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

Optimal. Leaf size=61 \[ -\frac {\text {Li}_2\left (-e^{2 \text {sech}^{-1}\left (a x^n\right )}\right )}{2 n}+\frac {\text {sech}^{-1}\left (a x^n\right )^2}{2 n}-\frac {\text {sech}^{-1}\left (a x^n\right ) \log \left (e^{2 \text {sech}^{-1}\left (a x^n\right )}+1\right )}{n} \]

[Out]

1/2*arcsech(a*x^n)^2/n-arcsech(a*x^n)*ln(1+(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)*(1/a/(x^n)+1)^(1/2))^2)/n-1/2*polylo
g(2,-(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)*(1/a/(x^n)+1)^(1/2))^2)/n

________________________________________________________________________________________

Rubi [A]  time = 0.11, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6281, 5660, 3718, 2190, 2279, 2391} \[ -\frac {\text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}\left (a x^n\right )}\right )}{2 n}+\frac {\text {sech}^{-1}\left (a x^n\right )^2}{2 n}-\frac {\text {sech}^{-1}\left (a x^n\right ) \log \left (e^{2 \text {sech}^{-1}\left (a x^n\right )}+1\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSech[a*x^n]^2/(2*n) - (ArcSech[a*x^n]*Log[1 + E^(2*ArcSech[a*x^n])])/n - PolyLog[2, -E^(2*ArcSech[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 5660

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

Rule 6281

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [B]  time = 0.98, size = 219, normalized size = 3.59 \[ \frac {\sqrt {\frac {1-a x^n}{a x^n+1}} \left (\sqrt {1-a^2 x^{2 n}} \left (-4 \text {Li}_2\left (\frac {1}{2}-\frac {1}{2} \sqrt {1-a^2 x^{2 n}}\right )+\log ^2\left (a^2 x^{2 n}\right )+2 \log ^2\left (\frac {1}{2} \left (\sqrt {1-a^2 x^{2 n}}+1\right )\right )-4 \log \left (\frac {1}{2} \left (\sqrt {1-a^2 x^{2 n}}+1\right )\right ) \log \left (a^2 x^{2 n}\right )\right )+4 \sqrt {a^2 x^{2 n}-1} \left (2 n \log (x)-\log \left (a^2 x^{2 n}\right )\right ) \tan ^{-1}\left (\sqrt {a^2 x^{2 n}-1}\right )\right )}{8 \left (n-a n x^n\right )}+\log (x) \text {sech}^{-1}\left (a x^n\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

ArcSech[a*x^n]*Log[x] + (Sqrt[(1 - a*x^n)/(1 + a*x^n)]*(4*Sqrt[-1 + a^2*x^(2*n)]*ArcTan[Sqrt[-1 + a^2*x^(2*n)]
]*(2*n*Log[x] - Log[a^2*x^(2*n)]) + Sqrt[1 - a^2*x^(2*n)]*(Log[a^2*x^(2*n)]^2 - 4*Log[a^2*x^(2*n)]*Log[(1 + Sq
rt[1 - a^2*x^(2*n)])/2] + 2*Log[(1 + Sqrt[1 - a^2*x^(2*n)])/2]^2 - 4*PolyLog[2, 1/2 - Sqrt[1 - a^2*x^(2*n)]/2]
)))/(8*(n - a*n*x^n))

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsech(a*x^n)/x, x)

________________________________________________________________________________________

maple [A]  time = 0.15, size = 116, normalized size = 1.90 \[ \frac {\mathrm {arcsech}\left (a \,x^{n}\right )^{2}}{2 n}-\frac {\mathrm {arcsech}\left (a \,x^{n}\right ) \ln \left (1+\left (\frac {x^{-n}}{a}+\sqrt {\frac {x^{-n}}{a}-1}\, \sqrt {\frac {x^{-n}}{a}+1}\right )^{2}\right )}{n}-\frac {\polylog \left (2, -\left (\frac {x^{-n}}{a}+\sqrt {\frac {x^{-n}}{a}-1}\, \sqrt {\frac {x^{-n}}{a}+1}\right )^{2}\right )}{2 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*arcsech(a*x^n)^2/n-arcsech(a*x^n)*ln(1+(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)*(1/a/(x^n)+1)^(1/2))^2)/n-1/2*polylo
g(2,-(1/a/(x^n)+(1/a/(x^n)-1)^(1/2)*(1/a/(x^n)+1)^(1/2))^2)/n

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsech(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*x^n + 1)*sqrt(-a*x^n + 1) - x), x)
+ 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(sqrt(a*x^n + 1)*sq
rt(-a*x^n + 1) + 1)*log(x) - log(a)*log(x) - log(x)*log(x^n)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(1/(a*x^n))/x,x)

[Out]

int(acosh(1/(a*x^n))/x, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asech(a*x**n)/x, x)

________________________________________________________________________________________