\(\int \frac {\text {arcsinh}(a x^n)}{x} \, dx\) [311]

Optimal result

Integrand size = 10, antiderivative size = 60 \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=-\frac {\text {arcsinh}\left (a x^n\right )^2}{2 n}+\frac {\text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{n}+\frac {\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{2 n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5869, 3797, 2221, 2317, 2438} \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{2 n}-\frac {\text {arcsinh}\left (a x^n\right )^2}{2 n}+\frac {\text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{n} \]

[In]

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

[Out]

-1/2*ArcSinh[a*x^n]^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 2221

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/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], 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 2317

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 2438

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 3797

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))/(1 + E^(2*((-I)*e + f*
fz*x))/E^(2*I*k*Pi))))/E^(2*I*k*Pi), x], x] /; FreeQ[{c, d, e, f, fz}, x] && IntegerQ[4*k] && IGtQ[m, 0]

Rule 5869

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]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int x \coth (x) \, dx,x,\text {arcsinh}\left (a x^n\right )\right )}{n} \\ & = -\frac {\text {arcsinh}\left (a x^n\right )^2}{2 n}-\frac {2 \text {Subst}\left (\int \frac {e^{2 x} x}{1-e^{2 x}} \, dx,x,\text {arcsinh}\left (a x^n\right )\right )}{n} \\ & = -\frac {\text {arcsinh}\left (a x^n\right )^2}{2 n}+\frac {\text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{n}-\frac {\text {Subst}\left (\int \log \left (1-e^{2 x}\right ) \, dx,x,\text {arcsinh}\left (a x^n\right )\right )}{n} \\ & = -\frac {\text {arcsinh}\left (a x^n\right )^2}{2 n}+\frac {\text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{n}-\frac {\text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{2 n} \\ & = -\frac {\text {arcsinh}\left (a x^n\right )^2}{2 n}+\frac {\text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{n}+\frac {\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{2 n} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.87 \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\frac {-\text {arcsinh}\left (a x^n\right ) \left (\text {arcsinh}\left (a x^n\right )-2 \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )\right )+\operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^n\right )}\right )}{2 n} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.26 (sec) , antiderivative size = 120, normalized size of antiderivative = 2.00

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arcsinh}\left (a \,x^{n}\right )^{2}}{2}+\operatorname {arcsinh}\left (a \,x^{n}\right ) \ln \left (1+a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {polylog}\left (2, -a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {arcsinh}\left (a \,x^{n}\right ) \ln \left (1-a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {polylog}\left (2, a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )}{n}\)	\(120\)
default	\(\frac {-\frac {\operatorname {arcsinh}\left (a \,x^{n}\right )^{2}}{2}+\operatorname {arcsinh}\left (a \,x^{n}\right ) \ln \left (1+a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {polylog}\left (2, -a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {arcsinh}\left (a \,x^{n}\right ) \ln \left (1-a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {polylog}\left (2, a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )}{n}\)	\(120\)

[In]

int(arcsinh(a*x^n)/x,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {asinh}{\left (a x^{n} \right )}}{x}\, dx \]

[In]

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

[Out]

Integral(asinh(a*x**n)/x, x)

Maxima [F]

\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x} \,d x } \]

[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)

Giac [F]

\[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arsinh}\left (a x^{n}\right )}{x} \,d x } \]

[In]

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

[Out]

integrate(arcsinh(a*x^n)/x, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {asinh}\left (a\,x^n\right )}{x} \,d x \]

[In]

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

[Out]

int(asinh(a*x^n)/x, x)