3.5.12 \(\int \frac {1}{(c-a^2 c x^2)^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx\) [412]

Optimal. Leaf size=110 \[ -\frac {2 \sqrt {-1+a x} \sqrt {1+a x}}{a \left (c-a^2 c x^2\right )^{3/2} \sqrt {\cosh ^{-1}(a x)}}+\frac {4 a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\cosh ^{-1}(a x)}},x\right )}{c \sqrt {c-a^2 c x^2}} \]

[Out]

-2*(a*x-1)^(1/2)*(a*x+1)^(1/2)/a/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(1/2)+4*a*(a*x-1)^(1/2)*(a*x+1)^(1/2)*Unint
egrable(x/(a^2*x^2-1)^2/arccosh(a*x)^(1/2),x)/c/(-a^2*c*x^2+c)^(1/2)

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Rubi [A]
time = 0.14, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

(-2*Sqrt[-1 + a*x]*Sqrt[1 + a*x])/(a*(c - a^2*c*x^2)^(3/2)*Sqrt[ArcCosh[a*x]]) + (4*a*Sqrt[-1 + a*x]*Sqrt[1 +
a*x]*Defer[Int][x/((-1 + a^2*x^2)^2*Sqrt[ArcCosh[a*x]]), x])/(c*Sqrt[c - a^2*c*x^2])

Rubi steps

\begin {align*} \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {1}{(-1+a x)^{3/2} (1+a x)^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ &=-\frac {2 \sqrt {-1+a x}}{a c (1-a x) \sqrt {1+a x} \sqrt {c-a^2 c x^2} \sqrt {\cosh ^{-1}(a x)}}+\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {x}{\left (-1+a^2 x^2\right )^2 \sqrt {\cosh ^{-1}(a x)}} \, dx}{c \sqrt {c-a^2 c x^2}}\\ \end {align*}

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Mathematica [A]
time = 1.30, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (c-a^2 c x^2\right )^{3/2} \cosh ^{-1}(a x)^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)),x]

[Out]

Integrate[1/((c - a^2*c*x^2)^(3/2)*ArcCosh[a*x]^(3/2)), x]

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Maple [A]
time = 4.14, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}} \mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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)^(3/2)/arccosh(a*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((-a^2*c*x^2 + c)^(3/2)*arccosh(a*x)^(3/2)), 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(1/(-a^2*c*x^2+c)^(3/2)/arccosh(a*x)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {3}{2}} \operatorname {acosh}^{\frac {3}{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)**(3/2)/acosh(a*x)**(3/2),x)

[Out]

Integral(1/((-c*(a*x - 1)*(a*x + 1))**(3/2)*acosh(a*x)**(3/2)), x)

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Giac [A]
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)^(3/2)/arccosh(a*x)^(3/2),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\mathrm {acosh}\left (a\,x\right )}^{3/2}\,{\left (c-a^2\,c\,x^2\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(3/2)),x)

[Out]

int(1/(acosh(a*x)^(3/2)*(c - a^2*c*x^2)^(3/2)), x)

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