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

Optimal. Leaf size=77 \[ -\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-4 b^3 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+4 b^4 x \]

[Out]

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

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Rubi [A]  time = 0.0582334, antiderivative size = 77, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {2168, 2158, 29} \[ -\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-4 b^3 \log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+4 b^4 x \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]^4/x^4,x]

[Out]

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

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 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))^4}{x^4} \, dx &=-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\frac{1}{3} (4 b) \int \frac{\tanh ^{-1}(\tanh (a+b x))^3}{x^3} \, dx\\ &=-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\left (2 b^2\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))^2}{x^2} \, dx\\ &=-\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}+\left (4 b^3\right ) \int \frac{\tanh ^{-1}(\tanh (a+b x))}{x} \, dx\\ &=4 b^4 x-\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-\left (4 b^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right )\right ) \int \frac{1}{x} \, dx\\ &=4 b^4 x-\frac{2 b^2 \tanh ^{-1}(\tanh (a+b x))^2}{x}-\frac{2 b \tanh ^{-1}(\tanh (a+b x))^3}{3 x^2}-\frac{\tanh ^{-1}(\tanh (a+b x))^4}{3 x^3}-4 b^3 \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0481872, size = 82, normalized size = 1.06 \[ -\frac{6 b^2 x^2 \tanh ^{-1}(\tanh (a+b x))^2-2 b^3 x^3 (6 \log (x)+11) \tanh ^{-1}(\tanh (a+b x))+2 b x \tanh ^{-1}(\tanh (a+b x))^3+\tanh ^{-1}(\tanh (a+b x))^4+2 b^4 x^4 (6 \log (x)+5)}{3 x^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(6*b^2*x^2*ArcTanh[Tanh[a + b*x]]^2 + 2*b*x*ArcTanh[Tanh[a + b*x]]^3 + ArcTanh[Tanh[a + b*x]]^4 + 2*b^4*x^4*(
5 + 6*Log[x]) - 2*b^3*x^3*ArcTanh[Tanh[a + b*x]]*(11 + 6*Log[x]))/(3*x^3)

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Maple [A]  time = 0.046, size = 76, normalized size = 1. \begin{align*} -{\frac{ \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{4}}{3\,{x}^{3}}}-{\frac{2\,b \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{3}}{3\,{x}^{2}}}-2\,{\frac{{b}^{2} \left ({\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \right ) ^{2}}{x}}-4\,\ln \left ( x \right ) x{b}^{4}+4\,{\it Artanh} \left ( \tanh \left ( bx+a \right ) \right ) \ln \left ( x \right ){b}^{3}+4\,{b}^{4}x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.61241, size = 123, normalized size = 1.6 \begin{align*} 2 \,{\left (2 \,{\left (b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right ) \log \left (x\right ) -{\left (b{\left (x + \frac{a}{b}\right )} \log \left (x\right ) - b{\left (x + \frac{a \log \left (x\right )}{b}\right )}\right )} b\right )} b - \frac{b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{2}}{x}\right )} b - \frac{2 \, b \operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{3}}{3 \, x^{2}} - \frac{\operatorname{artanh}\left (\tanh \left (b x + a\right )\right )^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.54661, size = 105, normalized size = 1.36 \begin{align*} \frac{3 \, b^{4} x^{4} + 12 \, a b^{3} x^{3} \log \left (x\right ) - 18 \, a^{2} b^{2} x^{2} - 6 \, a^{3} b x - a^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*b^4*x^4 + 12*a*b^3*x^3*log(x) - 18*a^2*b^2*x^2 - 6*a^3*b*x - a^4)/x^3

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{atanh}^{4}{\left (\tanh{\left (a + b x \right )} \right )}}{x^{4}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13538, size = 57, normalized size = 0.74 \begin{align*} b^{4} x + 4 \, a b^{3} \log \left ({\left | x \right |}\right ) - \frac{18 \, a^{2} b^{2} x^{2} + 6 \, a^{3} b x + a^{4}}{3 \, x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4*x + 4*a*b^3*log(abs(x)) - 1/3*(18*a^2*b^2*x^2 + 6*a^3*b*x + a^4)/x^3