3.4.87 \(\int \frac {\cosh ^{-1}(a x)^{3/2}}{(c-a^2 c x^2)^{3/2}} \, dx\) [387]

Optimal. Leaf size=95 \[ \frac {x \cosh ^{-1}(a x)^{3/2}}{c \sqrt {c-a^2 c x^2}}+\frac {3 a \sqrt {-1+a x} \sqrt {1+a x} \text {Int}\left (\frac {x \sqrt {\cosh ^{-1}(a x)}}{1-a^2 x^2},x\right )}{2 c \sqrt {c-a^2 c x^2}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.07, 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 {\cosh ^{-1}(a x)^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]
time = 1.52, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^{-1}(a x)^{3/2}}{\left (c-a^2 c x^2\right )^{3/2}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]
time = 4.34, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (a x \right )^{\frac {3}{2}}}{\left (-a^{2} c \,x^{2}+c \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________