\(\int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx\) [65]

Optimal result

Integrand size = 13, antiderivative size = 154 \[ \int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx=\frac {24 b^4 x^{5+m}}{(1+m) (2+m) (3+m) \left (20+9 m+m^2\right )}-\frac {24 b^3 x^{4+m} \text {arctanh}(\tanh (a+b x))}{(1+m) \left (24+26 m+9 m^2+m^3\right )}+\frac {12 b^2 x^{3+m} \text {arctanh}(\tanh (a+b x))^2}{6+11 m+6 m^2+m^3}-\frac {4 b x^{2+m} \text {arctanh}(\tanh (a+b x))^3}{2+3 m+m^2}+\frac {x^{1+m} \text {arctanh}(\tanh (a+b x))^4}{1+m} \] Output:

24*b^4*x^(5+m)/(1+m)/(2+m)/(3+m)/(m^2+9*m+20)-24*b^3*x^(4+m)*arctanh(tanh( 
b*x+a))/(1+m)/(m^3+9*m^2+26*m+24)+12*b^2*x^(3+m)*arctanh(tanh(b*x+a))^2/(m 
^3+6*m^2+11*m+6)-4*b*x^(2+m)*arctanh(tanh(b*x+a))^3/(m^2+3*m+2)+x^(1+m)*ar 
ctanh(tanh(b*x+a))^4/(1+m)
 

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.89 \[ \int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx=\frac {x^{1+m} \left (24 b^4 x^4-24 b^3 (5+m) x^3 \text {arctanh}(\tanh (a+b x))+12 b^2 \left (20+9 m+m^2\right ) x^2 \text {arctanh}(\tanh (a+b x))^2-4 b \left (60+47 m+12 m^2+m^3\right ) x \text {arctanh}(\tanh (a+b x))^3+\left (120+154 m+71 m^2+14 m^3+m^4\right ) \text {arctanh}(\tanh (a+b x))^4\right )}{(1+m) (2+m) (3+m) (4+m) (5+m)} \] Input:

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

Output:

(x^(1 + m)*(24*b^4*x^4 - 24*b^3*(5 + m)*x^3*ArcTanh[Tanh[a + b*x]] + 12*b^ 
2*(20 + 9*m + m^2)*x^2*ArcTanh[Tanh[a + b*x]]^2 - 4*b*(60 + 47*m + 12*m^2 
+ m^3)*x*ArcTanh[Tanh[a + b*x]]^3 + (120 + 154*m + 71*m^2 + 14*m^3 + m^4)* 
ArcTanh[Tanh[a + b*x]]^4))/((1 + m)*(2 + m)*(3 + m)*(4 + m)*(5 + m))
 

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 124, normalized size of antiderivative = 0.81, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2599, 2599, 2599, 2599, 15}

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.

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

\(\Big \downarrow \) 2599

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

\(\Big \downarrow \) 2599

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

\(\Big \downarrow \) 2599

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

\(\Big \downarrow \) 2599

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

\(\Big \downarrow \) 15

\(\displaystyle \frac {x^{m+1} \text {arctanh}(\tanh (a+b x))^4}{m+1}-\frac {4 b \left (\frac {x^{m+2} \text {arctanh}(\tanh (a+b x))^3}{m+2}-\frac {3 b \left (\frac {x^{m+3} \text {arctanh}(\tanh (a+b x))^2}{m+3}-\frac {2 b \left (\frac {x^{m+4} \text {arctanh}(\tanh (a+b x))}{m+4}-\frac {b x^{m+5}}{(m+4) (m+5)}\right )}{m+3}\right )}{m+2}\right )}{m+1}\)

Input:

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

Output:

(x^(1 + m)*ArcTanh[Tanh[a + b*x]]^4)/(1 + m) - (4*b*((x^(2 + m)*ArcTanh[Ta 
nh[a + b*x]]^3)/(2 + m) - (3*b*((x^(3 + m)*ArcTanh[Tanh[a + b*x]]^2)/(3 + 
m) - (2*b*(-((b*x^(5 + m))/((4 + m)*(5 + m))) + (x^(4 + m)*ArcTanh[Tanh[a 
+ b*x]])/(4 + m)))/(3 + m)))/(2 + m)))/(1 + m)
 

