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

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

[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 = 0.15, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {5713, 5676} \[ -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]

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 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 {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 \[ -\frac {2 \sqrt {a x-1} \sqrt {a x+1}}{3 a \sqrt {c-a^2 c x^2} \cosh ^{-1}(a x)^{3/2}} \]

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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fricas [A]  time = 0.66, size = 59, normalized size = 1.23 \[ \frac {2 \, \sqrt {-a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} - 1}}{3 \, {\left (a^{3} c x^{2} - a c\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{\frac {3}{2}}} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

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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maple [A]  time = 0.08, size = 41, normalized size = 0.85 \[ -\frac {2 \sqrt {a x -1}\, \sqrt {a x +1}}{3 \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}} a \sqrt {-\left (a x -1\right ) \left (a x +1\right ) c}} \]

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)

[Out]

-2/3/arccosh(a*x)^(3/2)/a/(-(a*x-1)*(a*x+1)*c)^(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 \[ \int \frac {1}{\sqrt {-a^{2} c x^{2} + c} \operatorname {arcosh}\left (a x\right )^{\frac {5}{2}}}\,{d x} \]

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

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

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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