3.65 \(\int x^m \tanh ^{-1}(\tanh (a+b x))^4 \, dx\)

Optimal. Leaf size=154 \[ -\frac {24 b^3 x^{m+4} \tanh ^{-1}(\tanh (a+b x))}{(m+1) \left (m^3+9 m^2+26 m+24\right )}+\frac {12 b^2 x^{m+3} \tanh ^{-1}(\tanh (a+b x))^2}{m^3+6 m^2+11 m+6}-\frac {4 b x^{m+2} \tanh ^{-1}(\tanh (a+b x))^3}{m^2+3 m+2}+\frac {x^{m+1} \tanh ^{-1}(\tanh (a+b x))^4}{m+1}+\frac {24 b^4 x^{m+5}}{(m+1) (m+2) (m+3) \left (m^2+9 m+20\right )} \]

[Out]

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)*
arctanh(tanh(b*x+a))^4/(1+m)

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Rubi [A]  time = 0.10, antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2168, 30} \[ \frac {12 b^2 x^{m+3} \tanh ^{-1}(\tanh (a+b x))^2}{m^3+6 m^2+11 m+6}-\frac {24 b^3 x^{m+4} \tanh ^{-1}(\tanh (a+b x))}{(m+1) \left (m^3+9 m^2+26 m+24\right )}-\frac {4 b x^{m+2} \tanh ^{-1}(\tanh (a+b x))^3}{m^2+3 m+2}+\frac {x^{m+1} \tanh ^{-1}(\tanh (a+b x))^4}{m+1}+\frac {24 b^4 x^{m+5}}{(m+1) (m+2) (m+3) \left (m^2+9 m+20\right )} \]

Antiderivative was successfully verified.

[In]

Int[x^m*ArcTanh[Tanh[a + b*x]]^4,x]

[Out]

(24*b^4*x^(5 + m))/((1 + m)*(2 + m)*(3 + m)*(20 + 9*m + m^2)) - (24*b^3*x^(4 + m)*ArcTanh[Tanh[a + b*x]])/((1
+ m)*(24 + 26*m + 9*m^2 + m^3)) + (12*b^2*x^(3 + m)*ArcTanh[Tanh[a + b*x]]^2)/(6 + 11*m + 6*m^2 + m^3) - (4*b*
x^(2 + m)*ArcTanh[Tanh[a + b*x]]^3)/(2 + 3*m + m^2) + (x^(1 + m)*ArcTanh[Tanh[a + b*x]]^4)/(1 + m)

Rule 30

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

Rule 2168

Int[(u_)^(m_)*(v_)^(n_.), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[(u^(m + 1)*v^
n)/(a*(m + 1)), x] - Dist[(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] && (Fra
ctionQ[m] || GeQ[2*n + m + 1, 0]))) || (IGtQ[n, 0] && IGtQ[m, 0] && LeQ[n, m]) || (IGtQ[n, 0] &&  !IntegerQ[m]
) || (ILtQ[m, 0] &&  !IntegerQ[n]))

