\(\int \frac {x^2}{\text {arccosh}(a x)^2} \, dx\) [53]

Optimal result

Integrand size = 10, antiderivative size = 59 \[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=-\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{4 a^3}+\frac {3 \text {Chi}(3 \text {arccosh}(a x))}{4 a^3} \]

[Out]

1/4*Chi(arccosh(a*x))/a^3+3/4*Chi(3*arccosh(a*x))/a^3-x^2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5885, 3382} \[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\frac {\text {Chi}(\text {arccosh}(a x))}{4 a^3}+\frac {3 \text {Chi}(3 \text {arccosh}(a x))}{4 a^3}-\frac {x^2 \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)} \]

[In]

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

[Out]

-((x^2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*ArcCosh[a*x])) + CoshIntegral[ArcCosh[a*x]]/(4*a^3) + (3*CoshIntegral[
3*ArcCosh[a*x]])/(4*a^3)

Rule 3382

Int[sin[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[CoshIntegral[c*f*(fz/d)
+ f*fz*x]/d, x] /; FreeQ[{c, d, e, f, fz}, x] && EqQ[d*(e - Pi/2) - c*f*fz*I, 0]

Rule 5885

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_)*(x_)^(m_.), x_Symbol] :> Simp[x^m*Sqrt[1 + c*x]*Sqrt[-1 + c*x]*((
a + b*ArcCosh[c*x])^(n + 1)/(b*c*(n + 1))), x] + Dist[1/(b^2*c^(m + 1)*(n + 1)), Subst[Int[ExpandTrigReduce[x^
(n + 1), Cosh[-a/b + x/b]^(m - 1)*(m - (m + 1)*Cosh[-a/b + x/b]^2), x], x], x, a + b*ArcCosh[c*x]], x] /; Free
Q[{a, b, c}, x] && IGtQ[m, 0] && GeQ[n, -2] && LtQ[n, -1]

Rubi steps \begin{align*} \text {integral}& = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}-\frac {\text {Subst}\left (\int \left (-\frac {\cosh (x)}{4 x}-\frac {3 \cosh (3 x)}{4 x}\right ) \, dx,x,\text {arccosh}(a x)\right )}{a^3} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{4 a^3}+\frac {3 \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\text {arccosh}(a x)\right )}{4 a^3} \\ & = -\frac {x^2 \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(\text {arccosh}(a x))}{4 a^3}+\frac {3 \text {Chi}(3 \text {arccosh}(a x))}{4 a^3} \\ \end{align*}

Mathematica [A] (warning: unable to verify)

Time = 0.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.98 \[ \int \frac {x^2}{\text {arccosh}(a x)^2} \, dx=\frac {-\frac {4 a^2 x^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{\text {arccosh}(a x)}+\text {Chi}(\text {arccosh}(a x))+3 \text {Chi}(3 \text {arccosh}(a x))}{4 a^3} \]

[In]

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

[Out]

((-4*a^2*x^2*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x))/ArcCosh[a*x] + CoshIntegral[ArcCosh[a*x]] + 3*CoshIntegral[
3*ArcCosh[a*x]])/(4*a^3)

Maple [A] (verified)

Time = 0.05 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.00

method	result	size
derivativedivides	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{4}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4}}{a^{3}}\)	\(59\)
default	\(\frac {-\frac {\sqrt {a x -1}\, \sqrt {a x +1}}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {\operatorname {Chi}\left (\operatorname {arccosh}\left (a x \right )\right )}{4}-\frac {\sinh \left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4 \,\operatorname {arccosh}\left (a x \right )}+\frac {3 \,\operatorname {Chi}\left (3 \,\operatorname {arccosh}\left (a x \right )\right )}{4}}{a^{3}}\)	\(59\)

[In]

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

[Out]

1/a^3*(-1/4/arccosh(a*x)*(a*x-1)^(1/2)*(a*x+1)^(1/2)+1/4*Chi(arccosh(a*x))-1/4/arccosh(a*x)*sinh(3*arccosh(a*x
))+3/4*Chi(3*arccosh(a*x)))

Fricas [F]

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

[In]

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

[Out]

integral(x^2/arccosh(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x**2/acosh(a*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

-(a^3*x^5 - a*x^3 + (a^2*x^4 - x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))/((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))) + integrate((3*a^5*x^6 - 6*a^3*x^4 + (3*a^3*x^4 - a*x^2)*(a*x +
1)*(a*x - 1) + 3*a*x^2 + (6*a^4*x^5 - 7*a^2*x^3 + 2*x)*sqrt(a*x + 1)*sqrt(a*x - 1))/((a^5*x^4 + (a*x + 1)*(a*x
- 1)*a^3*x^2 - 2*a^3*x^2 + 2*(a^4*x^3 - a^2*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + a)*log(a*x + sqrt(a*x + 1)*sqrt(
a*x - 1))), x)

Giac [F]

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

[In]

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

[Out]

integrate(x^2/arccosh(a*x)^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/acosh(a*x)^2, x)