\(\int \frac {\text {arccosh}(a x^n)}{x} \, dx\) [239]

Optimal result

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

[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

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 = {6011, 3799, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (a x^n\right )}\right )}{2 n}-\frac {\text {arccosh}\left (a x^n\right )^2}{2 n}+\frac {\text {arccosh}\left (a x^n\right ) \log \left (e^{2 \text {arccosh}\left (a x^n\right )}+1\right )}{n} \]

[In]

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

[Out]

-1/2*ArcCosh[a*x^n]^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 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 3799

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 6011

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(179\) vs. \(2(60)=120\).

Time = 0.39 (sec) , antiderivative size = 179, normalized size of antiderivative = 2.98 \[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\text {arccosh}\left (a x^n\right ) \log (x)+\frac {a \sqrt {1-a^2 x^{2 n}} \left (\text {arcsinh}\left (\sqrt {-a^2} x^n\right )^2+2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right )}\right )-2 n \log (x) \log \left (\sqrt {-a^2} x^n+\sqrt {1-a^2 x^{2 n}}\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right )}\right )\right )}{2 \sqrt {-a^2} n \sqrt {-1+a x^n} \sqrt {1+a x^n}} \]

[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])

Maple [A] (verified)

Time = 0.46 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.43

method	result	size
derivativedivides	\(\frac {-\frac {\operatorname {arccosh}\left (a \,x^{n}\right )^{2}}{2}+\operatorname {arccosh}\left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{2}}{n}\)	\(86\)
default	\(\frac {-\frac {\operatorname {arccosh}\left (a \,x^{n}\right )^{2}}{2}+\operatorname {arccosh}\left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{2}}{n}\)	\(86\)

[In]

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

[Out]

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

Fricas [F(-2)]

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

[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)

Sympy [F]

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

[In]

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

[Out]

Integral(acosh(a*x**n)/x, x)

Maxima [F]

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

[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)

Giac [F]

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

[In]

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

[Out]

integrate(arccosh(a*x^n)/x, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(acosh(a*x^n)/x, x)