Integrand size = 8, antiderivative size = 43 \[ \int \frac {\text {arccosh}(a x)}{x} \, dx=-\frac {1}{2} \text {arccosh}(a x)^2+\text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right ) \]
-1/2*arccosh(a*x)^2+arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2) +1/2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)
Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\frac {1}{2} \left (\text {arccosh}(a x) \left (\text {arccosh}(a x)+2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )\right ) \]
(ArcCosh[a*x]*(ArcCosh[a*x] + 2*Log[1 + E^(-2*ArcCosh[a*x])]) - PolyLog[2, -E^(-2*ArcCosh[a*x])])/2
Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.33, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.875, Rules used = {6297, 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.
\(\displaystyle \int \frac {\text {arccosh}(a x)}{x} \, dx\) |
\(\Big \downarrow \) 6297 |
\(\displaystyle \int \frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \text {arccosh}(a x)}{a x}d\text {arccosh}(a x)\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \text {arccosh}(a x) \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \text {arccosh}(a x) \tan (i \text {arccosh}(a x))d\text {arccosh}(a x)\) |
\(\Big \downarrow \) 4201 |
\(\displaystyle -i \left (2 i \int \frac {e^{2 \text {arccosh}(a x)} \text {arccosh}(a x)}{1+e^{2 \text {arccosh}(a x)}}d\text {arccosh}(a x)-\frac {1}{2} i \text {arccosh}(a x)^2\right )\) |
\(\Big \downarrow \) 2620 |
\(\displaystyle -i \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {1}{2} \int \log \left (1+e^{2 \text {arccosh}(a x)}\right )d\text {arccosh}(a x)\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )\) |
\(\Big \downarrow \) 2715 |
\(\displaystyle -i \left (2 i \left (\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )-\frac {1}{4} \int e^{-2 \text {arccosh}(a x)} \log \left (1+e^{2 \text {arccosh}(a x)}\right )de^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )\) |
\(\Big \downarrow \) 2838 |
\(\displaystyle -i \left (2 i \left (\frac {1}{4} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )\right )-\frac {1}{2} i \text {arccosh}(a x)^2\right )\) |
(-I)*((-1/2*I)*ArcCosh[a*x]^2 + (2*I)*((ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[ a*x])])/2 + PolyLog[2, -E^(2*ArcCosh[a*x])]/4))
3.1.6.3.1 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[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Simp[1/b Subst[Int[x^n*Tanh[-a/b + x/b], x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a , b, c}, x] && IGtQ[n, 0]
Time = 0.20 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.53
method | result | size |
derivativedivides | \(-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{2}+\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\) | \(66\) |
default | \(-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{2}+\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\) | \(66\) |
-1/2*arccosh(a*x)^2+arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2) +1/2*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)
\[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{x} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\int \frac {\operatorname {acosh}{\left (a x \right )}}{x}\, dx \]
\[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{x} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )}{x} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\int \frac {\mathrm {acosh}\left (a\,x\right )}{x} \,d x \]