3.93 \(\int \frac{\coth ^{-1}(a x^5)}{x} \, dx\)

Optimal. Leaf size=28 \[ \frac{1}{10} \text{PolyLog}\left (2,-\frac{1}{a x^5}\right )-\frac{1}{10} \text{PolyLog}\left (2,\frac{1}{a x^5}\right ) \]

[Out]

PolyLog[2, -(1/(a*x^5))]/10 - PolyLog[2, 1/(a*x^5)]/10

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Rubi [A]  time = 0.0217677, antiderivative size = 28, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6096, 5913} \[ \frac{1}{10} \text{PolyLog}\left (2,-\frac{1}{a x^5}\right )-\frac{1}{10} \text{PolyLog}\left (2,\frac{1}{a x^5}\right ) \]

Antiderivative was successfully verified.

[In]

Int[ArcCoth[a*x^5]/x,x]

[Out]

PolyLog[2, -(1/(a*x^5))]/10 - PolyLog[2, 1/(a*x^5)]/10

Rule 6096

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

Rule 5913

Int[((a_.) + ArcCoth[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> Simp[a*Log[x], x] + (Simp[(b*PolyLog[2, -(c*x)^(-1)
])/2, x] - Simp[(b*PolyLog[2, 1/(c*x)])/2, x]) /; FreeQ[{a, b, c}, x]

Rubi steps

\begin{align*} \int \frac{\coth ^{-1}\left (a x^5\right )}{x} \, dx &=\frac{1}{5} \operatorname{Subst}\left (\int \frac{\coth ^{-1}(a x)}{x} \, dx,x,x^5\right )\\ &=\frac{1}{10} \text{Li}_2\left (-\frac{1}{a x^5}\right )-\frac{1}{10} \text{Li}_2\left (\frac{1}{a x^5}\right )\\ \end{align*}

Mathematica [A]  time = 0.0134413, size = 26, normalized size = 0.93 \[ \frac{1}{10} \left (\text{PolyLog}\left (2,-\frac{1}{a x^5}\right )-\text{PolyLog}\left (2,\frac{1}{a x^5}\right )\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[ArcCoth[a*x^5]/x,x]

[Out]

(PolyLog[2, -(1/(a*x^5))] - PolyLog[2, 1/(a*x^5)])/10

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Maple [C]  time = 0.146, size = 85, normalized size = 3. \begin{align*} \ln \left ( x \right ){\rm arccoth} \left (a{x}^{5}\right )+{\frac{1}{2}\sum _{{\it \_R1}={\it RootOf} \left ( a{{\it \_Z}}^{5}-1 \right ) }\ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) }-{\frac{1}{2}\sum _{{\it \_R1}={\it RootOf} \left ( a{{\it \_Z}}^{5}+1 \right ) }\ln \left ( x \right ) \ln \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) +{\it dilog} \left ({\frac{{\it \_R1}-x}{{\it \_R1}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccoth(a*x^5)/x,x)

[Out]

ln(x)*arccoth(a*x^5)+1/2*sum(ln(x)*ln((_R1-x)/_R1)+dilog((_R1-x)/_R1),_R1=RootOf(_Z^5*a-1))-1/2*sum(ln(x)*ln((
_R1-x)/_R1)+dilog((_R1-x)/_R1),_R1=RootOf(_Z^5*a+1))

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Maxima [B]  time = 0.990651, size = 140, normalized size = 5. \begin{align*} -\frac{1}{2} \, a{\left (\frac{\log \left (a x^{5} + 1\right )}{a} - \frac{\log \left (a x^{5} - 1\right )}{a}\right )} \log \left (x\right ) - \frac{1}{10} \, a{\left (\frac{\log \left (a x^{5} - 1\right ) \log \left (a x^{5}\right ) +{\rm Li}_2\left (-a x^{5} + 1\right )}{a} - \frac{\log \left (a x^{5} + 1\right ) \log \left (-a x^{5}\right ) +{\rm Li}_2\left (a x^{5} + 1\right )}{a}\right )} + \operatorname{arcoth}\left (a x^{5}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x^5)/x,x, algorithm="maxima")

[Out]

-1/2*a*(log(a*x^5 + 1)/a - log(a*x^5 - 1)/a)*log(x) - 1/10*a*((log(a*x^5 - 1)*log(a*x^5) + dilog(-a*x^5 + 1))/
a - (log(a*x^5 + 1)*log(-a*x^5) + dilog(a*x^5 + 1))/a) + arccoth(a*x^5)*log(x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arcoth}\left (a x^{5}\right )}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x^5)/x,x, algorithm="fricas")

[Out]

integral(arccoth(a*x^5)/x, x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: KeyError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acoth(a*x**5)/x,x)

[Out]

Exception raised: KeyError

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{arcoth}\left (a x^{5}\right )}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccoth(a*x^5)/x,x, algorithm="giac")

[Out]

integrate(arccoth(a*x^5)/x, x)