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

Optimal. Leaf size=26 \[ -\frac{1}{2} b \text{PolyLog}(2,-c x)+\frac{1}{2} b \text{PolyLog}(2,c x)+a \log (x) \]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x)])/2 + (b*PolyLog[2, c*x])/2

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Rubi [A]  time = 0.0148648, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {5912} \[ -\frac{1}{2} b \text{PolyLog}(2,-c x)+\frac{1}{2} b \text{PolyLog}(2,c x)+a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x])/x,x]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x)])/2 + (b*PolyLog[2, c*x])/2

Rule 5912

Int[((a_.) + ArcTanh[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (-Simp[(b*PolyLog[2, -(c*x)])/2
, x] + Simp[(b*PolyLog[2, c*x])/2, x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx &=a \log (x)-\frac{1}{2} b \text{Li}_2(-c x)+\frac{1}{2} b \text{Li}_2(c x)\\ \end{align*}

Mathematica [A]  time = 0.0093242, size = 24, normalized size = 0.92 \[ \frac{1}{2} b (\text{PolyLog}(2,c x)-\text{PolyLog}(2,-c x))+a \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x])/x,x]

[Out]

a*Log[x] + (b*(-PolyLog[2, -(c*x)] + PolyLog[2, c*x]))/2

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Maple [B]  time = 0.013, size = 47, normalized size = 1.8 \begin{align*} a\ln \left ( cx \right ) +b\ln \left ( cx \right ){\it Artanh} \left ( cx \right ) -{\frac{b{\it dilog} \left ( cx \right ) }{2}}-{\frac{b{\it dilog} \left ( cx+1 \right ) }{2}}-{\frac{b\ln \left ( cx \right ) \ln \left ( cx+1 \right ) }{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*ln(c*x)+b*ln(c*x)*arctanh(c*x)-1/2*b*dilog(c*x)-1/2*b*dilog(c*x+1)-1/2*b*ln(c*x)*ln(c*x+1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{2} \, b \int \frac{\log \left (c x + 1\right ) - \log \left (-c x + 1\right )}{x}\,{d x} + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*b*integrate((log(c*x + 1) - log(-c*x + 1))/x, x) + a*log(x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \operatorname{artanh}\left (c x\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*arctanh(c*x) + a)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*atanh(c*x))/x, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (c x\right ) + a}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((b*arctanh(c*x) + a)/x, x)