3.82 \(\int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^{10}} \, dx\)

Optimal. Leaf size=80 \[ -\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}-\frac{b^4}{630 x^5} \]

[Out]

-b^4/(630*x^5) - (b^3*ArcTanh[Tanh[a + b*x]])/(126*x^6) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(42*x^7) - (b*ArcTanh
[Tanh[a + b*x]]^3)/(18*x^8) - ArcTanh[Tanh[a + b*x]]^4/(9*x^9)

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Rubi [A]  time = 0.0554779, antiderivative size = 80, normalized size of antiderivative = 1., 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} \[ -\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}-\frac{b^4}{630 x^5} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^4/x^10,x]

[Out]

-b^4/(630*x^5) - (b^3*ArcTanh[Tanh[a + b*x]])/(126*x^6) - (b^2*ArcTanh[Tanh[a + b*x]]^2)/(42*x^7) - (b*ArcTanh
[Tanh[a + b*x]]^3)/(18*x^8) - ArcTanh[Tanh[a + b*x]]^4/(9*x^9)

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]))

Rule 30

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

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^{10}} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{9} (4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^9} \, dx\\ &=-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{6} b^2 \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^8} \, dx\\ &=-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{21} b^3 \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x^7} \, dx\\ &=-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}+\frac{1}{126} b^4 \int \frac{1}{x^6} \, dx\\ &=-\frac{b^4}{630 x^5}-\frac{b^3 \tanh ^{-1}(\tanh (a+b x))}{126 x^6}-\frac{b^2 \tanh ^{-1}(\tanh (a+b x))^2}{42 x^7}-\frac{b \tanh ^{-1}(\tanh (a+b x))^3}{18 x^8}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{9 x^9}\\ \end{align*}

Mathematica [A]  time = 0.0599034, size = 71, normalized size = 0.89 \[ -\frac{5 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+15 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+35 b x \tanh ^{-1}(\tanh (a+b x))^3+70 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{630 x^9} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^4/x^10,x]

[Out]

-(b^4*x^4 + 5*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 15*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 + 35*b*x*ArcTanh[Tanh[a + b
*x]]^3 + 70*ArcTanh[Tanh[a + b*x]]^4)/(630*x^9)

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Maple [A]  time = 0.04, size = 74, normalized size = 0.9 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{9\,{x}^{9}}}+{\frac{4\,b}{9} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{8\,{x}^{8}}}+{\frac{3\,b}{8} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{7\,{x}^{7}}}+{\frac{2\,b}{7} \left ( -{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{6\,{x}^{6}}}-{\frac{b}{30\,{x}^{5}}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^4/x^10,x)

[Out]

-1/9*arctanh(tanh(b*x+a))^4/x^9+4/9*b*(-1/8/x^8*arctanh(tanh(b*x+a))^3+3/8*b*(-1/7/x^7*arctanh(tanh(b*x+a))^2+
2/7*b*(-1/6/x^6*arctanh(tanh(b*x+a))-1/30/x^5*b)))

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Maxima [A]  time = 1.78264, size = 97, normalized size = 1.21 \begin{align*} -\frac{1}{630} \,{\left (b{\left (\frac{b^{2}}{x^{5}} + \frac{5 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{6}}\right )} + \frac{15 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{7}}\right )} b - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{18 \, x^{8}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{9 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^10,x, algorithm="maxima")

[Out]

-1/630*(b*(b^2/x^5 + 5*b*arctanh(tanh(b*x + a))/x^6) + 15*b*arctanh(tanh(b*x + a))^2/x^7)*b - 1/18*b*arctanh(t
anh(b*x + a))^3/x^8 - 1/9*arctanh(tanh(b*x + a))^4/x^9

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Fricas [A]  time = 1.46187, size = 112, normalized size = 1.4 \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^10,x, algorithm="fricas")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9

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Sympy [A]  time = 25.8709, size = 76, normalized size = 0.95 \begin{align*} - \frac{b^{4}}{630 x^{5}} - \frac{b^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{126 x^{6}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{42 x^{7}} - \frac{b \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{18 x^{8}} - \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{9 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**4/x**10,x)

[Out]

-b**4/(630*x**5) - b**3*atanh(tanh(a + b*x))/(126*x**6) - b**2*atanh(tanh(a + b*x))**2/(42*x**7) - b*atanh(tan
h(a + b*x))**3/(18*x**8) - atanh(tanh(a + b*x))**4/(9*x**9)

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Giac [A]  time = 1.12085, size = 62, normalized size = 0.78 \begin{align*} -\frac{126 \, b^{4} x^{4} + 420 \, a b^{3} x^{3} + 540 \, a^{2} b^{2} x^{2} + 315 \, a^{3} b x + 70 \, a^{4}}{630 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^10,x, algorithm="giac")

[Out]

-1/630*(126*b^4*x^4 + 420*a*b^3*x^3 + 540*a^2*b^2*x^2 + 315*a^3*b*x + 70*a^4)/x^9