3.311 \(\int \frac{\sinh ^{-1}(a x^n)}{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0662234, 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 = {5890, 3716, 2190, 2279, 2391} \[ \frac{\text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac{\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5890

Int[ArcSinh[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Dist[1/p, Subst[Int[x^n*Coth[x], x], x, ArcSinh[a*x^p]],
x] /; FreeQ[{a, p}, 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{\sinh ^{-1}\left (a x^n\right )}{x} \, dx &=\frac{\operatorname{Subst}\left (\int x \coth (x) \, dx,x,\sinh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac{\sinh ^{-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,\sinh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac{\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n}-\frac{\operatorname{Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\sinh ^{-1}\left (a x^n\right )\right )}{n}\\ &=-\frac{\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n}-\frac{\operatorname{Subst}\left (\int \frac{\log (1-x)}{x} \, dx,x,e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ &=-\frac{\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n}+\frac{\text{Li}_2\left (e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}\\ \end{align*}

Mathematica [A]  time = 0.0076814, size = 60, normalized size = 1. \[ \frac{\text{PolyLog}\left (2,e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{2 n}-\frac{\sinh ^{-1}\left (a x^n\right )^2}{2 n}+\frac{\sinh ^{-1}\left (a x^n\right ) \log \left (1-e^{2 \sinh ^{-1}\left (a x^n\right )}\right )}{n} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 133, normalized size = 2.2 \begin{align*} -{\frac{ \left ({\it Arcsinh} \left ( a{x}^{n} \right ) \right ) ^{2}}{2\,n}}+{\frac{{\it Arcsinh} \left ( a{x}^{n} \right ) }{n}\ln \left ( 1+a{x}^{n}+\sqrt{1+{a}^{2} \left ({x}^{n} \right ) ^{2}} \right ) }+{\frac{1}{n}{\it polylog} \left ( 2,-a{x}^{n}-\sqrt{1+{a}^{2} \left ({x}^{n} \right ) ^{2}} \right ) }+{\frac{{\it Arcsinh} \left ( a{x}^{n} \right ) }{n}\ln \left ( 1-a{x}^{n}-\sqrt{1+{a}^{2} \left ({x}^{n} \right ) ^{2}} \right ) }+{\frac{1}{n}{\it polylog} \left ( 2,a{x}^{n}+\sqrt{1+{a}^{2} \left ({x}^{n} \right ) ^{2}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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^{2} x^{2 \, n} + 1}}\,{d x} - \frac{1}{2} \, n \log \left (x\right )^{2} + n \int \frac{\log \left (x\right )}{a^{2} x x^{2 \, n} + x}\,{d x} + \log \left (a x^{n} + \sqrt{a^{2} x^{2 \, n} + 1}\right ) \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arcsinh(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^2*x^(2*n) + 1)), x) - 1/2*n*lo
g(x)^2 + n*integrate(log(x)/(a^2*x*x^(2*n) + x), x) + log(a*x^n + sqrt(a^2*x^(2*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(arcsinh(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{asinh}{\left (a x^{n} \right )}}{x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(asinh(a*x**n)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(arcsinh(a*x^n)/x, x)