3.4.68 \(\int \frac {1}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^3} \, dx\) [368]

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

[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 = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5892} \begin {gather*} -\frac {\sqrt {a x-1}}{2 a \sqrt {1-a x} \cosh ^{-1}(a x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5892

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

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 = 0.02, size = 45, normalized size = 1.41 \begin {gather*} -\frac {\sqrt {-1+a x} \sqrt {1+a x}}{2 a \sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2} \end {gather*}

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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Maple [A]
time = 1.42, size = 51, normalized size = 1.59

method	result	size
default	\(\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}}{2 a \left (a^{2} x^{2}-1\right ) \mathrm {arccosh}\left (a x \right )^{2}}\)	\(51\)

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,method=_RETURNVERBOSE)

[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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (26) = 52\).
time = 0.34, size = 56, normalized size = 1.75 \begin {gather*} \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}} \end {gather*}

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

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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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(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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Mupad [B]
time = 0.41, size = 48, normalized size = 1.50 \begin {gather*} \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 )} \end {gather*}

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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