh}(\tanh (a+b x))^4-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{3} \text {arctanh}(\tanh (a+b x))^3-(b x-\text {arctanh}(\tanh (a+b x))) \left (\frac {1}{2} \text {arctanh}(\tanh (a+b x))^2-(b x-\text {arctanh}(\tanh (a+b x))) (b x-\log (x) (b x-\text {arctanh}(\tanh (a+b x))))\right )\right )\)

Input:

Int[ArcTanh[Tanh[a + b*x]]^4/x,x]
 

Output:

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

Defintions of rubi rules used

rule 14

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

rule 2589

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

rule 2590

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(264\) vs. \(2(99)=198\).

Time = 0.31 (sec) , antiderivative size = 265, normalized size of antiderivative = 2.52

method	result	size
default	\(\ln \left (x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4}-4 b \left (b^{3} \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )+3 a \,b^{2} \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+3 b^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+3 a^{2} b \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+6 a b \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+3 b \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+a^{3} \left (x \ln \left (x \right )-x \right )+3 a^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (x \ln \left (x \right )-x \right )+3 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \left (x \ln \left (x \right )-x \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3} \left (x \ln \left (x \right )-x \right )\right )\)	\(265\)
parts	\(\ln \left (x \right ) \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4}-4 b \left (b^{3} \left (\frac {x^{4} \ln \left (x \right )}{4}-\frac {x^{4}}{16}\right )+3 a \,b^{2} \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+3 b^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (\frac {x^{3} \ln \left (x \right )}{3}-\frac {x^{3}}{9}\right )+3 a^{2} b \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+6 a b \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+3 b \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \left (\frac {x^{2} \ln \left (x \right )}{2}-\frac {x^{2}}{4}\right )+a^{3} \left (x \ln \left (x \right )-x \right )+3 a^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right ) \left (x \ln \left (x \right )-x \right )+3 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2} \left (x \ln \left (x \right )-x \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3} \left (x \ln \left (x \right )-x \right )\right )\)	\(265\)
risch	\(\text {Expression too large to display}\)	\(2511\)

Input:

int(arctanh(tanh(b*x+a))^4/x,x,method=_RETURNVERBOSE)
 

Output:

ln(x)*arctanh(tanh(b*x+a))^4-4*b*(b^3*(1/4*x^4*ln(x)-1/16*x^4)+3*a*b^2*(1/ 
3*x^3*ln(x)-1/9*x^3)+3*b^2*(arctanh(tanh(b*x+a))-b*x-a)*(1/3*x^3*ln(x)-1/9 
*x^3)+3*a^2*b*(1/2*x^2*ln(x)-1/4*x^2)+6*a*b*(arctanh(tanh(b*x+a))-b*x-a)*( 
1/2*x^2*ln(x)-1/4*x^2)+3*b*(arctanh(tanh(b*x+a))-b*x-a)^2*(1/2*x^2*ln(x)-1 
/4*x^2)+a^3*(x*ln(x)-x)+3*a^2*(arctanh(tanh(b*x+a))-b*x-a)*(x*ln(x)-x)+3*a 
*(arctanh(tanh(b*x+a))-b*x-a)^2*(x*ln(x)-x)+(arctanh(tanh(b*x+a))-b*x-a)^3 
*(x*ln(x)-x))
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.40 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4} \log \left (x\right ) \] Input:

integrate(arctanh(tanh(b*x+a))^4/x,x, algorithm="fricas")
 

Output:

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

Sympy [F]

\[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\int \frac {\operatorname {atanh}^{4}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \] Input:

integrate(atanh(tanh(b*x+a))**4/x,x)
 

Output:

Integral(atanh(tanh(a + b*x))**4/x, x)
 

Maxima [A] (verification not implemented)

Time = 0.33 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.40 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4} \log \left (x\right ) \] Input:

integrate(arctanh(tanh(b*x+a))^4/x,x, algorithm="maxima")
 

Output:

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

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.41 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\frac {1}{4} \, b^{4} x^{4} + \frac {4}{3} \, a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 4 \, a^{3} b x + a^{4} \log \left ({\left | x \right |}\right ) \] Input:

integrate(arctanh(tanh(b*x+a))^4/x,x, algorithm="giac")
 

Output:

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

Mupad [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 423, normalized size of antiderivative = 4.03 \[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\ln \left (x\right )\,\left (\frac {{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4}{16}+\frac {3\,a^2\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{2}+a^4-\frac {a\,{\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{2}-2\,a^3\,\left (2\,a-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\right )+\frac {b^4\,x^4}{4}-\frac {2\,b^3\,x^3\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}{3}+\frac {3\,b^2\,x^2\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2}{4}-\frac {b\,x\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3}{2} \] Input:

int(atanh(tanh(a + b*x))^4/x,x)
 

Output:

log(x)*((2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + lo 
g(2/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^4/16 + (3*a^2*(2*a - log((2*exp(2* 
a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + log(2/(exp(2*a)*exp(2*b*x) + 1 
)) + 2*b*x)^2)/2 + a^4 - (a*(2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*e 
xp(2*b*x) + 1)) + log(2/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^3)/2 - 2*a^3*( 
2*a - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + log(2/(exp( 
2*a)*exp(2*b*x) + 1)) + 2*b*x)) + (b^4*x^4)/4 - (2*b^3*x^3*(log(2/(exp(2*a 
)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1) 
) + 2*b*x))/3 + (3*b^2*x^2*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp( 
2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1)) + 2*b*x)^2)/4 - (b*x*(log(2/(e 
xp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x 
) + 1)) + 2*b*x)^3)/2
 

Reduce [F]

\[ \int \frac {\text {arctanh}(\tanh (a+b x))^4}{x} \, dx=\int \frac {\mathit {atanh} \left (\tanh \left (b x +a \right )\right )^{4}}{x}d x \] Input:

int(atanh(tanh(b*x+a))^4/x,x)
 

Output:

int(atanh(tanh(a + b*x))**4/x,x)