3.368 \(\int \frac {1}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3} \, dx\)

Optimal. Leaf size=32 \[ -\frac {\sqrt {a x-1}}{2 a \sqrt {1-a x} \cosh ^{-1}(a x)^2} \]

[Out]

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

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Rubi [A]  time = 0.15, antiderivative size = 45, normalized size of antiderivative = 1.41, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {5713, 5676} \[ -\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3),x]

[Out]

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

Rule 5676

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

Rule 5713

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((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[(1 + c*x)^p*(-1 + c*x)^p*(a + b*Ar
cCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[c^2*d + e, 0] &&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^3} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {\sqrt {-1+a x} \sqrt {1+a x}}{2 a \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 45, normalized size = 1.41 \[ -\frac {\sqrt {a x-1} \sqrt {a x+1}}{2 a \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[1 - a^2*x^2]*ArcCosh[a*x]^3),x]

[Out]

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

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fricas [B]  time = 0.60, size = 56, normalized size = 1.75 \[ \frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1}}{2 \, {\left (a^{3} x^{2} - a\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{3}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-a^2*x^2 + 1)*arccosh(a*x)^3), x)

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maple [A]  time = 0.07, size = 51, normalized size = 1.59 \[ \frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}}{2 a \left (a^{2} x^{2}-1\right ) \mathrm {arccosh}\left (a x \right )^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x)

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ -\frac {a^{7} x^{7} - 3 \, a^{5} x^{5} + 3 \, a^{3} x^{3} + {\left (a^{4} x^{4} - a^{2} x^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} + {\left (3 \, a^{5} x^{5} - 5 \, a^{3} x^{3} + 2 \, a x\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (3 \, a^{6} x^{6} - 7 \, a^{4} x^{4} + 5 \, a^{2} x^{2} - 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} - a x - {\left (a^{5} x^{5} - 2 \, a^{3} x^{3} - {\left (a^{2} x^{2} - 1\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} - {\left (a^{3} x^{3} - a x\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + {\left (a^{4} x^{4} - 2 \, a^{2} x^{2} + 1\right )} \sqrt {a x + 1} \sqrt {a x - 1} + a x\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{2 \, {\left ({\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{\frac {3}{2}} a^{4} x^{3} + 3 \, {\left (a^{5} x^{4} - a^{3} x^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )} + 3 \, {\left (a^{6} x^{5} - 2 \, a^{4} x^{3} + a^{2} x\right )} {\left (a x + 1\right )} \sqrt {a x - 1} + {\left (a^{7} x^{6} - 3 \, a^{5} x^{4} + 3 \, a^{3} x^{2} - a\right )} \sqrt {a x + 1}\right )} \sqrt {-a x + 1} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}} - \int -\frac {2 \, a^{6} x^{6} - 3 \, a^{4} x^{4} - {\left (2 \, a^{2} x^{2} - 3\right )} {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{2} - 4 \, {\left (a^{3} x^{3} - a x\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )}^{\frac {3}{2}} - 4 \, {\left (a^{2} x^{2} - 1\right )} {\left (a x + 1\right )} {\left (a x - 1\right )} + 4 \, {\left (a^{5} x^{5} - 2 \, a^{3} x^{3} + a x\right )} \sqrt {a x + 1} \sqrt {a x - 1} + 1}{2 \, {\left ({\left (a x + 1\right )}^{\frac {5}{2}} {\left (a x - 1\right )}^{2} a^{4} x^{4} + 4 \, {\left (a^{5} x^{5} - a^{3} x^{3}\right )} {\left (a x + 1\right )}^{2} {\left (a x - 1\right )}^{\frac {3}{2}} + 6 \, {\left (a^{6} x^{6} - 2 \, a^{4} x^{4} + a^{2} x^{2}\right )} {\left (a x + 1\right )}^{\frac {3}{2}} {\left (a x - 1\right )} + 4 \, {\left (a^{7} x^{7} - 3 \, a^{5} x^{5} + 3 \, a^{3} x^{3} - a x\right )} {\left (a x + 1\right )} \sqrt {a x - 1} + {\left (a^{8} x^{8} - 4 \, a^{6} x^{6} + 6 \, a^{4} x^{4} - 4 \, a^{2} x^{2} + 1\right )} \sqrt {a x + 1}\right )} \sqrt {-a x + 1} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/arccosh(a*x)^3/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

-1/2*(a^7*x^7 - 3*a^5*x^5 + 3*a^3*x^3 + (a^4*x^4 - a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (3*a^5*x^5 - 5*a
^3*x^3 + 2*a*x)*(a*x + 1)*(a*x - 1) + (3*a^6*x^6 - 7*a^4*x^4 + 5*a^2*x^2 - 1)*sqrt(a*x + 1)*sqrt(a*x - 1) - a*
x - (a^5*x^5 - 2*a^3*x^3 - (a^2*x^2 - 1)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - (a^3*x^3 - a*x)*(a*x + 1)*(a*x - 1)
+ (a^4*x^4 - 2*a^2*x^2 + 1)*sqrt(a*x + 1)*sqrt(a*x - 1) + a*x)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1)))/(((a*x
+ 1)^2*(a*x - 1)^(3/2)*a^4*x^3 + 3*(a^5*x^4 - a^3*x^2)*(a*x + 1)^(3/2)*(a*x - 1) + 3*(a^6*x^5 - 2*a^4*x^3 + a
^2*x)*(a*x + 1)*sqrt(a*x - 1) + (a^7*x^6 - 3*a^5*x^4 + 3*a^3*x^2 - a)*sqrt(a*x + 1))*sqrt(-a*x + 1)*log(a*x +
sqrt(a*x + 1)*sqrt(a*x - 1))^2) - integrate(-1/2*(2*a^6*x^6 - 3*a^4*x^4 - (2*a^2*x^2 - 3)*(a*x + 1)^2*(a*x - 1
)^2 - 4*(a^3*x^3 - a*x)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) - 4*(a^2*x^2 - 1)*(a*x + 1)*(a*x - 1) + 4*(a^5*x^5 - 2
*a^3*x^3 + a*x)*sqrt(a*x + 1)*sqrt(a*x - 1) + 1)/(((a*x + 1)^(5/2)*(a*x - 1)^2*a^4*x^4 + 4*(a^5*x^5 - a^3*x^3)
*(a*x + 1)^2*(a*x - 1)^(3/2) + 6*(a^6*x^6 - 2*a^4*x^4 + a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1) + 4*(a^7*x^7 - 3*a^
5*x^5 + 3*a^3*x^3 - a*x)*(a*x + 1)*sqrt(a*x - 1) + (a^8*x^8 - 4*a^6*x^6 + 6*a^4*x^4 - 4*a^2*x^2 + 1)*sqrt(a*x
+ 1))*sqrt(-a*x + 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

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mupad [B]  time = 0.41, size = 48, normalized size = 1.50 \[ \frac {\sqrt {1-a^2\,x^2}\,\sqrt {a\,x-1}\,\sqrt {a\,x+1}}{a\,{\mathrm {acosh}\left (a\,x\right )}^2\,\left (2\,a^2\,x^2-2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acosh(a*x)^3*(1 - a^2*x^2)^(1/2)),x)

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{3}{\left (a x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/acosh(a*x)**3/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(1/(sqrt(-(a*x - 1)*(a*x + 1))*acosh(a*x)**3), x)

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