3.291 \(\int \frac{x}{\sqrt{-1+x} \sqrt{1+x} \cosh ^{-1}(x)} \, dx\)

Optimal. Leaf size=3 \[ \text{Chi}\left (\cosh ^{-1}(x)\right ) \]

[Out]

CoshIntegral[ArcCosh[x]]

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Rubi [A]  time = 0.120613, antiderivative size = 3, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {5781, 3301} \[ \text{Chi}\left (\cosh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[-1 + x]*Sqrt[1 + x]*ArcCosh[x]),x]

[Out]

CoshIntegral[ArcCosh[x]]

Rule 5781

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(-(d1*d2))^p/c^(m + 1), Subst[Int[(a + b*x)^n*Cosh[x]^m*Sinh[x]^(2*p + 1), x], x, ArcCos
h[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[p
+ 1/2] && GtQ[p, -1] && IGtQ[m, 0] && (GtQ[d1, 0] && LtQ[d2, 0])

Rule 3301

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]

Rubi steps

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

Mathematica [B]  time = 0.0543266, size = 19, normalized size = 6.33 \[ \frac{1}{2} (x-1) \text{Chi}\left (\cosh ^{-1}(x)\right ) \text{csch}^2\left (\frac{1}{2} \cosh ^{-1}(x)\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[-1 + x]*Sqrt[1 + x]*ArcCosh[x]),x]

[Out]

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

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Maple [B]  time = 0.14, size = 78, normalized size = 26. \begin{align*} -{\frac{{\it Ei} \left ( 1,{\rm arccosh} \left (x\right ) \right ) }{4\,{x}^{2}-4}\sqrt{2+2\,x}\sqrt{-2+2\,x}\sqrt{-1+x}\sqrt{1+x}}-{\frac{{\it Ei} \left ( 1,-{\rm arccosh} \left (x\right ) \right ) }{4\,{x}^{2}-4}\sqrt{2+2\,x}\sqrt{-2+2\,x}\sqrt{-1+x}\sqrt{1+x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(2+2*x)^(1/2)*(-2+2*x)^(1/2)*(-1+x)^(1/2)*(1+x)^(1/2)*Ei(1,arccosh(x))/(x^2-1)-1/4*(2+2*x)^(1/2)*(-2+2*x)
^(1/2)*(-1+x)^(1/2)*(1+x)^(1/2)*Ei(1,-arccosh(x))/(x^2-1)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x + 1} \sqrt{x - 1} \operatorname{arcosh}\left (x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{x + 1} \sqrt{x - 1} x}{{\left (x^{2} - 1\right )} \operatorname{arcosh}\left (x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x - 1} \sqrt{x + 1} \operatorname{acosh}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x}{\sqrt{x + 1} \sqrt{x - 1} \operatorname{arcosh}\left (x\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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