\(\int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx\) [230]

Optimal result

Integrand size = 24, antiderivative size = 183 \[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\frac {2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {-1+a x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {-1+a x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \]

[Out]

2*arccosh(a*x)^2*arctan(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-2*I*arccosh(a*x)*polylog
(2,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+2*I*arccosh(a*x)*polylog(2,I*(a*x+(a*x-1
)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)+2*I*polylog(3,-I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-
1)^(1/2)/(-a*x+1)^(1/2)-2*I*polylog(3,I*(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2)))*(a*x-1)^(1/2)/(-a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {5946, 4265, 2611, 2320, 6724} \[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\frac {2 \sqrt {a x-1} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {a x-1} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {a x-1} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {a x-1} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {a x-1} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \]

[In]

Int[ArcCosh[a*x]^2/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

(2*Sqrt[-1 + a*x]*ArcCosh[a*x]^2*ArcTan[E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((2*I)*Sqrt[-1 + a*x]*ArcCosh[a*x]*Po
lyLog[2, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] + ((2*I)*Sqrt[-1 + a*x]*ArcCosh[a*x]*PolyLog[2, I*E^ArcCosh[a*x]]
)/Sqrt[1 - a*x] + ((2*I)*Sqrt[-1 + a*x]*PolyLog[3, (-I)*E^ArcCosh[a*x]])/Sqrt[1 - a*x] - ((2*I)*Sqrt[-1 + a*x]
*PolyLog[3, I*E^ArcCosh[a*x]])/Sqrt[1 - a*x]

Rule 2320

Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Dist[v/D[v, x], Subst[Int[FunctionOfExponentialFu
nction[u, x]/x, x], x, v], x]] /; FunctionOfExponentialQ[u, x] &&  !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; F
reeQ[{a, m, n}, x] && IntegerQ[m*n]] &&  !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x))*(F_)[v_] /; FreeQ[{a, b, c}, x
] && InverseFunctionQ[F[x]]]

Rule 2611

Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(
f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Dist[g*(m/(b*c*n*Log[F])), Int[(f + g*
x)^(m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e, f, g, n}, x] && GtQ[m, 0]

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 5946

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Dist[(1/c^(m
+ 1))*Simp[Sqrt[1 + c*x]*(Sqrt[-1 + c*x]/Sqrt[d + e*x^2])], Subst[Int[(a + b*x)^n*Cosh[x]^m, x], x, ArcCosh[c
*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c^2*d + e, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 6724

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+a x} \text {Subst}\left (\int x^2 \text {sech}(x) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (2 i \sqrt {-1+a x}\right ) \text {Subst}\left (\int x \log \left (1-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}}+\frac {\left (2 i \sqrt {-1+a x}\right ) \text {Subst}\left (\int x \log \left (1+i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (2 i \sqrt {-1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}}-\frac {\left (2 i \sqrt {-1+a x}\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^x\right ) \, dx,x,\text {arccosh}(a x)\right )}{\sqrt {1-a x}} \\ & = \frac {2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {\left (2 i \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {\left (2 i \sqrt {-1+a x}\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \\ & = \frac {2 \sqrt {-1+a x} \text {arccosh}(a x)^2 \arctan \left (e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {-1+a x} \text {arccosh}(a x) \operatorname {PolyLog}\left (2,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {2 i \sqrt {-1+a x} \operatorname {PolyLog}\left (3,-i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {2 i \sqrt {-1+a x} \operatorname {PolyLog}\left (3,i e^{\text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.83 \[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\frac {i \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (-\text {arccosh}(a x)^2 \left (\log \left (1-i e^{-\text {arccosh}(a x)}\right )-\log \left (1+i e^{-\text {arccosh}(a x)}\right )\right )-2 \text {arccosh}(a x) \left (\operatorname {PolyLog}\left (2,-i e^{-\text {arccosh}(a x)}\right )-\operatorname {PolyLog}\left (2,i e^{-\text {arccosh}(a x)}\right )\right )-2 \operatorname {PolyLog}\left (3,-i e^{-\text {arccosh}(a x)}\right )+2 \operatorname {PolyLog}\left (3,i e^{-\text {arccosh}(a x)}\right )\right )}{\sqrt {1-a^2 x^2}} \]

[In]

Integrate[ArcCosh[a*x]^2/(x*Sqrt[1 - a^2*x^2]),x]

[Out]

(I*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(-(ArcCosh[a*x]^2*(Log[1 - I/E^ArcCosh[a*x]] - Log[1 + I/E^ArcCosh[a*x
]])) - 2*ArcCosh[a*x]*(PolyLog[2, (-I)/E^ArcCosh[a*x]] - PolyLog[2, I/E^ArcCosh[a*x]]) - 2*PolyLog[3, (-I)/E^A
rcCosh[a*x]] + 2*PolyLog[3, I/E^ArcCosh[a*x]]))/Sqrt[1 - a^2*x^2]

Maple [F]

\[\int \frac {\operatorname {arccosh}\left (a x \right )^{2}}{x \sqrt {-a^{2} x^{2}+1}}d x\]

[In]

int(arccosh(a*x)^2/x/(-a^2*x^2+1)^(1/2),x)

[Out]

int(arccosh(a*x)^2/x/(-a^2*x^2+1)^(1/2),x)

Fricas [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

[In]

integrate(arccosh(a*x)^2/x/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^2/(a^2*x^3 - x), x)

Sympy [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

[In]

integrate(acosh(a*x)**2/x/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(acosh(a*x)**2/(x*sqrt(-(a*x - 1)*(a*x + 1))), x)

Maxima [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

[In]

integrate(arccosh(a*x)^2/x/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(arccosh(a*x)^2/(sqrt(-a^2*x^2 + 1)*x), x)

Giac [F]

\[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x} \,d x } \]

[In]

integrate(arccosh(a*x)^2/x/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^2/(sqrt(-a^2*x^2 + 1)*x), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}(a x)^2}{x \sqrt {1-a^2 x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x\,\sqrt {1-a^2\,x^2}} \,d x \]

[In]

int(acosh(a*x)^2/(x*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(acosh(a*x)^2/(x*(1 - a^2*x^2)^(1/2)), x)