Integrand size = 10, antiderivative size = 60 \[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=-\frac {\text {arccosh}\left (a x^n\right )^2}{2 n}+\frac {\text {arccosh}\left (a x^n\right ) \log \left (1+e^{2 \text {arccosh}\left (a x^n\right )}\right )}{n}+\frac {\operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (a x^n\right )}\right )}{2 n} \]
-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
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 \[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\text {arccosh}\left (a x^n\right ) \log (x)+\frac {a \sqrt {1-a^2 x^{2 n}} \left (\text {arcsinh}\left (\sqrt {-a^2} x^n\right )^2+2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right ) \log \left (1-e^{-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right )}\right )-2 n \log (x) \log \left (\sqrt {-a^2} x^n+\sqrt {1-a^2 x^{2 n}}\right )-\operatorname {PolyLog}\left (2,e^{-2 \text {arcsinh}\left (\sqrt {-a^2} x^n\right )}\right )\right )}{2 \sqrt {-a^2} n \sqrt {-1+a x^n} \sqrt {1+a x^n}} \]
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*Ar cSinh[Sqrt[-a^2]*x^n])]))/(2*Sqrt[-a^2]*n*Sqrt[-1 + a*x^n]*Sqrt[1 + a*x^n] )
Result contains complex when optimal does not.
Time = 0.35 (sec) , antiderivative size = 68, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.700, Rules used = {6426, 3042, 26, 4201, 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.
\(\displaystyle \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx\) |
\(\Big \downarrow \) 6426 |
\(\displaystyle \frac {\int \frac {x^{-n} \sqrt {\frac {a x^n-1}{a x^n+1}} \left (a x^n+1\right ) \text {arccosh}\left (a x^n\right )}{a}d\text {arccosh}\left (a x^n\right )}{n}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int -i \text {arccosh}\left (a x^n\right ) \tan \left (i \text {arccosh}\left (a x^n\right )\right )d\text {arccosh}\left (a x^n\right )}{n}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {i \int \text {arccosh}\left (a x^n\right ) \tan \left (i \text {arccosh}\left (a x^n\right )\right )d\text {arccosh}\left (a x^n\right )}{n}\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -\frac {i \left (2 i \int \frac {e^{2 \text {arccosh}\left (a x^n\right )} \text {arccosh}\left (a x^n\right )}{1+e^{2 \text {arccosh}\left (a x^n\right )}}d\text {arccosh}\left (a x^n\right )-\frac {1}{2} i \text {arccosh}\left (a x^n\right )^2\right )}{n}\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (a x^n\right ) \log \left (e^{2 \text {arccosh}\left (a x^n\right )}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}\left (a x^n\right )}\right )d\text {arccosh}\left (a x^n\right )\right )-\frac {1}{2} i \text {arccosh}\left (a x^n\right )^2\right )}{n}\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{2} \text {arccosh}\left (a x^n\right ) \log \left (e^{2 \text {arccosh}\left (a x^n\right )}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}\left (a x^n\right )} \log \left (1+e^{2 \text {arccosh}\left (a x^n\right )}\right )de^{2 \text {arccosh}\left (a x^n\right )}\right )-\frac {1}{2} i \text {arccosh}\left (a x^n\right )^2\right )}{n}\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -\frac {i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}\left (a x^n\right )}\right )+\frac {1}{2} \text {arccosh}\left (a x^n\right ) \log \left (e^{2 \text {arccosh}\left (a x^n\right )}+1\right )\right )-\frac {1}{2} i \text {arccosh}\left (a x^n\right )^2\right )}{n}\) |
((-I)*((-1/2*I)*ArcCosh[a*x^n]^2 + (2*I)*((ArcCosh[a*x^n]*Log[1 + E^(2*Arc Cosh[a*x^n])])/2 + PolyLog[2, -E^(2*ArcCosh[a*x^n])]/4)))/n
3.3.39.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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]
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]
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]
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] + Simp[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]
Int[ArcCosh[(a_.)*(x_)^(p_)]^(n_.)/(x_), x_Symbol] :> Simp[1/p Subst[Int[ x^n*Tanh[x], x], x, ArcCosh[a*x^p]], x] /; FreeQ[{a, p}, x] && IGtQ[n, 0]
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 | \(\frac {-\frac {\operatorname {arccosh}\left (a \,x^{n}\right )^{2}}{2}+\operatorname {arccosh}\left (a \,x^{n}\right ) \ln \left (1+\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (a \,x^{n}+\sqrt {a \,x^{n}-1}\, \sqrt {a \,x^{n}+1}\right )^{2}\right )}{2}}{n}\) | \(86\) |
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))
Exception generated. \[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\int \frac {\operatorname {acosh}{\left (a x^{n} \right )}}{x}\, dx \]
\[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x^{n}\right )}{x} \,d x } \]
a*n*integrate(x^n*log(x)/(a^3*x*x^(3*n) - a*x*x^n + (a^2*x*x^(2*n) - x)*sq rt(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)
\[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x^{n}\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}\left (a x^n\right )}{x} \, dx=\int \frac {\mathrm {acosh}\left (a\,x^n\right )}{x} \,d x \]