3.3.95 \(\int \frac {x}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx\) [295]

Optimal. Leaf size=28 \[ \frac {\sqrt {-1+a x} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {1-a x}} \]

[Out]

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

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Rubi [A]
time = 0.06, antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {5952, 3382} \begin {gather*} \frac {\sqrt {a x-1} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]

[Out]

(Sqrt[-1 + a*x]*CoshIntegral[ArcCosh[a*x]])/(a^2*Sqrt[1 - a*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 5952

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[(1/(b*
c^(m + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)], Subst[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, d, e, n}, x] && EqQ[c^2*d + e, 0] && IGtQ[2*p + 2
, 0] && IGtQ[m, 0]

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 50, normalized size = 1.79 \begin {gather*} -\frac {\sqrt {-((-1+a x) (1+a x))} \text {Chi}\left (\cosh ^{-1}(a x)\right )}{a^2 \sqrt {\frac {-1+a x}{1+a x}} (1+a x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]),x]

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(99\) vs. \(2(24)=48\).
time = 3.01, size = 100, normalized size = 3.57

method	result	size
default	\(\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \expIntegral \left (1, \mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \expIntegral \left (1, -\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}\)	\(100\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(x/(sqrt(-(a*x - 1)*(a*x + 1))*acosh(a*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(a*x)/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {x}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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