3.23 \(\int \frac{\text{csch}^{-1}(a x^n)}{x} \, dx\)

Optimal. Leaf size=61 \[ -\frac{\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (a x^n\right )}\right )}{2 n}+\frac{\text{csch}^{-1}\left (a x^n\right )^2}{2 n}-\frac{\text{csch}^{-1}\left (a x^n\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^n\right )}\right )}{n} \]

[Out]

ArcCsch[a*x^n]^2/(2*n) - (ArcCsch[a*x^n]*Log[1 - E^(2*ArcCsch[a*x^n])])/n - PolyLog[2, E^(2*ArcCsch[a*x^n])]/(
2*n)

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Rubi [A]  time = 0.107918, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {6282, 5659, 3716, 2190, 2279, 2391} \[ -\frac{\text{PolyLog}\left (2,e^{2 \text{csch}^{-1}\left (a x^n\right )}\right )}{2 n}+\frac{\text{csch}^{-1}\left (a x^n\right )^2}{2 n}-\frac{\text{csch}^{-1}\left (a x^n\right ) \log \left (1-e^{2 \text{csch}^{-1}\left (a x^n\right )}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcCsch[a*x^n]^2/(2*n) - (ArcCsch[a*x^n]*Log[1 - E^(2*ArcCsch[a*x^n])])/n - PolyLog[2, E^(2*ArcCsch[a*x^n])]/(
2*n)

Rule 6282

Int[((a_.) + ArcCsch[(c_.)*(x_)]*(b_.))/(x_), x_Symbol] :> -Subst[Int[(a + b*ArcSinh[x/c])/x, x], x, 1/x] /; F
reeQ[{a, b, c}, x]

Rule 5659

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

Rule 3716

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c
+ d*x)^(m + 1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(E^(2*I*k*Pi)*(1 + E^(2*
(-(I*e) + f*fz*x))/E^(2*I*k*Pi))), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

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

Mathematica [C]  time = 0.0859943, size = 64, normalized size = 1.05 \[ \log (x) \left (\text{csch}^{-1}\left (a x^n\right )-\sinh ^{-1}\left (\frac{x^{-n}}{a}\right )\right )-\frac{x^{-n} \text{HypergeometricPFQ}\left (\left \{\frac{1}{2},\frac{1}{2},\frac{1}{2}\right \},\left \{\frac{3}{2},\frac{3}{2}\right \},-\frac{x^{-2 n}}{a^2}\right )}{a n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, -(1/(a^2*x^(2*n)))]/(a*n*x^n)) + (ArcCsch[a*x^n] - ArcSinh[1/
(a*x^n)])*Log[x]

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Maple [F]  time = 0.378, size = 0, normalized size = 0. \begin{align*} \int{\frac{{\rm arccsch} \left (a{x}^{n}\right )}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{2} n \int \frac{x^{2 \, n} \log \left (x\right )}{a^{2} x x^{2 \, n} +{\left (a^{2} x x^{2 \, n} + x\right )} \sqrt{a^{2} x^{2 \, n} + 1} + x}\,{d x} + n \int \frac{\log \left (x\right )}{a^{2} x x^{2 \, n} + x}\,{d x} - \log \left (a\right ) \log \left (x\right ) - \log \left (x\right ) \log \left (x^{n}\right ) + \log \left (x\right ) \log \left (\sqrt{a^{2} x^{2 \, n} + 1} + 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a^2*n*integrate(x^(2*n)*log(x)/(a^2*x*x^(2*n) + (a^2*x*x^(2*n) + x)*sqrt(a^2*x^(2*n) + 1) + x), x) + n*integra
te(log(x)/(a^2*x*x^(2*n) + x), x) - log(a)*log(x) - log(x)*log(x^n) + log(x)*log(sqrt(a^2*x^(2*n) + 1) + 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: UnboundLocalError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acsch(a*x**n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccsch(a*x^n)/x, x)