3.59 \(\int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x} \, dx\)

Optimal. Leaf size=77 \[ b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]

[Out]

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

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Rubi [A]  time = 0.0809734, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2159, 2158, 29} \[ b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \tanh ^{-1}(\tanh (a+b x))^2 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^3/x,x]

[Out]

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

Rule 2159

Int[(v_)^(n_)/(u_), x_Symbol] :> With[{a = Simplify[D[u, x]], b = Simplify[D[v, x]]}, Simp[v^n/(a*n), x] - Dis
t[(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] && Ne
Q[n, 1]

Rule 2158

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

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rubi steps

\begin{align*} \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x} \, dx &=\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x} \, dx\\ &=-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3+\left (\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2 \left (-b x+\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=b x \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^2-\frac{1}{2} \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \tanh ^{-1}(\tanh (a+b x))^2+\frac{1}{3} \tanh ^{-1}(\tanh (a+b x))^3-\left (b x-\tanh ^{-1}(\tanh (a+b x))\right )^3 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0576073, size = 104, normalized size = 1.35 \[ (a+b x) \left (a^2-3 a \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )+3 \left (-\tanh ^{-1}(\tanh (a+b x))+a+b x\right )^2\right )+\frac{1}{3} (a+b x)^3-\frac{1}{2} (a+b x)^2 \left (-3 \tanh ^{-1}(\tanh (a+b x))+2 a+3 b x\right )+\log (b x) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )^3 \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]^3/x,x]

[Out]

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

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Maple [A]  time = 0.037, size = 92, normalized size = 1.2 \begin{align*} \ln \left ( x \right ) \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}+3\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \ln \left ( x \right ){x}^{2}{b}^{2}-{\frac{9\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ){x}^{2}{b}^{2}}{2}}-3\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}\ln \left ( x \right ) x-{b}^{3}{x}^{3}\ln \left ( x \right ) +{\frac{11\,{x}^{3}{b}^{3}}{6}}+3\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(tanh(b*x+a))^3/x,x)

[Out]

ln(x)*arctanh(tanh(b*x+a))^3+3*arctanh(tanh(b*x+a))*ln(x)*x^2*b^2-9/2*arctanh(tanh(b*x+a))*x^2*b^2-3*b*arctanh
(tanh(b*x+a))^2*ln(x)*x-b^3*x^3*ln(x)+11/6*x^3*b^3+3*b*arctanh(tanh(b*x+a))^2*x

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Maxima [A]  time = 2.44981, size = 42, normalized size = 0.55 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^3/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.57665, size = 73, normalized size = 0.95 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^3/x,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{3}{\left (\tanh{\left (a + b x \right )} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(tanh(b*x+a))**3/x,x)

[Out]

Integral(atanh(tanh(a + b*x))**3/x, x)

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Giac [A]  time = 1.15529, size = 43, normalized size = 0.56 \begin{align*} \frac{1}{3} \, b^{3} x^{3} + \frac{3}{2} \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(tanh(b*x+a))^3/x,x, algorithm="giac")

[Out]

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