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

Optimal. Leaf size=28 \[ \frac {1}{10} \text {Li}_2\left (-\frac {1}{a x^5}\right )-\frac {1}{10} \text {Li}_2\left (\frac {1}{a x^5}\right ) \]

[Out]

1/10*polylog(2,-1/a/x^5)-1/10*polylog(2,1/a/x^5)

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Rubi [A]  time = 0.02, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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 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]

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]

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*}

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Mathematica [A]  time = 0.01, size = 26, normalized size = 0.93 \[ \frac {1}{10} \left (\text {Li}_2\left (-\frac {1}{a x^5}\right )-\text {Li}_2\left (\frac {1}{a x^5}\right )\right ) \]

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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fricas [F]  time = 0.51, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arcoth}\left (a x^{5}\right )}{x}, x\right ) \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcoth}\left (a x^{5}\right )}{x}\,{d x} \]

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)

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maple [C]  time = 0.17, size = 85, normalized size = 3.04 \[ \ln \relax (x ) \mathrm {arccoth}\left (a \,x^{5}\right )-\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (a \,\textit {\_Z}^{5}+1\right )}{\sum }\left (\ln \relax (x ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2}+\frac {\left (\munderset {\textit {\_R1} =\RootOf \left (a \,\textit {\_Z}^{5}-1\right )}{\sum }\left (\ln \relax (x ) \ln \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )+\dilog \left (\frac {\textit {\_R1} -x}{\textit {\_R1}}\right )\right )\right )}{2} \]

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 = 0.31, size = 104, normalized size = 3.71 \[ -\frac {1}{2} \, a {\left (\frac {\log \left (a x^{5} + 1\right )}{a} - \frac {\log \left (a x^{5} - 1\right )}{a}\right )} \log \relax (x) - \frac {1}{10} \, a {\left (\frac {\log \left (a x^{5} - 1\right ) \log \left (a x^{5}\right ) + {\rm Li}_2\left (-a x^{5} + 1\right )}{a} - \frac {\log \left (a x^{5} + 1\right ) \log \left (-a x^{5}\right ) + {\rm Li}_2\left (a x^{5} + 1\right )}{a}\right )} + \operatorname {arcoth}\left (a x^{5}\right ) \log \relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {\mathrm {acoth}\left (a\,x^5\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acoth(a*x^5)/x, x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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