3.5.16 \(\int \frac {1}{\sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}} \, dx\) [416]

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

[Out]

-2/3*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/arccosh(a*x)^(3/2)/(-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}}{3 a \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2)),x]

[Out]

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

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 {c-a^2 c x^2} \cosh ^{-1}(a x)^{5/2}} \, 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)^{5/2}} \, dx}{\sqrt {c-a^2 c x^2}}\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 48, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{3 a \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[c - a^2*c*x^2]*ArcCosh[a*x]^(5/2)),x]

[Out]

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

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Maple [A]
time = 1.40, size = 41, normalized size = 0.85

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),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(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(1/(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^(5/2)), x)

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Fricas [A]
time = 0.36, size = 59, normalized size = 1.23 \begin {gather*} \frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} - 1}}{3 \, {\left (a^{3} c x^{2} - a c\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}^{\frac {5}{2}}{\left (a x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(-a**2*c*x**2+c)**(1/2)/acosh(a*x)**(5/2),x)

[Out]

Integral(1/(sqrt(-c*(a*x - 1)*(a*x + 1))*acosh(a*x)**(5/2)), 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/(-a^2*c*x^2+c)^(1/2)/arccosh(a*x)^(5/2),x, algorithm="giac")

[Out]

integrate(1/(sqrt(-a^2*c*x^2 + c)*arccosh(a*x)^(5/2)), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{5/2}\,\sqrt {c-a^2\,c\,x^2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acosh(a*x)^(5/2)*(c - a^2*c*x^2)^(1/2)),x)

[Out]

int(1/(acosh(a*x)^(5/2)*(c - a^2*c*x^2)^(1/2)), x)

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