Defintions of rubi rules used

rule 15

Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ 
{a, m}, x] && NeQ[m, -1]
 

rule 2599

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Sim 
plify[D[v, x]]}, Simp[u^(m + 1)*(v^n/(a*(m + 1))), x] - Simp[b*(n/(a*(m + 1 
)))   Int[u^(m + 1)*v^(n - 1), x], x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n} 
, x] && PiecewiseLinearQ[u, v, x] && NeQ[m, -1] && ((LtQ[m, -1] && GtQ[n, 0 
] &&  !(ILtQ[m + n, -2] && (FractionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ 
[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]) || (ILt 
Q[m, 0] &&  !IntegerQ[n]))
 

Maple [A] (verified)

Time = 46.25 (sec) , antiderivative size = 278, normalized size of antiderivative = 1.81

method	result	size
default	\(\frac {b^{4} x^{5} {\mathrm e}^{m \ln \left (x \right )}}{5+m}+\frac {\left (a^{4}+4 a^{3} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+6 a^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+4 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}\right ) x \,{\mathrm e}^{m \ln \left (x \right )}}{1+m}+\frac {4 b \left (a^{3}+3 a^{2} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+3 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}\right ) x^{2} {\mathrm e}^{m \ln \left (x \right )}}{2+m}+\frac {6 b^{2} \left (a^{2}+2 a \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) x^{3} {\mathrm e}^{m \ln \left (x \right )}}{3+m}+\frac {4 b^{3} \left (\operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )-b x \right ) x^{4} {\mathrm e}^{m \ln \left (x \right )}}{4+m}\)	\(278\)
parallelrisch	\(-\frac {240 b \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{3} x^{m} x^{2}-240 b^{2} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{2} x^{m} x^{3}+120 b^{3} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right ) x^{m} x^{4}+188 x^{2} x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{3} b m -24 b^{4} x^{m} x^{5}-120 \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4} x \,x^{m}+24 x^{4} x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right ) b^{3} m -12 x^{3} x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{2} b^{2} m^{2}+4 x^{2} x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{3} b \,m^{3}-108 x^{3} x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{2} b^{2} m +48 x^{2} x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{3} b \,m^{2}-x \,x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4} m^{4}-14 x \,x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4} m^{3}-71 x \,x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4} m^{2}-154 x \,x^{m} \operatorname {arctanh}\left (\tanh \left (b x +a \right )\right )^{4} m}{\left (1+m \right ) \left (m^{3}+9 m^{2}+26 m +24\right ) \left (5+m \right )}\)	\(305\)
risch	\(\text {Expression too large to display}\)	\(94967\)

Input:

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

Output:

b^4/(5+m)*x^5*exp(m*ln(x))+(a^4+4*a^3*(arctanh(tanh(b*x+a))-b*x-a)+6*a^2*( 
arctanh(tanh(b*x+a))-b*x-a)^2+4*a*(arctanh(tanh(b*x+a))-b*x-a)^3+(arctanh( 
tanh(b*x+a))-b*x-a)^4)/(1+m)*x*exp(m*ln(x))+4*b*(a^3+3*a^2*(arctanh(tanh(b 
*x+a))-b*x-a)+3*a*(arctanh(tanh(b*x+a))-b*x-a)^2+(arctanh(tanh(b*x+a))-b*x 
-a)^3)/(2+m)*x^2*exp(m*ln(x))+6*b^2*(a^2+2*a*(arctanh(tanh(b*x+a))-b*x-a)+ 
(arctanh(tanh(b*x+a))-b*x-a)^2)/(3+m)*x^3*exp(m*ln(x))+4*b^3*(arctanh(tanh 
(b*x+a))-b*x)/(4+m)*x^4*exp(m*ln(x))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 483 vs. \(2 (154) = 308\).

