3.3.91 \(\int \frac {x}{\sqrt {-1+x} \sqrt {1+x} \cosh ^{-1}(x)} \, dx\) [291]

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

[Out]

Chi(arccosh(x))

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Rubi [A]
time = 0.08, antiderivative size = 3, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {5953, 3382} \begin {gather*} \text {Chi}\left (\cosh ^{-1}(x)\right ) \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(19\) vs. \(2(3)=6\).
time = 0.04, size = 19, normalized size = 6.33 \begin {gather*} \frac {1}{2} (-1+x) \text {Chi}\left (\cosh ^{-1}(x)\right ) \text {csch}^2\left (\frac {1}{2} \cosh ^{-1}(x)\right ) \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. \(77\) vs. \(2(3)=6\).
time = 0.22, size = 78, normalized size = 26.00

method	result	size
default	\(-\frac {\sqrt {-2+2 x}\, \sqrt {2+2 x}\, \sqrt {-1+x}\, \sqrt {1+x}\, \expIntegral \left (1, \mathrm {arccosh}\left (x \right )\right )}{4 \left (x^{2}-1\right )}-\frac {\sqrt {-2+2 x}\, \sqrt {2+2 x}\, \sqrt {-1+x}\, \sqrt {1+x}\, \expIntegral \left (1, -\mathrm {arccosh}\left (x \right )\right )}{4 \left (x^{2}-1\right )}\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x}{\sqrt {x - 1} \sqrt {x + 1} \operatorname {acosh}{\left (x \right )}}\, dx \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.33 \begin {gather*} \int \frac {x}{\mathrm {acosh}\left (x\right )\,\sqrt {x-1}\,\sqrt {x+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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