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

Optimal. Leaf size=21 \[ b x-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \]

[Out]

b*x-(b*x-arctanh(tanh(b*x+a)))*ln(x)

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Rubi [A]  time = 0.04, antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {2158, 29} \[ b x-\log (x) \left (b x-\tanh ^{-1}(\tanh (a+b x))\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcTanh[Tanh[a + b*x]]/x,x]

[Out]

b*x - (b*x - ArcTanh[Tanh[a + b*x]])*Log[x]

Rule 29

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

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]

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 19, normalized size = 0.90 \[ \log (x) \left (\tanh ^{-1}(\tanh (a+b x))-b x\right )+b x \]

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[Tanh[a + b*x]]/x,x]

[Out]

b*x + (-(b*x) + ArcTanh[Tanh[a + b*x]])*Log[x]

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fricas [A]  time = 0.66, size = 8, normalized size = 0.38 \[ b x + a \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(x)

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giac [A]  time = 0.19, size = 9, normalized size = 0.43 \[ b x + a \log \left ({\left | x \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x + a*log(abs(x))

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maple [A]  time = 0.15, size = 21, normalized size = 1.00 \[ \ln \relax (x ) \arctanh \left (\tanh \left (b x +a \right )\right )-\ln \relax (x ) x b +b x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

ln(x)*arctanh(tanh(b*x+a))-ln(x)*x*b+b*x

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maxima [A]  time = 0.33, size = 34, normalized size = 1.62 \[ -b {\left (x + \frac {a}{b}\right )} \log \relax (x) + b {\left (x + \frac {a \log \relax (x)}{b}\right )} + \operatorname {artanh}\left (\tanh \left (b x + a\right )\right ) \log \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b*(x + a/b)*log(x) + b*(x + a*log(x)/b) + arctanh(tanh(b*x + a))*log(x)

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mupad [B]  time = 1.10, size = 58, normalized size = 2.76 \[ b\,x-\frac {\ln \left (\frac {1}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)}{2}+\frac {\ln \left (\frac {{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}}{{\mathrm {e}}^{2\,a}\,{\mathrm {e}}^{2\,b\,x}+1}\right )\,\ln \relax (x)}{2}-b\,x\,\ln \relax (x) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(tanh(a + b*x))/x,x)

[Out]

b*x - (log(1/(exp(2*a)*exp(2*b*x) + 1))*log(x))/2 + (log((exp(2*a)*exp(2*b*x))/(exp(2*a)*exp(2*b*x) + 1))*log(
x))/2 - b*x*log(x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {atanh}{\left (\tanh {\left (a + b x \right )} \right )}}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(atanh(tanh(a + b*x))/x, x)

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