3.54 \(\int \frac{a+b \tanh ^{-1}(c x^2)}{x} \, dx\)

Optimal. Leaf size=30 \[ -\frac{1}{4} b \text{PolyLog}\left (2,-c x^2\right )+\frac{1}{4} b \text{PolyLog}\left (2,c x^2\right )+a \log (x) \]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^2)])/4 + (b*PolyLog[2, c*x^2])/4

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Rubi [A]  time = 0.0331072, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {6095, 5912} \[ -\frac{1}{4} b \text{PolyLog}\left (2,-c x^2\right )+\frac{1}{4} b \text{PolyLog}\left (2,c x^2\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^2])/x,x]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^2)])/4 + (b*PolyLog[2, c*x^2])/4

Rule 6095

Int[((a_.) + ArcTanh[(c_.)*(x_)^(n_)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/n, Subst[Int[(a + b*ArcTanh[c*x])
^p/x, x], x, x^n], x] /; FreeQ[{a, b, c, n}, x] && IGtQ[p, 0]

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}\left (c x^2\right )}{x} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^2\right )\\ &=a \log (x)-\frac{1}{4} b \text{Li}_2\left (-c x^2\right )+\frac{1}{4} b \text{Li}_2\left (c x^2\right )\\ \end{align*}

Mathematica [A]  time = 0.0133833, size = 28, normalized size = 0.93 \[ \frac{1}{4} b \left (\text{PolyLog}\left (2,c x^2\right )-\text{PolyLog}\left (2,-c x^2\right )\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^2])/x,x]

[Out]

a*Log[x] + (b*(-PolyLog[2, -(c*x^2)] + PolyLog[2, c*x^2]))/4

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Maple [B]  time = 0.029, size = 124, normalized size = 4.1 \begin{align*} a\ln \left ( x \right ) +b\ln \left ( x \right ){\it Artanh} \left ( c{x}^{2} \right ) -{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1+x\sqrt{-c} \right ) }-{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1-x\sqrt{-c} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1+x\sqrt{-c} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1-x\sqrt{-c} \right ) }+{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1-x\sqrt{c} \right ) }+{\frac{b\ln \left ( x \right ) }{2}\ln \left ( 1+x\sqrt{c} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1-x\sqrt{c} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1+x\sqrt{c} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^2))/x,x)

[Out]

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

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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^{2} + 1\right ) - \log \left (-c x^{2} + 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^2))/x,x, algorithm="maxima")

[Out]

1/2*b*integrate((log(c*x^2 + 1) - log(-c*x^2 + 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^{2}\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^2))/x,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x^2) + 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^{2} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**2))/x,x)

[Out]

Integral((a + b*atanh(c*x**2))/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^{2}\right ) + a}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^2))/x,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^2) + a)/x, x)