Time = 0.09 (sec) , antiderivative size = 483, normalized size of antiderivative = 3.14 \[ \int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx=\frac {{\left ({\left (b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}\right )} x^{5} + 4 \, {\left (a b^{3} m^{4} + 11 \, a b^{3} m^{3} + 41 \, a b^{3} m^{2} + 61 \, a b^{3} m + 30 \, a b^{3}\right )} x^{4} + 6 \, {\left (a^{2} b^{2} m^{4} + 12 \, a^{2} b^{2} m^{3} + 49 \, a^{2} b^{2} m^{2} + 78 \, a^{2} b^{2} m + 40 \, a^{2} b^{2}\right )} x^{3} + 4 \, {\left (a^{3} b m^{4} + 13 \, a^{3} b m^{3} + 59 \, a^{3} b m^{2} + 107 \, a^{3} b m + 60 \, a^{3} b\right )} x^{2} + {\left (a^{4} m^{4} + 14 \, a^{4} m^{3} + 71 \, a^{4} m^{2} + 154 \, a^{4} m + 120 \, a^{4}\right )} x\right )} \cosh \left (m \log \left (x\right )\right ) + {\left ({\left (b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}\right )} x^{5} + 4 \, {\left (a b^{3} m^{4} + 11 \, a b^{3} m^{3} + 41 \, a b^{3} m^{2} + 61 \, a b^{3} m + 30 \, a b^{3}\right )} x^{4} + 6 \, {\left (a^{2} b^{2} m^{4} + 12 \, a^{2} b^{2} m^{3} + 49 \, a^{2} b^{2} m^{2} + 78 \, a^{2} b^{2} m + 40 \, a^{2} b^{2}\right )} x^{3} + 4 \, {\left (a^{3} b m^{4} + 13 \, a^{3} b m^{3} + 59 \, a^{3} b m^{2} + 107 \, a^{3} b m + 60 \, a^{3} b\right )} x^{2} + {\left (a^{4} m^{4} + 14 \, a^{4} m^{3} + 71 \, a^{4} m^{2} + 154 \, a^{4} m + 120 \, a^{4}\right )} x\right )} \sinh \left (m \log \left (x\right )\right )}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \] Input:

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

Output:

(((b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)*x^5 + 4*(a*b^3*m 
^4 + 11*a*b^3*m^3 + 41*a*b^3*m^2 + 61*a*b^3*m + 30*a*b^3)*x^4 + 6*(a^2*b^2 
*m^4 + 12*a^2*b^2*m^3 + 49*a^2*b^2*m^2 + 78*a^2*b^2*m + 40*a^2*b^2)*x^3 + 
4*(a^3*b*m^4 + 13*a^3*b*m^3 + 59*a^3*b*m^2 + 107*a^3*b*m + 60*a^3*b)*x^2 + 
 (a^4*m^4 + 14*a^4*m^3 + 71*a^4*m^2 + 154*a^4*m + 120*a^4)*x)*cosh(m*log(x 
)) + ((b^4*m^4 + 10*b^4*m^3 + 35*b^4*m^2 + 50*b^4*m + 24*b^4)*x^5 + 4*(a*b 
^3*m^4 + 11*a*b^3*m^3 + 41*a*b^3*m^2 + 61*a*b^3*m + 30*a*b^3)*x^4 + 6*(a^2 
*b^2*m^4 + 12*a^2*b^2*m^3 + 49*a^2*b^2*m^2 + 78*a^2*b^2*m + 40*a^2*b^2)*x^ 
3 + 4*(a^3*b*m^4 + 13*a^3*b*m^3 + 59*a^3*b*m^2 + 107*a^3*b*m + 60*a^3*b)*x 
^2 + (a^4*m^4 + 14*a^4*m^3 + 71*a^4*m^2 + 154*a^4*m + 120*a^4)*x)*sinh(m*l 
og(x)))/(m^5 + 15*m^4 + 85*m^3 + 225*m^2 + 274*m + 120)
 

Sympy [F]

\[ \int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx=\text {Too large to display} \] Input:

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

Output:

Piecewise((b**4*log(x) - b**3*atanh(tanh(a + b*x))/x - b**2*atanh(tanh(a + 
 b*x))**2/(2*x**2) - b*atanh(tanh(a + b*x))**3/(3*x**3) - atanh(tanh(a + b 
*x))**4/(4*x**4), Eq(m, -5)), (Integral(atanh(tanh(a + b*x))**4/x**4, x), 
Eq(m, -4)), (Integral(atanh(tanh(a + b*x))**4/x**3, x), Eq(m, -3)), (Integ 
ral(atanh(tanh(a + b*x))**4/x**2, x), Eq(m, -2)), (Integral(atanh(tanh(a + 
 b*x))**4/x, x), Eq(m, -1)), (24*b**4*x**5*x**m/(m**5 + 15*m**4 + 85*m**3 
+ 225*m**2 + 274*m + 120) - 24*b**3*m*x**4*x**m*atanh(tanh(a + b*x))/(m**5 
 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) - 120*b**3*x**4*x**m*atanh( 
tanh(a + b*x))/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 12*b* 
*2*m**2*x**3*x**m*atanh(tanh(a + b*x))**2/(m**5 + 15*m**4 + 85*m**3 + 225* 
m**2 + 274*m + 120) + 108*b**2*m*x**3*x**m*atanh(tanh(a + b*x))**2/(m**5 + 
 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 240*b**2*x**3*x**m*atanh(ta 
nh(a + b*x))**2/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) - 4*b* 
m**3*x**2*x**m*atanh(tanh(a + b*x))**3/(m**5 + 15*m**4 + 85*m**3 + 225*m** 
2 + 274*m + 120) - 48*b*m**2*x**2*x**m*atanh(tanh(a + b*x))**3/(m**5 + 15* 
m**4 + 85*m**3 + 225*m**2 + 274*m + 120) - 188*b*m*x**2*x**m*atanh(tanh(a 
+ b*x))**3/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) - 240*b*x** 
2*x**m*atanh(tanh(a + b*x))**3/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274* 
m + 120) + m**4*x*x**m*atanh(tanh(a + b*x))**4/(m**5 + 15*m**4 + 85*m**3 + 
 225*m**2 + 274*m + 120) + 14*m**3*x*x**m*atanh(tanh(a + b*x))**4/(m**5...
 

