\(\int \frac {\text {arccosh}(a x)^2}{x^2} \, dx\) [18]

Optimal result

Integrand size = 10, antiderivative size = 60 \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=-\frac {\text {arccosh}(a x)^2}{x}+4 a \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-2 i a \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+2 i a \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right ) \]

[Out]

-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)))

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5883, 5947, 4265, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=4 a \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-2 i a \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )+2 i a \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )-\frac {\text {arccosh}(a x)^2}{x} \]

[In]

Int[ArcCosh[a*x]^2/x^2,x]

[Out]

-(ArcCosh[a*x]^2/x) + 4*a*ArcCosh[a*x]*ArcTan[E^ArcCosh[a*x]] - (2*I)*a*PolyLog[2, (-I)*E^ArcCosh[a*x]] + (2*I
)*a*PolyLog[2, I*E^ArcCosh[a*x]]

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 4265

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] + (-Dist[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] + Dist[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]

Rule 5883

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*ArcC
osh[c*x])^n/(d*(m + 1))), x] - Dist[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]

Rule 5947

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]
), x_Symbol] :> Dist[(1/c^(m + 1))*Simp[Sqrt[1 + c*x]/Sqrt[d1 + e1*x]]*Simp[Sqrt[-1 + c*x]/Sqrt[d2 + e2*x]], S
ubst[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*d
1] && EqQ[e2, (-c)*d2] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {arccosh}(a x)^2}{x}+(2 a) \int \frac {\text {arccosh}(a x)}{x \sqrt {-1+a x} \sqrt {1+a x}} \, dx \\ & = -\frac {\text {arccosh}(a x)^2}{x}+(2 a) \text {Subst}(\int x \text {sech}(x) \, dx,x,\text {arccosh}(a x)) \\ & = -\frac {\text {arccosh}(a x)^2}{x}+4 a \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-(2 i a) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )+(2 i a) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right ) \\ & = -\frac {\text {arccosh}(a x)^2}{x}+4 a \text {arccosh}(a x) \arctan \left (e^{\text {arccosh}(a x)}\right )-(2 i a) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )+(2 i a) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right ) \\ & = -\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 ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.53 \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=-i a \left (\text {arccosh}(a x) \left (-\frac {i \text {arccosh}(a x)}{a x}+2 \log \left (1-i e^{-\text {arccosh}(a x)}\right )-2 \log \left (1+i e^{-\text {arccosh}(a x)}\right )\right )+2 \operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-2 \operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right ) \]

[In]

Integrate[ArcCosh[a*x]^2/x^2,x]

[Out]

(-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]])

Maple [A] (verified)

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	\(a \left (-\frac {\operatorname {arccosh}\left (a x \right )^{2}}{a x}-2 i \operatorname {arccosh}\left (a x \right ) \ln \left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i \operatorname {arccosh}\left (a x \right ) \ln \left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )-2 i \operatorname {dilog}\left (1+i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )+2 i \operatorname {dilog}\left (1-i \left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )\right )\right )\)	\(138\)

[In]

int(arccosh(a*x)^2/x^2,x,method=_RETURNVERBOSE)

[Out]

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*dilog(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))))

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(arccosh(a*x)^2/x^2,x, algorithm="fricas")

[Out]

integral(arccosh(a*x)^2/x^2, x)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{2}}\, dx \]

[In]

integrate(acosh(a*x)**2/x**2,x)

[Out]

Integral(acosh(a*x)**2/x**2, x)

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(arccosh(a*x)^2/x^2,x, algorithm="maxima")

[Out]

-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)*lo
g(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)

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{x^{2}} \,d x } \]

[In]

integrate(arccosh(a*x)^2/x^2,x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^2/x^2, x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x^2} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^2} \,d x \]

[In]

int(acosh(a*x)^2/x^2,x)

[Out]

int(acosh(a*x)^2/x^2, x)