3.79 \(\int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^7} \, dx\)

Optimal. Leaf size=64 \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{30 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]

[Out]

(b*ArcTanh[Tanh[a + b*x]]^5)/(30*x^5*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^5/(6*x^6*(b*x
- ArcTanh[Tanh[a + b*x]]))

________________________________________________________________________________________

Rubi [A]  time = 0.0313652, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {2171, 2167} \[ \frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{30 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^4/x^7,x]

[Out]

(b*ArcTanh[Tanh[a + b*x]]^5)/(30*x^5*(b*x - ArcTanh[Tanh[a + b*x]])^2) + ArcTanh[Tanh[a + b*x]]^5/(6*x^6*(b*x
- ArcTanh[Tanh[a + b*x]]))

Rule 2171

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^
(n + 1))/((m + 1)*(b*u - a*v)), x] + Dist[(b*(m + n + 2))/((m + 1)*(b*u - a*v)), Int[u^(m + 1)*v^n, x], x] /;
NeQ[b*u - a*v, 0]] /; PiecewiseLinearQ[u, v, x] && NeQ[m + n + 2, 0] && LtQ[m, -1]

Rule 2167

Int[(u_)^(m_)*(v_)^(n_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, -Simp[(u^(m + 1)*v^
(n + 1))/((m + 1)*(b*u - a*v)), x] /; NeQ[b*u - a*v, 0]] /; FreeQ[{m, n}, x] && PiecewiseLinearQ[u, v, x] && E
qQ[m + n + 2, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^7} \, dx &=\frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}+\frac{b \int \frac{\tanh ^{-1}(\tanh (a+b x))^4}{x^6} \, dx}{6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ &=\frac{b \tanh ^{-1}(\tanh (a+b x))^5}{30 x^5 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2}+\frac{\tanh ^{-1}(\tanh (a+b x))^5}{6 x^6 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )}\\ \end{align*}

Mathematica [A]  time = 0.0313331, size = 71, normalized size = 1.11 \[ -\frac{2 b^3 x^3 \tanh ^{-1}(\tanh (a+b x))+3 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2+4 b x \tanh ^{-1}(\tanh (a+b x))^3+5 \tanh ^{-1}(\tanh (a+b x))^4+b^4 x^4}{30 x^6} \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^4/x^7,x]

[Out]

-(b^4*x^4 + 2*b^3*x^3*ArcTanh[Tanh[a + b*x]] + 3*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 + 4*b*x*ArcTanh[Tanh[a + b*x
]]^3 + 5*ArcTanh[Tanh[a + b*x]]^4)/(30*x^6)

________________________________________________________________________________________

Maple [A]  time = 0.042, size = 74, normalized size = 1.2 \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{6\,{x}^{6}}}+{\frac{2\,b}{3} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{5\,{x}^{5}}}+{\frac{3\,b}{5} \left ( -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{4\,{x}^{4}}}+{\frac{b}{2} \left ( -{\frac{b}{6\,{x}^{2}}}-{\frac{{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) }{3\,{x}^{3}}} \right ) } \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^4/x^7,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.79142, size = 97, normalized size = 1.52 \begin{align*} -\frac{1}{30} \,{\left (b{\left (\frac{b^{2}}{x^{2}} + \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )}{x^{3}}\right )} + \frac{3 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x^{4}}\right )} b - \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{15 \, x^{5}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{6 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^7,x, algorithm="maxima")

[Out]

-1/30*(b*(b^2/x^2 + 2*b*arctanh(tanh(b*x + a))/x^3) + 3*b*arctanh(tanh(b*x + a))^2/x^4)*b - 2/15*b*arctanh(tan
h(b*x + a))^3/x^5 - 1/6*arctanh(tanh(b*x + a))^4/x^6

________________________________________________________________________________________

Fricas [A]  time = 1.45738, size = 104, normalized size = 1.62 \begin{align*} -\frac{15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^7,x, algorithm="fricas")

[Out]

-1/30*(15*b^4*x^4 + 40*a*b^3*x^3 + 45*a^2*b^2*x^2 + 24*a^3*b*x + 5*a^4)/x^6

________________________________________________________________________________________

Sympy [A]  time = 6.67547, size = 78, normalized size = 1.22 \begin{align*} - \frac{b^{4}}{30 x^{2}} - \frac{b^{3} \operatorname{atanh}{\left (\tanh{\left (a + b x \right )} \right )}}{15 x^{3}} - \frac{b^{2} \operatorname{atanh}^{2}{\left (\tanh{\left (a + b x \right )} \right )}}{10 x^{4}} - \frac{2 b \operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{15 x^{5}} - \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{6 x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**4/x**7,x)

[Out]

-b**4/(30*x**2) - b**3*atanh(tanh(a + b*x))/(15*x**3) - b**2*atanh(tanh(a + b*x))**2/(10*x**4) - 2*b*atanh(tan
h(a + b*x))**3/(15*x**5) - atanh(tanh(a + b*x))**4/(6*x**6)

________________________________________________________________________________________

Giac [A]  time = 1.16793, size = 62, normalized size = 0.97 \begin{align*} -\frac{15 \, b^{4} x^{4} + 40 \, a b^{3} x^{3} + 45 \, a^{2} b^{2} x^{2} + 24 \, a^{3} b x + 5 \, a^{4}}{30 \, x^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^4/x^7,x, algorithm="giac")

[Out]

-1/30*(15*b^4*x^4 + 40*a*b^3*x^3 + 45*a^2*b^2*x^2 + 24*a^3*b*x + 5*a^4)/x^6