\(\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \text {arccosh}(x)} \, dx\) [291]

Optimal result

Integrand size = 20, antiderivative size = 3 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \text {arccosh}(x)} \, dx=\text {Chi}(\text {arccosh}(x)) \]

[Out]

Chi(arccosh(x))

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5953, 3382} \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \text {arccosh}(x)} \, dx=\text {Chi}(\text {arccosh}(x)) \]

[In]

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

[Out]

CoshIntegral[ArcCosh[x]]

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 5953

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_
.), x_Symbol] :> Dist[(1/(b*c^(m + 1)))*Simp[(d1 + e1*x)^p/(1 + c*x)^p]*Simp[(d2 + e2*x)^p/(-1 + c*x)^p], Subs
t[Int[x^n*Cosh[-a/b + x/b]^m*Sinh[-a/b + x/b]^(2*p + 1), x], x, a + b*ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1,
e1, d2, e2, n}, x] && EqQ[e1, c*d1] && EqQ[e2, (-c)*d2] && IGtQ[p + 3/2, 0] && IGtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\text {arccosh}(x)\right ) \\ & = \text {Chi}(\text {arccosh}(x)) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 3, normalized size of antiderivative = 1.00 \[ \int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \text {arccosh}(x)} \, dx=\text {Chi}(\text {arccosh}(x)) \]

[In]

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

[Out]

CoshIntegral[ArcCosh[x]]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(44\) vs. \(2(3)=6\).

Time = 1.60 (sec) , antiderivative size = 45, normalized size of antiderivative = 15.00

method	result	size
default	\(-\frac {\sqrt {2 x -2}\, \sqrt {2+2 x}\, \sqrt {x -1}\, \sqrt {1+x}\, \left (\operatorname {Ei}_{1}\left (\operatorname {arccosh}\left (x \right )\right )+\operatorname {Ei}_{1}\left (-\operatorname {arccosh}\left (x \right )\right )\right )}{4 \left (x^{2}-1\right )}\)	\(45\)

[In]

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

[Out]

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

Fricas [F]

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

[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)

Sympy [F]

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

[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)

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

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

[In]

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

[Out]

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