Rubi steps

\begin {align*} \int x^m \tanh ^{-1}(\tanh (a+b x))^4 \, dx &=\frac {x^{1+m} \tanh ^{-1}(\tanh (a+b x))^4}{1+m}-\frac {(4 b) \int x^{1+m} \tanh ^{-1}(\tanh (a+b x))^3 \, dx}{1+m}\\ &=-\frac {4 b x^{2+m} \tanh ^{-1}(\tanh (a+b x))^3}{2+3 m+m^2}+\frac {x^{1+m} \tanh ^{-1}(\tanh (a+b x))^4}{1+m}+\frac {\left (12 b^2\right ) \int x^{2+m} \tanh ^{-1}(\tanh (a+b x))^2 \, dx}{2+3 m+m^2}\\ &=\frac {12 b^2 x^{3+m} \tanh ^{-1}(\tanh (a+b x))^2}{6+11 m+6 m^2+m^3}-\frac {4 b x^{2+m} \tanh ^{-1}(\tanh (a+b x))^3}{2+3 m+m^2}+\frac {x^{1+m} \tanh ^{-1}(\tanh (a+b x))^4}{1+m}-\frac {\left (24 b^3\right ) \int x^{3+m} \tanh ^{-1}(\tanh (a+b x)) \, dx}{6+11 m+6 m^2+m^3}\\ &=-\frac {24 b^3 x^{4+m} \tanh ^{-1}(\tanh (a+b x))}{(4+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {12 b^2 x^{3+m} \tanh ^{-1}(\tanh (a+b x))^2}{6+11 m+6 m^2+m^3}-\frac {4 b x^{2+m} \tanh ^{-1}(\tanh (a+b x))^3}{2+3 m+m^2}+\frac {x^{1+m} \tanh ^{-1}(\tanh (a+b x))^4}{1+m}+\frac {\left (24 b^4\right ) \int x^{4+m} \, dx}{(4+m) \left (6+11 m+6 m^2+m^3\right )}\\ &=\frac {24 b^4 x^{5+m}}{(4+m) (5+m) \left (6+11 m+6 m^2+m^3\right )}-\frac {24 b^3 x^{4+m} \tanh ^{-1}(\tanh (a+b x))}{(4+m) \left (6+11 m+6 m^2+m^3\right )}+\frac {12 b^2 x^{3+m} \tanh ^{-1}(\tanh (a+b x))^2}{6+11 m+6 m^2+m^3}-\frac {4 b x^{2+m} \tanh ^{-1}(\tanh (a+b x))^3}{2+3 m+m^2}+\frac {x^{1+m} \tanh ^{-1}(\tanh (a+b x))^4}{1+m}\\ \end {align*}

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Mathematica [A]  time = 0.41, size = 137, normalized size = 0.89 \[ \frac {x^{m+1} \left (-24 b^3 (m+5) x^3 \tanh ^{-1}(\tanh (a+b x))+12 b^2 \left (m^2+9 m+20\right ) x^2 \tanh ^{-1}(\tanh (a+b x))^2-4 b \left (m^3+12 m^2+47 m+60\right ) x \tanh ^{-1}(\tanh (a+b x))^3+\left (m^4+14 m^3+71 m^2+154 m+120\right ) \tanh ^{-1}(\tanh (a+b x))^4+24 b^4 x^4\right )}{(m+1) (m+2) (m+3) (m+4) (m+5)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(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))

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fricas [B]  time = 0.64, size = 483, normalized size = 3.14 \[ \frac {{\left ({\left (b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}\right )} x^{5} + 4 \, {\left (a b^{3} m^{4} + 11 \, a b^{3} m^{3} + 41 \, a b^{3} m^{2} + 61 \, a b^{3} m + 30 \, a b^{3}\right )} x^{4} + 6 \, {\left (a^{2} b^{2} m^{4} + 12 \, a^{2} b^{2} m^{3} + 49 \, a^{2} b^{2} m^{2} + 78 \, a^{2} b^{2} m + 40 \, a^{2} b^{2}\right )} x^{3} + 4 \, {\left (a^{3} b m^{4} + 13 \, a^{3} b m^{3} + 59 \, a^{3} b m^{2} + 107 \, a^{3} b m + 60 \, a^{3} b\right )} x^{2} + {\left (a^{4} m^{4} + 14 \, a^{4} m^{3} + 71 \, a^{4} m^{2} + 154 \, a^{4} m + 120 \, a^{4}\right )} x\right )} \cosh \left (m \log \relax (x)\right ) + {\left ({\left (b^{4} m^{4} + 10 \, b^{4} m^{3} + 35 \, b^{4} m^{2} + 50 \, b^{4} m + 24 \, b^{4}\right )} x^{5} + 4 \, {\left (a b^{3} m^{4} + 11 \, a b^{3} m^{3} + 41 \, a b^{3} m^{2} + 61 \, a b^{3} m + 30 \, a b^{3}\right )} x^{4} + 6 \, {\left (a^{2} b^{2} m^{4} + 12 \, a^{2} b^{2} m^{3} + 49 \, a^{2} b^{2} m^{2} + 78 \, a^{2} b^{2} m + 40 \, a^{2} b^{2}\right )} x^{3} + 4 \, {\left (a^{3} b m^{4} + 13 \, a^{3} b m^{3} + 59 \, a^{3} b m^{2} + 107 \, a^{3} b m + 60 \, a^{3} b\right )} x^{2} + {\left (a^{4} m^{4} + 14 \, a^{4} m^{3} + 71 \, a^{4} m^{2} + 154 \, a^{4} m + 120 \, a^{4}\right )} x\right )} \sinh \left (m \log \relax (x)\right )}{m^{5} + 15 \, m^{4} + 85 \, m^{3} + 225 \, m^{2} + 274 \, m + 120} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(((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 + 6
1*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*log(x)))/(m^5 + 15*m^4 + 85*m^3
+ 225*m^2 + 274*m + 120)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.15, size = 278, normalized size = 1.81 \[ \frac {b^{4} x^{5} {\mathrm e}^{m \ln \relax (x )}}{5+m}+\frac {\left (a^{4}+4 a^{3} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+6 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+4 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{4}\right ) x \,{\mathrm e}^{m \ln \relax (x )}}{1+m}+\frac {4 b \left (a^{3}+3 a^{2} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+3 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{3}\right ) x^{2} {\mathrm e}^{m \ln \relax (x )}}{2+m}+\frac {6 b^{2} \left (a^{2}+2 a \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )+\left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x -a \right )^{2}\right ) x^{3} {\mathrm e}^{m \ln \relax (x )}}{3+m}+\frac {4 b^{3} \left (\arctanh \left (\tanh \left (b x +a \right )\right )-b x \right ) x^{4} {\mathrm e}^{m \ln \relax (x )}}{4+m} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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*(a
rctanh(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))

