\(\int \frac {\text {arcsinh}(a x^n)}{x} \, dx\) [246]

Optimal result

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

-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
 

Mathematica [A] (verified)

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

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

Output:

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

Rubi [C] (verified)

Result contains complex when optimal does not.

Time = 0.51 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6284, 3042, 26, 4199, 25, 2620, 2715, 2838}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\text {arcsinh}\left (a x^n\right )}{x} \, dx\)

\(\Big \downarrow \) 6284

\(\displaystyle \frac {\int \frac {x^{-n} \sqrt {a^2 x^{2 n}+1} \text {arcsinh}\left (a x^n\right )}{a}d\text {arcsinh}\left (a x^n\right )}{n}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {\int -i \text {arcsinh}\left (a x^n\right ) \tan \left (i \text {arcsinh}\left (a x^n\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (a x^n\right )}{n}\)

\(\Big \downarrow \) 26

\(\displaystyle -\frac {i \int \text {arcsinh}\left (a x^n\right ) \tan \left (i \text {arcsinh}\left (a x^n\right )+\frac {\pi }{2}\right )d\text {arcsinh}\left (a x^n\right )}{n}\)

\(\Big \downarrow \) 4199

\(\displaystyle -\frac {i \left (2 i \int -\frac {e^{2 \text {arcsinh}\left (a x^n\right )} \text {arcsinh}\left (a x^n\right )}{1-e^{2 \text {arcsinh}\left (a x^n\right )}}d\text {arcsinh}\left (a x^n\right )-\frac {1}{2} i \text {arcsinh}\left (a x^n\right )^2\right )}{n}\)

\(\Big \downarrow \) 25

\(\displaystyle -\frac {i \left (-2 i \int \frac {e^{2 \text {arcsinh}\left (a x^n\right )} \text {arcsinh}\left (a x^n\right )}{1-e^{2 \text {arcsinh}\left (a x^n\right )}}d\text {arcsinh}\left (a x^n\right )-\frac {1}{2} i \text {arcsinh}\left (a x^n\right )^2\right )}{n}\)

\(\Big \downarrow \) 2620

\(\displaystyle -\frac {i \left (-2 i \left (\frac {1}{2} \int \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )d\text {arcsinh}\left (a x^n\right )-\frac {1}{2} \text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (a x^n\right )^2\right )}{n}\)

\(\Big \downarrow \) 2715

\(\displaystyle -\frac {i \left (-2 i \left (\frac {1}{4} \int e^{-2 \text {arcsinh}\left (a x^n\right )} \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )de^{2 \text {arcsinh}\left (a x^n\right )}-\frac {1}{2} \text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (a x^n\right )^2\right )}{n}\)

\(\Big \downarrow \) 2838

\(\displaystyle -\frac {i \left (-2 i \left (-\frac {1}{4} \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}\left (a x^n\right )}\right )-\frac {1}{2} \text {arcsinh}\left (a x^n\right ) \log \left (1-e^{2 \text {arcsinh}\left (a x^n\right )}\right )\right )-\frac {1}{2} i \text {arcsinh}\left (a x^n\right )^2\right )}{n}\)

Input:

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

Output:

((-I)*((-1/2*I)*ArcSinh[a*x^n]^2 - (2*I)*(-1/2*(ArcSinh[a*x^n]*Log[1 - E^( 
2*ArcSinh[a*x^n])]) - PolyLog[2, E^(2*ArcSinh[a*x^n])]/4)))/n
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 2620

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] - Si 
mp[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 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[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 2838

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 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4199

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] + Simp 
[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] && In 
tegerQ[4*k] && IGtQ[m, 0]
 

rule 6284

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

Maple [A] (verified)

Time = 0.37 (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	\(\frac {-\frac {\operatorname {arcsinh}\left (a \,x^{n}\right )^{2}}{2}+\operatorname {arcsinh}\left (a \,x^{n}\right ) \ln \left (1-a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {polylog}\left (2, a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {arcsinh}\left (a \,x^{n}\right ) \ln \left (1+a \,x^{n}+\sqrt {1+a^{2} x^{2 n}}\right )+\operatorname {polylog}\left (2, -a \,x^{n}-\sqrt {1+a^{2} x^{2 n}}\right )}{n}\)	\(120\)

Input:

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

Output:

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

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

Output:

Exception raised: TypeError >>  Error detected within library code:   inte 
grate: implementation incomplete (constant residues)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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