3.239 \(\int \frac{\cosh ^{-1}(a x^n)}{x} \, dx\)

Optimal. Leaf size=60 \[ \frac{\text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac{\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cosh ^{-1}\left (a x^n\right ) \log \left (e^{2 \cosh ^{-1}\left (a x^n\right )}+1\right )}{n} \]

[Out]

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

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Rubi [A]  time = 0.0654295, antiderivative size = 60, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {5891, 3718, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac{\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cosh ^{-1}\left (a x^n\right ) \log \left (e^{2 \cosh ^{-1}\left (a x^n\right )}+1\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5891

Int[ArcCosh[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Dist[1/p, Subst[Int[x^n*Tanh[x], x], x, ArcCosh[a*x^p]],
x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (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)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && 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{\cosh ^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac{\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{2 \operatorname{Subst}\left (\int \frac{e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac{\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}-\frac{\operatorname{Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac{\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}-\frac{\operatorname{Subst}\left (\int \frac{\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ &=-\frac{\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\cosh ^{-1}\left (a x^n\right ) \log \left (1+e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{n}+\frac{\text{Li}_2\left (-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ \end{align*}

Mathematica [B]  time = 0.468645, size = 179, normalized size = 2.98 \[ \frac{a \sqrt{1-a^2 x^{2 n}} \left (-\text{PolyLog}\left (2,e^{-2 \sinh ^{-1}\left (\sqrt{-a^2} x^n\right )}\right )-2 n \log (x) \log \left (\sqrt{-a^2} x^n+\sqrt{1-a^2 x^{2 n}}\right )+\sinh ^{-1}\left (\sqrt{-a^2} x^n\right )^2+2 \sinh ^{-1}\left (\sqrt{-a^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt{-a^2} x^n\right )}\right )\right )}{2 \sqrt{-a^2} n \sqrt{a x^n-1} \sqrt{a x^n+1}}+\log (x) \cosh ^{-1}\left (a x^n\right ) \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

ArcCosh[a*x^n]*Log[x] + (a*Sqrt[1 - a^2*x^(2*n)]*(ArcSinh[Sqrt[-a^2]*x^n]^2 + 2*ArcSinh[Sqrt[-a^2]*x^n]*Log[1
- E^(-2*ArcSinh[Sqrt[-a^2]*x^n])] - 2*n*Log[x]*Log[Sqrt[-a^2]*x^n + Sqrt[1 - a^2*x^(2*n)]] - PolyLog[2, E^(-2*
ArcSinh[Sqrt[-a^2]*x^n])]))/(2*Sqrt[-a^2]*n*Sqrt[-1 + a*x^n]*Sqrt[1 + a*x^n])

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Maple [A]  time = 0.049, size = 91, normalized size = 1.5 \begin{align*} -{\frac{ \left ({\rm arccosh} \left (a{x}^{n}\right ) \right ) ^{2}}{2\,n}}+{\frac{{\rm arccosh} \left (a{x}^{n}\right )}{n}\ln \left ( 1+ \left ( a{x}^{n}+\sqrt{a{x}^{n}-1}\sqrt{a{x}^{n}+1} \right ) ^{2} \right ) }+{\frac{1}{2\,n}{\it polylog} \left ( 2,- \left ( a{x}^{n}+\sqrt{a{x}^{n}-1}\sqrt{a{x}^{n}+1} \right ) ^{2} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a n \int \frac{x^{n} \log \left (x\right )}{a^{3} x x^{3 \, n} - a x x^{n} +{\left (a^{2} x x^{2 \, n} - x\right )} \sqrt{a x^{n} + 1} \sqrt{a x^{n} - 1}}\,{d x} - \frac{1}{2} \, n \log \left (x\right )^{2} + n \int \frac{\log \left (x\right )}{2 \,{\left (a x x^{n} + x\right )}}\,{d x} - n \int \frac{\log \left (x\right )}{2 \,{\left (a x x^{n} - x\right )}}\,{d x} + \log \left (a x^{n} + \sqrt{a x^{n} + 1} \sqrt{a x^{n} - 1}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*n*integrate(x^n*log(x)/(a^3*x*x^(3*n) - a*x*x^n + (a^2*x*x^(2*n) - x)*sqrt(a*x^n + 1)*sqrt(a*x^n - 1)), x) -
1/2*n*log(x)^2 + n*integrate(1/2*log(x)/(a*x*x^n + x), x) - n*integrate(1/2*log(x)/(a*x*x^n - x), x) + log(a*
x^n + sqrt(a*x^n + 1)*sqrt(a*x^n - 1))*log(x)

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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(arccosh(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{acosh}{\left (a x^{n} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(acosh(a*x**n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arccosh(a*x^n)/x, x)