3.217 \(\int \frac{a+b \tanh ^{-1}(c x^{3/2})}{x} \, dx\)

Optimal. Leaf size=34 \[ -\frac{1}{3} b \text{PolyLog}\left (2,-c x^{3/2}\right )+\frac{1}{3} b \text{PolyLog}\left (2,c x^{3/2}\right )+a \log (x) \]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^(3/2))])/3 + (b*PolyLog[2, c*x^(3/2)])/3

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Rubi [A]  time = 0.033444, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {6095, 5912} \[ -\frac{1}{3} b \text{PolyLog}\left (2,-c x^{3/2}\right )+\frac{1}{3} b \text{PolyLog}\left (2,c x^{3/2}\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b*ArcTanh[c*x^(3/2)])/x,x]

[Out]

a*Log[x] - (b*PolyLog[2, -(c*x^(3/2))])/3 + (b*PolyLog[2, c*x^(3/2)])/3

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^{3/2}\right )}{x} \, dx &=\frac{2}{3} \operatorname{Subst}\left (\int \frac{a+b \tanh ^{-1}(c x)}{x} \, dx,x,x^{3/2}\right )\\ &=a \log (x)-\frac{1}{3} b \text{Li}_2\left (-c x^{3/2}\right )+\frac{1}{3} b \text{Li}_2\left (c x^{3/2}\right )\\ \end{align*}

Mathematica [A]  time = 0.0196923, size = 32, normalized size = 0.94 \[ \frac{1}{3} b \left (\text{PolyLog}\left (2,c x^{3/2}\right )-\text{PolyLog}\left (2,-c x^{3/2}\right )\right )+a \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*ArcTanh[c*x^(3/2)])/x,x]

[Out]

a*Log[x] + (b*(-PolyLog[2, -(c*x^(3/2))] + PolyLog[2, c*x^(3/2)]))/3

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Maple [B]  time = 0.036, size = 63, normalized size = 1.9 \begin{align*}{\frac{2\,a}{3}\ln \left ( c{x}^{{\frac{3}{2}}} \right ) }+{\frac{2\,b}{3}\ln \left ( c{x}^{{\frac{3}{2}}} \right ){\it Artanh} \left ( c{x}^{{\frac{3}{2}}} \right ) }-{\frac{b}{3}{\it dilog} \left ( c{x}^{{\frac{3}{2}}} \right ) }-{\frac{b}{3}{\it dilog} \left ( c{x}^{{\frac{3}{2}}}+1 \right ) }-{\frac{b}{3}\ln \left ( c{x}^{{\frac{3}{2}}} \right ) \ln \left ( c{x}^{{\frac{3}{2}}}+1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*arctanh(c*x^(3/2)))/x,x)

[Out]

2/3*a*ln(c*x^(3/2))+2/3*b*ln(c*x^(3/2))*arctanh(c*x^(3/2))-1/3*b*dilog(c*x^(3/2))-1/3*b*dilog(c*x^(3/2)+1)-1/3
*b*ln(c*x^(3/2))*ln(c*x^(3/2)+1)

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Maxima [B]  time = 1.59449, size = 84, normalized size = 2.47 \begin{align*} -\frac{1}{3} \,{\left (\log \left (c x^{\frac{3}{2}}\right ) \log \left (-c x^{\frac{3}{2}} + 1\right ) +{\rm Li}_2\left (-c x^{\frac{3}{2}} + 1\right )\right )} b + \frac{1}{3} \,{\left (\log \left (c x^{\frac{3}{2}} + 1\right ) \log \left (-c x^{\frac{3}{2}}\right ) +{\rm Li}_2\left (c x^{\frac{3}{2}} + 1\right )\right )} b + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*arctanh(c*x^(3/2)))/x,x, algorithm="maxima")

[Out]

-1/3*(log(c*x^(3/2))*log(-c*x^(3/2) + 1) + dilog(-c*x^(3/2) + 1))*b + 1/3*(log(c*x^(3/2) + 1)*log(-c*x^(3/2))
+ dilog(c*x^(3/2) + 1))*b + 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^{\frac{3}{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^(3/2)))/x,x, algorithm="fricas")

[Out]

integral((b*arctanh(c*x^(3/2)) + a)/x, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*atanh(c*x**(3/2)))/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{b \operatorname{artanh}\left (c x^{\frac{3}{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^(3/2)))/x,x, algorithm="giac")

[Out]

integrate((b*arctanh(c*x^(3/2)) + a)/x, x)