3.3.96 \(\int \frac {1}{\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)} \, dx\) [296]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 28, 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 = {5890} \begin {gather*} \frac {\sqrt {a x-1} \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt {1-a x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 5890

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

Rubi steps

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

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Mathematica [A]
time = 0.04, size = 47, normalized size = 1.68 \begin {gather*} \frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \log \left (\cosh ^{-1}(a x)\right )}{a \sqrt {-((-1+a x) (1+a x))}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 3.64, size = 48, normalized size = 1.71

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\mathrm {arccosh}\left (a x \right )\right )}{a \left (a^{2} x^{2}-1\right )}\)	\(48\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (24) = 48\).
time = 0.36, size = 55, normalized size = 1.96 \begin {gather*} -\frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )\right )}{a^{3} x^{2} - a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

integrate(1/(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 {1}{\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(1/(acosh(a*x)*(1 - a^2*x^2)^(1/2)),x)

[Out]

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

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