3.4.81 \(\int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx\) [381]

Optimal. Leaf size=48 \[ \frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt {c-a^2 c x^2}} \]

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892} \begin {gather*} \frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[ArcCosh[a*x]]/Sqrt[c - a^2*c*x^2],x]

[Out]

(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[c - a^2*c*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 {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {c-a^2 c x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\sqrt {\cosh ^{-1}(a x)}}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 48, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^{3/2}}{3 a \sqrt {c-a^2 c x^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[ArcCosh[a*x]]/Sqrt[c - a^2*c*x^2],x]

[Out]

(2*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^(3/2))/(3*a*Sqrt[c - a^2*c*x^2])

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Maple [A]
time = 1.25, size = 41, normalized size = 0.85

method	result	size
default	\(\frac {2 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}{3 a \sqrt {-c \left (a x -1\right ) \left (a x +1\right )}}\)	\(41\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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Fricas [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

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

[Out]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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