Maxima [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.94 \[ \int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx=-\frac {4 \, b x^{2} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{{\left (m + 2\right )} {\left (m + 1\right )}} + \frac {x^{m + 1} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{m + 1} + \frac {12 \, {\left (\frac {b x^{3} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{{\left (m + 3\right )} {\left (m + 2\right )}} + \frac {2 \, {\left (\frac {b^{2} x^{5} x^{m}}{{\left (m + 5\right )} {\left (m + 4\right )} {\left (m + 3\right )}} - \frac {b x^{4} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{{\left (m + 4\right )} {\left (m + 3\right )}}\right )} b}{m + 2}\right )} b}{m + 1} \] Input:

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

Output:

-4*b*x^2*x^m*arctanh(tanh(b*x + a))^3/((m + 2)*(m + 1)) + x^(m + 1)*arctan 
h(tanh(b*x + a))^4/(m + 1) + 12*(b*x^3*x^m*arctanh(tanh(b*x + a))^2/((m + 
3)*(m + 2)) + 2*(b^2*x^5*x^m/((m + 5)*(m + 4)*(m + 3)) - b*x^4*x^m*arctanh 
(tanh(b*x + a))/((m + 4)*(m + 3)))*b/(m + 2))*b/(m + 1)
 

Giac [F]

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

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

Output:

integrate(x^m*arctanh(tanh(b*x + a))^4, x)
 

Mupad [B] (verification not implemented)

Time = 3.66 (sec) , antiderivative size = 479, normalized size of antiderivative = 3.11 \[ \int x^m \text {arctanh}(\tanh (a+b x))^4 \, dx=\frac {x\,x^m\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4\,\left (m^4+14\,m^3+71\,m^2+154\,m+120\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}+\frac {16\,b^4\,x^m\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}+\frac {24\,b^2\,x^m\,x^3\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2\,\left (m^4+12\,m^3+49\,m^2+78\,m+40\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}-\frac {32\,b^3\,x^m\,x^4\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\,\left (m^4+11\,m^3+41\,m^2+61\,m+30\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}-\frac {8\,b\,x^m\,x^2\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3\,\left (m^4+13\,m^3+59\,m^2+107\,m+60\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920} \] Input:

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

Output:

(x*x^m*(log(2/(exp(2*a)*exp(2*b*x) + 1)) - log((2*exp(2*a)*exp(2*b*x))/(ex 
p(2*a)*exp(2*b*x) + 1)) + 2*b*x)^4*(154*m + 71*m^2 + 14*m^3 + m^4 + 120))/ 
(4384*m + 3600*m^2 + 1360*m^3 + 240*m^4 + 16*m^5 + 1920) + (16*b^4*x^m*x^5 
*(50*m + 35*m^2 + 10*m^3 + m^4 + 24))/(4384*m + 3600*m^2 + 1360*m^3 + 240* 
m^4 + 16*m^5 + 1920) + (24*b^2*x^m*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)^2*(78*m + 
 49*m^2 + 12*m^3 + m^4 + 40))/(4384*m + 3600*m^2 + 1360*m^3 + 240*m^4 + 16 
*m^5 + 1920) - (32*b^3*x^m*x^4*(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)*(61*m + 41*m^2 + 
11*m^3 + m^4 + 30))/(4384*m + 3600*m^2 + 1360*m^3 + 240*m^4 + 16*m^5 + 192 
0) - (8*b*x^m*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)^3*(107*m + 59*m^2 + 13*m^3 + m 
^4 + 60))/(4384*m + 3600*m^2 + 1360*m^3 + 240*m^4 + 16*m^5 + 1920)
 

Reduce [F]

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

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

Output:

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