3.3.36 \(\int \frac {\tanh ^{-1}(a x^n)}{x} \, dx\) [236]

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

[Out]

-1/2*polylog(2,-a*x^n)/n+1/2*polylog(2,a*x^n)/n

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Rubi [A]
time = 0.02, antiderivative size = 30, 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 = {6035, 6031} \begin {gather*} \frac {\text {Li}_2\left (a x^n\right )}{2 n}-\frac {\text {Li}_2\left (-a x^n\right )}{2 n} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[ArcTanh[a*x^n]/x,x]

[Out]

-1/2*PolyLog[2, -(a*x^n)]/n + PolyLog[2, a*x^n]/(2*n)

Rule 6031

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

Rule 6035

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]

Rubi steps

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

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 0.03, size = 33, normalized size = 1.10 \begin {gather*} \frac {a x^n \, _3F_2\left (\frac {1}{2},\frac {1}{2},1;\frac {3}{2},\frac {3}{2};a^2 x^{2 n}\right )}{n} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[ArcTanh[a*x^n]/x,x]

[Out]

(a*x^n*HypergeometricPFQ[{1/2, 1/2, 1}, {3/2, 3/2}, a^2*x^(2*n)])/n

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Maple [A]
time = 0.13, size = 53, normalized size = 1.77

method	result	size
risch	\(\frac {\dilog \left (1-a \,x^{n}\right )}{2 n}-\frac {\dilog \left (a \,x^{n}+1\right )}{2 n}\)	\(29\)
derivativedivides	\(\frac {\ln \left (a \,x^{n}\right ) \arctanh \left (a \,x^{n}\right )-\frac {\dilog \left (a \,x^{n}\right )}{2}-\frac {\dilog \left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\)	\(53\)
default	\(\frac {\ln \left (a \,x^{n}\right ) \arctanh \left (a \,x^{n}\right )-\frac {\dilog \left (a \,x^{n}\right )}{2}-\frac {\dilog \left (a \,x^{n}+1\right )}{2}-\frac {\ln \left (a \,x^{n}\right ) \ln \left (a \,x^{n}+1\right )}{2}}{n}\)	\(53\)
meijerg	\(-\frac {i \left (\frac {2 i a \,x^{n} \polylog \left (2, \sqrt {x^{2 n} a^{2}}\right )}{\sqrt {x^{2 n} a^{2}}}-\frac {2 i a \,x^{n} \polylog \left (2, -\sqrt {x^{2 n} a^{2}}\right )}{\sqrt {x^{2 n} a^{2}}}\right )}{4 n}\)	\(72\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arctanh(a*x^n)/x,x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (24) = 48\).
time = 0.32, size = 147, normalized size = 4.90 \begin {gather*} -\frac {1}{2} \, a n {\left (\frac {\log \left (\frac {a x^{n} + 1}{a}\right )}{a n} - \frac {\log \left (\frac {a x^{n} - 1}{a}\right )}{a n}\right )} \log \left (x\right ) + \frac {1}{2} \, a n {\left (\frac {\log \left (a x^{n} + 1\right ) \log \left (x\right ) - \log \left (a x^{n} - 1\right ) \log \left (x\right )}{a n} - \frac {n \log \left (a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (-a x^{n}\right )}{a n^{2}} + \frac {n \log \left (-a x^{n} + 1\right ) \log \left (x\right ) + {\rm Li}_2\left (a x^{n}\right )}{a n^{2}}\right )} + \operatorname {artanh}\left (a x^{n}\right ) \log \left (x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x^n)/x,x, algorithm="maxima")

[Out]

-1/2*a*n*(log((a*x^n + 1)/a)/(a*n) - log((a*x^n - 1)/a)/(a*n))*log(x) + 1/2*a*n*((log(a*x^n + 1)*log(x) - log(
a*x^n - 1)*log(x))/(a*n) - (n*log(a*x^n + 1)*log(x) + dilog(-a*x^n))/(a*n^2) + (n*log(-a*x^n + 1)*log(x) + dil
og(a*x^n))/(a*n^2)) + arctanh(a*x^n)*log(x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 129 vs. \(2 (24) = 48\).
time = 0.36, size = 129, normalized size = 4.30 \begin {gather*} -\frac {n \log \left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right ) + 1\right ) \log \left (x\right ) - n \log \left (x\right ) \log \left (-\frac {a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) + 1}{a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right ) - 1}\right ) - {\rm Li}_2\left (a \cosh \left (n \log \left (x\right )\right ) + a \sinh \left (n \log \left (x\right )\right )\right ) + {\rm Li}_2\left (-a \cosh \left (n \log \left (x\right )\right ) - a \sinh \left (n \log \left (x\right )\right )\right )}{2 \, n} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x^n)/x,x, algorithm="fricas")

[Out]

-1/2*(n*log(a*cosh(n*log(x)) + a*sinh(n*log(x)) + 1)*log(x) - n*log(-a*cosh(n*log(x)) - a*sinh(n*log(x)) + 1)*
log(x) - n*log(x)*log(-(a*cosh(n*log(x)) + a*sinh(n*log(x)) + 1)/(a*cosh(n*log(x)) + a*sinh(n*log(x)) - 1)) -
dilog(a*cosh(n*log(x)) + a*sinh(n*log(x))) + dilog(-a*cosh(n*log(x)) - a*sinh(n*log(x))))/n

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (a x^{n} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(atanh(a*x**n)/x,x)

[Out]

Integral(atanh(a*x**n)/x, x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arctanh(a*x^n)/x,x, algorithm="giac")

[Out]

integrate(arctanh(a*x^n)/x, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {atanh}\left (a\,x^n\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(a*x^n)/x,x)

[Out]

int(atanh(a*x^n)/x, x)

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