\(\int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\) [231]

Optimal result

Integrand size = 24, antiderivative size = 124 \[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {-1+a x} \text {arccosh}(a x)^2}{\sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}-\frac {2 a \sqrt {-1+a x} \text {arccosh}(a x) \log \left (1+e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}-\frac {a \sqrt {-1+a x} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.12 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5917, 5882, 3799, 2221, 2317, 2438} \[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=-\frac {\sqrt {1-a^2 x^2} \text {arccosh}(a x)^2}{x}-\frac {a \sqrt {a x-1} \operatorname {PolyLog}\left (2,-e^{2 \text {arccosh}(a x)}\right )}{\sqrt {1-a x}}+\frac {a \sqrt {a x-1} \text {arccosh}(a x)^2}{\sqrt {1-a x}}-\frac {2 a \sqrt {a x-1} \text {arccosh}(a x) \log \left (e^{2 \text {arccosh}(a x)}+1\right )}{\sqrt {1-a x}} \]

[In]

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

[Out]

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

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]

Rule 5917

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp
[(f*x)^(m + 1)*(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(d*f*(m + 1))), x] + Dist[b*c*(n/(f*(m + 1)))*Simp[
(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Int[(f*x)^(m + 1)*(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*A
rcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && EqQ[m
+ 2*p + 3, 0] && NeQ[m, -1]

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

Mathematica [A] (verified)

Time = 0.37 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.90 \[ \int \frac {\text {arccosh}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx=\frac {a \sqrt {\frac {-1+a x}{1+a x}} (1+a x) \left (\text {arccosh}(a x) \left (-\text {arccosh}(a x)+\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \text {arccosh}(a x)}{a x}-2 \log \left (1+e^{-2 \text {arccosh}(a x)}\right )\right )+\operatorname {PolyLog}\left (2,-e^{-2 \text {arccosh}(a x)}\right )\right )}{\sqrt {-((-1+a x) (1+a x))}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.90 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.94

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x -1}\, \sqrt {a x +1}\, a x -1\right ) \operatorname {arccosh}\left (a x \right )^{2}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{2} a}{a^{2} x^{2}-1}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {polylog}\left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}\)	\(241\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

integrate(arccosh(a*x)^2/x^2/(-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^4 - x^2), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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