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maxima [A]  time = 0.54, size = 145, normalized size = 0.94 \[ -\frac {4 \, b x^{2} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{{\left (m + 2\right )} {\left (m + 1\right )}} + \frac {x^{m + 1} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{m + 1} + \frac {12 \, {\left (\frac {b x^{3} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{{\left (m + 3\right )} {\left (m + 2\right )}} + \frac {2 \, {\left (\frac {b^{2} x^{5} x^{m}}{{\left (m + 5\right )} {\left (m + 4\right )} {\left (m + 3\right )}} - \frac {b x^{4} x^{m} \operatorname {artanh}\left (\tanh \left (b x + a\right )\right )}{{\left (m + 4\right )} {\left (m + 3\right )}}\right )} b}{m + 2}\right )} b}{m + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.34, size = 479, normalized size = 3.11 \[ \frac {x\,x^m\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^4\,\left (m^4+14\,m^3+71\,m^2+154\,m+120\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}+\frac {16\,b^4\,x^m\,x^5\,\left (m^4+10\,m^3+35\,m^2+50\,m+24\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}+\frac {24\,b^2\,x^m\,x^3\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^2\,\left (m^4+12\,m^3+49\,m^2+78\,m+40\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}-\frac {32\,b^3\,x^m\,x^4\,\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )\,\left (m^4+11\,m^3+41\,m^2+61\,m+30\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920}-\frac {8\,b\,x^m\,x^2\,{\left (\ln \left (\frac {2}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )-\ln \left (\frac {2\,{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )+2\,b\,x\right )}^3\,\left (m^4+13\,m^3+59\,m^2+107\,m+60\right )}{16\,m^5+240\,m^4+1360\,m^3+3600\,m^2+4384\,m+1920} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*atanh(tanh(a + b*x))^4,x)

[Out]

(x*x^m*(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)^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 + 1920) - (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)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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)), (Integral(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(tanh(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 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 71*m**2*x*x**m
*atanh(tanh(a + b*x))**4/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 154*m*x*x**m*atanh(tanh(a + b*x
))**4/(m**5 + 15*m**4 + 85*m**3 + 225*m**2 + 274*m + 120) + 120*x*x**m*atanh(tanh(a + b*x))**4/(m**5 + 15*m**4
+ 85*m**3 + 225*m**2 + 274*m + 120), True))

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