\(\int \frac {\sqrt {\text {arccosh}(\frac {x}{a})}}{(a^2-x^2)^{3/2}} \, dx\) [395]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2 \sqrt {a^2-x^2}}+\frac {\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {Int}\left (\frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}},x\right )}{2 a^3 \sqrt {a^2-x^2}} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \frac {x \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{a^2 \sqrt {a^2-x^2}}+\frac {\left (\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}\right ) \int \frac {x}{\left (1-\frac {x^2}{a^2}\right ) \sqrt {\text {arccosh}\left (\frac {x}{a}\right )}} \, dx}{2 a^3 \sqrt {a^2-x^2}} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 7.05 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.85 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83

\[\int \frac {\sqrt {\operatorname {arccosh}\left (\frac {x}{a}\right )}}{\left (a^{2}-x^{2}\right )^{\frac {3}{2}}}d x\]

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 4.75 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\operatorname {acosh}{\left (\frac {x}{a} \right )}}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(acosh(x/a)**(1/2)/(a**2-x**2)**(3/2),x)

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(3/2),x, algorithm="maxima")

[Out]

integrate(sqrt(arccosh(x/a))/(a^2 - x^2)^(3/2), x)

Giac [N/A]

Not integrable

Time = 1.35 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int { \frac {\sqrt {\operatorname {arcosh}\left (\frac {x}{a}\right )}}{{\left (a^{2} - x^{2}\right )}^{\frac {3}{2}}} \,d x } \]

[In]

integrate(arccosh(x/a)^(1/2)/(a^2-x^2)^(3/2),x, algorithm="giac")

[Out]

integrate(sqrt(arccosh(x/a))/(a^2 - x^2)^(3/2), x)

Mupad [N/A]

Not integrable

Time = 3.02 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92 \[ \int \frac {\sqrt {\text {arccosh}\left (\frac {x}{a}\right )}}{\left (a^2-x^2\right )^{3/2}} \, dx=\int \frac {\sqrt {\mathrm {acosh}\left (\frac {x}{a}\right )}}{{\left (a^2-x^2\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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