3.3.28 \(\int \frac {x \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}} \, dx\) [228]

Optimal. Leaf size=79 \[ -\frac {2 \sqrt {1-a x} \sqrt {1+a x}}{a^2}-\frac {2 x \sqrt {-1+a x} \cosh ^{-1}(a x)}{a \sqrt {1-a x}}-\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{a^2} \]

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {5914, 5879, 75} \begin {gather*} -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{a^2}-\frac {2 \sqrt {1-a x} \sqrt {a x+1}}{a^2}-\frac {2 x \sqrt {a x-1} \cosh ^{-1}(a x)}{a \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 75

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

Rule 5879

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*ArcCosh[c*x])^n, x] - Dist[b*c*n, In
t[x*((a + b*ArcCosh[c*x])^(n - 1)/(Sqrt[1 + c*x]*Sqrt[-1 + c*x])), x], x] /; FreeQ[{a, b, c}, x] && GtQ[n, 0]

Rule 5914

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x^2)^
(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Dist[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*
(-1 + c*x)^p)], Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x])^(n - 1), x], x] /; FreeQ[{a,
b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]

Rubi steps

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

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Mathematica [A]
time = 0.07, size = 54, normalized size = 0.68 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (-2+\frac {2 a x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}}-\cosh ^{-1}(a x)^2\right )}{a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 2.33, size = 139, normalized size = 1.76

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (\sqrt {a x +1}\, \sqrt {a x -1}\, a x +a^{2} x^{2}-1\right ) \left (\mathrm {arccosh}\left (a x \right )^{2}-2 \,\mathrm {arccosh}\left (a x \right )+2\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \left (\mathrm {arccosh}\left (a x \right )^{2}+2 \,\mathrm {arccosh}\left (a x \right )+2\right )}{2 a^{2} \left (a^{2} x^{2}-1\right )}\)	\(139\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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Maxima [C] Result contains complex when optimal does not.
time = 0.28, size = 50, normalized size = 0.63 \begin {gather*} \frac {2 i \, x \operatorname {arcosh}\left (a x\right )}{a} - \frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2}} - \frac {2 i \, \sqrt {a^{2} x^{2} - 1}}{a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.35, size = 114, normalized size = 1.44 \begin {gather*} \frac {2 \, \sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + {\left (-a^{2} x^{2} + 1\right )}^{\frac {3}{2}} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} - 2 \, {\left (a^{2} x^{2} - 1\right )} \sqrt {-a^{2} x^{2} + 1}}{a^{4} x^{2} - a^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x \operatorname {acosh}^{2}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [C] Result contains complex when optimal does not.
time = 0.42, size = 76, normalized size = 0.96 \begin {gather*} -\frac {\sqrt {-a^{2} x^{2} + 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{a^{2}} - \frac {2 i \, {\left (x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {\sqrt {a^{2} x^{2} - 1}}{a}\right )}}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x\,{\mathrm {acosh}\left (a\,x\right )}^2}{\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)^2)/(1 - a^2*x^2)^(1/2),x)

[Out]

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

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