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

Optimal. Leaf size=60 \[ \frac {\text {Li}_2\left (-e^{2 \cosh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac {\cosh ^{-1}\left (a x^n\right )^2}{2 n}+\frac {\cosh ^{-1}\left (a x^n\right ) \log \left (e^{2 \cosh ^{-1}\left (a x^n\right )}+1\right )}{n} \]

[Out]

-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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Rubi [A]  time = 0.07, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, 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 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]

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 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]

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*}

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Mathematica [B]  time = 0.51, size = 179, normalized size = 2.98 \[ \frac {a \sqrt {1-a^2 x^{2 n}} \left (-\text {Li}_2\left (e^{-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right )}\right )+\sinh ^{-1}\left (\sqrt {-a^2} x^n\right )^2+2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right ) \log \left (1-e^{-2 \sinh ^{-1}\left (\sqrt {-a^2} x^n\right )}\right )-2 n \log (x) \log \left (\sqrt {1-a^2 x^{2 n}}+\sqrt {-a^2} x^n\right )\right )}{2 \sqrt {-a^2} n \sqrt {a x^n-1} \sqrt {a x^n+1}}+\log (x) \cosh ^{-1}\left (a x^n\right ) \]

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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fricas [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

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)

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maple [A]  time = 0.02, size = 91, normalized size = 1.52 \[ -\frac {\mathrm {arccosh}\left (a \,x^{n}\right )^{2}}{2 n}+\frac {\mathrm {arccosh}\left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{n}+\frac {\polylog \left (2, -\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{2 n} \]

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 = 0.00, size = 0, normalized size = 0.00 \[ a n \int \frac {x^{n} \log \relax (x)}{a^{3} x x^{3 \, n} - a x x^{n} + {\left (a^{2} x x^{2 \, n} - x\right )} \sqrt {a x^{n} + 1} \sqrt {a x^{n} - 1}}\,{d x} - \frac {1}{2} \, n \log \relax (x)^{2} + n \int \frac {\log \relax (x)}{2 \, {\left (a x x^{n} + x\right )}}\,{d x} - n \int \frac {\log \relax (x)}{2 \, {\left (a x x^{n} - x\right )}}\,{d x} + \log \left (a x^{n} + \sqrt {a x^{n} + 1} \sqrt {a x^{n} - 1}\right ) \log \relax (x) \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\mathrm {acosh}\left (a\,x^n\right )}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int(acosh(a*x^n)/x, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}{\left (a x^{n} \right )}}{x}\, dx \]

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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