3.138 \(\int \frac {x \cosh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)}{a^2}-\frac {x \sqrt {a x-1}}{a \sqrt {1-a x}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.18, antiderivative size = 73, normalized size of antiderivative = 1.49, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5798, 5718, 8} \[ -\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)}{a^2 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 5718

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x_))^(p_.), x_
Symbol] :> Simp[((d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(2*e1*e2*(p + 1)), x] - Dist[
(b*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(2*c*(p + 1)*(1 + c*x)^FracPart[p]
*(-1 + c*x)^FracPart[p]), Int[(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c,
d1, e1, d2, e2, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && GtQ[n, 0] && NeQ[p, -1] && IntegerQ[p + 1
/2]

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.09, size = 55, normalized size = 1.12 \[ \frac {\left (a^2 x^2-1\right ) \cosh ^{-1}(a x)-a x \sqrt {a x-1} \sqrt {a x+1}}{a^2 \sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.61, size = 72, normalized size = 1.47 \[ \frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{4} x^{2} - a^{2}} \]

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]

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

________________________________________________________________________________________

giac [C]  time = 0.73, size = 40, normalized size = 0.82 \[ -\frac {i \, x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{2}} \]

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]

-I*x/a - sqrt(-a^2*x^2 + 1)*log(a*x + sqrt(a^2*x^2 - 1))/a^2

________________________________________________________________________________________

maple [B]  time = 0.21, size = 123, normalized size = 2.51 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (-1+\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (1+\mathrm {arccosh}\left (a x \right )\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [C]  time = 0.66, size = 28, normalized size = 0.57 \[ \frac {i \, x}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )}{a^{2}} \]

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]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {x\,\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \]

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)

________________________________________________________________________________________

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

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

________________________________________________________________________________________