\(\int \frac {\text {arccosh}(a x)}{x} \, dx\) [6]

Optimal result

Integrand size = 8, antiderivative size = 43 \[ \int \frac {\text {arccosh}(a x)}{x} \, dx=-\frac {1}{2} \text {arccosh}(a x)^2+\text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )+\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right ) \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.625, Rules used = {5882, 3799, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\frac {1}{2} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )-\frac {1}{2} \text {arccosh}(a x)^2+\text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right ) \]

[In]

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

[Out]

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

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 5882

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Dist[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]

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 0.98 \[ \int \frac {\text {arccosh}(a x)}{x} \, dx=\frac {1}{2} \left (\text {arccosh}(a x) \left (\text {arccosh}(a x)+2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )-\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )\right ) \]

[In]

Integrate[ArcCosh[a*x]/x,x]

[Out]

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

Maple [A] (verified)

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	\(-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{2}+\operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )+\frac {\operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{2}\)	\(66\)

[In]

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

[Out]

-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)

Fricas [F]

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

[In]

integrate(arccosh(a*x)/x,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)/x, x)

Sympy [F]

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

[In]

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

[Out]

Integral(acosh(a*x)/x, x)

Maxima [F]

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

[In]

integrate(arccosh(a*x)/x,x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)/x, x)

Giac [F]

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

[In]

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

[Out]

integrate(arccosh(a*x)/x, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(acosh(a*x)/x, x)