\(\int \frac {x}{\text {arccosh}(a x)^2} \, dx\) [54]

Optimal result

Integrand size = 8, antiderivative size = 42 \[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \text {arccosh}(a x)}+\frac {\text {Chi}(2 \text {arccosh}(a x))}{a^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5885, 3382} \[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\frac {\text {Chi}(2 \text {arccosh}(a x))}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \text {arccosh}(a x)} \]

[In]

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

[Out]

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

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

Mathematica [A] (warning: unable to verify)

Time = 0.19 (sec) , antiderivative size = 44, normalized size of antiderivative = 1.05 \[ \int \frac {x}{\text {arccosh}(a x)^2} \, dx=\frac {-\frac {a x \sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{\text {arccosh}(a x)}+\text {Chi}(2 \text {arccosh}(a x))}{a^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.67

method	result	size
derivativedivides	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{2}}\)	\(28\)
default	\(\frac {-\frac {\sinh \left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{2 \,\operatorname {arccosh}\left (a x \right )}+\operatorname {Chi}\left (2 \,\operatorname {arccosh}\left (a x \right )\right )}{a^{2}}\)	\(28\)

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

integral(x/arccosh(a*x)^2, x)

Sympy [F]

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

[In]

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

[Out]

Integral(x/acosh(a*x)**2, x)

Maxima [F]

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

[In]

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

[Out]

-(a^3*x^4 - a*x^2 + (a^2*x^3 - x)*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((2*a^5*x^5 + 2*(a*x + 1)*(a*x - 1)*a^3*x^3 - 4*a^3*x^3
+ (4*a^4*x^4 - 4*a^2*x^2 + 1)*sqrt(a*x + 1)*sqrt(a*x - 1) + 2*a*x)/((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]

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

[In]

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

[Out]

integrate(x/arccosh(a*x)^2, x)

Mupad [F(-1)]

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

[In]

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

[Out]

int(x/acosh(a*x)^2, x)