\(\int \frac {\text {arccosh}(\frac {x}{a})^{3/2}}{\sqrt {a^2-x^2}} \, dx\) [399]

Optimal result

Integrand size = 24, antiderivative size = 50 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

[Out]

2/5*a*arccosh(x/a)^(5/2)*(-1+x/a)^(1/2)*(1+x/a)^(1/2)/(a^2-x^2)^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892} \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

[In]

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

[Out]

(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(5/2))/(5*Sqrt[a^2 - x^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*} \text {integral}& = \frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]

[In]

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

[Out]

(2*a*Sqrt[-1 + x/a]*Sqrt[1 + x/a]*ArcCosh[x/a]^(5/2))/(5*Sqrt[a^2 - x^2])

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88

method	result	size
default	\(\frac {2 \operatorname {arccosh}\left (\frac {x}{a}\right )^{\frac {5}{2}} \sqrt {-\frac {a -x}{a}}\, \sqrt {\frac {a +x}{a}}\, a}{5 \sqrt {\left (a -x \right ) \left (a +x \right )}}\)	\(44\)

[In]

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

[Out]

2/5*arccosh(x/a)^(5/2)/((a-x)*(a+x))^(1/2)*(-(a-x)/a)^(1/2)*((a+x)/a)^(1/2)*a

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {\operatorname {acosh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\sqrt {a^{2} - x^{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\sqrt {a^{2} - x^{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]

[In]

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

[Out]

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