Integrand size = 10, antiderivative size = 60 \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=-\frac {\text {arccosh}(a x)^2}{x}+4 a \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-2 i a \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+2 i a \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right ) \]
-arccosh(a*x)^2/x+4*a*arccosh(a*x)*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)) -2*I*a*polylog(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+2*I*a*polylog(2,I*( a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=-i a \left (\text {arccosh}(a x) \left (-\frac {i \text {arccosh}(a x)}{a x}+2 \log \left (1-i e^{-\text {arccosh}(a x)}\right )-2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right ) \]
(-I)*a*(ArcCosh[a*x]*(((-I)*ArcCosh[a*x])/(a*x) + 2*Log[1 - I/E^ArcCosh[a* x]] - 2*Log[1 + I/E^ArcCosh[a*x]]) + 2*PolyLog[2, (-I)/E^ArcCosh[a*x]] - 2 *PolyLog[2, I/E^ArcCosh[a*x]])
Time = 0.53 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.02, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used = {6298, 6362, 3042, 4668, 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.
| \(\displaystyle \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx\) |
| \(\Big \downarrow \) 6298 |
| \(\displaystyle 2 a \int \frac {\text {arccosh}(a x)}{x \sqrt {a x-1} \sqrt {a x+1}}dx-\frac {\text {arccosh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 6362 |
| \(\displaystyle 2 a \int \frac {\text {arccosh}(a x)}{a x}d\text {arccosh}(a x)-\frac {\text {arccosh}(a x)^2}{x}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^2}{x}+2 a \int \text {arccosh}(a x) \csc \left (i \text {arccosh}(a x)+\frac {\pi }{2}\right )d\text {arccosh}(a x)\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^2}{x}+2 a \left (-i \int \log \left (1-i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+i \int \log \left (1+i e^{\text {arccosh}(a x)}\right )d\text {arccosh}(a x)+2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )\right )\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^2}{x}+2 a \left (-i \int e^{-\text {arccosh}(a x)} \log \left (1-i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+i \int e^{-\text {arccosh}(a x)} \log \left (1+i e^{\text {arccosh}(a x)}\right )de^{\text {arccosh}(a x)}+2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )\right )\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {\text {arccosh}(a x)^2}{x}+2 a \left (2 \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-i \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+i \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )\right )\) |
-(ArcCosh[a*x]^2/x) + 2*a*(2*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]] - I*PolyL og[2, (-I)*E^ArcCosh[a*x]] + I*PolyLog[2, I*E^ArcCosh[a*x]])
3.1.18.3.1 Defintions of rubi rules used
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[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(d*(m + 1))), x] - Simp[b*c* (n/(d*(m + 1))) Int[(d*x)^(m + 1)*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] & & NeQ[m, -1]
Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1 _.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[(1/c^(m + 1))*Simp[ Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]] Subst [Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && Inte gerQ[m]
Time = 0.16 (sec) , antiderivative size = 138, normalized size of antiderivative = 2.30
| method | result | size |
| derivativedivides | \(a \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{a x}-2 i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )\right )\) | \(138\) |
| default | \(a \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{a x}-2 i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )\right )\) | \(138\) |
a*(-arccosh(a*x)^2/a/x-2*I*arccosh(a*x)*ln(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^ (1/2)))+2*I*arccosh(a*x)*ln(1-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))-2*I*dil og(1+I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))+2*I*dilog(1-I*(a*x+(a*x-1)^(1/2) *(a*x+1)^(1/2))))
\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]
\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
-log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/x + integrate(2*(a^3*x^2 + sqrt( a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/( a^3*x^4 - a*x^2 + (a^2*x^3 - x)*sqrt(a*x + 1)*sqrt(a*x - 1)), x)
\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}} \,d x } \]
Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^2} \,d x \]