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

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

[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 = 0.0327894, 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} \[ \frac{1}{4} b \text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\frac{1}{4} b \text{PolyLog}\left (2,\frac{c}{x^2}\right )+a \log (x) \]

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

Mathematica [A]  time = 0.0139595, size = 28, normalized size = 0.93 \[ \frac{1}{4} b \left (\text{PolyLog}\left (2,-\frac{c}{x^2}\right )-\text{PolyLog}\left (2,\frac{c}{x^2}\right )\right )+a \log (x) \]

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 = 0.03, size = 154, normalized size = 5.1 \begin{align*} -a\ln \left ({x}^{-1} \right ) -b\ln \left ({x}^{-1} \right ){\it Artanh} \left ({\frac{c}{{x}^{2}}} \right ) +{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1+{\frac{1}{x}\sqrt{-c}} \right ) }+{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1-{\frac{1}{x}\sqrt{-c}} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1+{\frac{1}{x}\sqrt{-c}} \right ) }+{\frac{b}{2}{\it dilog} \left ( 1-{\frac{1}{x}\sqrt{-c}} \right ) }-{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1-{\frac{1}{x}\sqrt{c}} \right ) }-{\frac{b\ln \left ({x}^{-1} \right ) }{2}\ln \left ( 1+{\frac{1}{x}\sqrt{c}} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1-{\frac{1}{x}\sqrt{c}} \right ) }-{\frac{b}{2}{\it dilog} \left ( 1+{\frac{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(1/x)-b*ln(1/x)*arctanh(c/x^2)+1/2*b*ln(1/x)*ln(1+(-c)^(1/2)/x)+1/2*b*ln(1/x)*ln(1-(-c)^(1/2)/x)+1/2*b*di
log(1+(-c)^(1/2)/x)+1/2*b*dilog(1-(-c)^(1/2)/x)-1/2*b*ln(1/x)*ln(1-1/x*c^(1/2))-1/2*b*ln(1/x)*ln(1+1/x*c^(1/2)
)-1/2*b*dilog(1-1/x*c^(1/2))-1/2*b*dilog(1+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 (\frac{c}{x^{2}} + 1\right ) - \log \left (-\frac{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 (\frac{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 (\frac{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 (\frac{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)