Optimal. Leaf size=98 \[ \frac {x \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}+\frac {3 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {Int}\left (\frac {x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{1-\frac {x^2}{a^2}},x\right )}{2 a^3 \sqrt {a^2-x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, 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}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx &=-\frac {\left (\sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}\right ) \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\left (-1+\frac {x}{a}\right )^{3/2} \left (1+\frac {x}{a}\right )^{3/2}} \, dx}{a^2 \sqrt {a^2-x^2}}\\ &=\frac {x \cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{a^2 \sqrt {a^2-x^2}}+\frac {\left (3 \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}}\right ) \int \frac {x \sqrt {\cosh ^{-1}\left (\frac {x}{a}\right )}}{1-\frac {x^2}{a^2}} \, dx}{2 a^3 \sqrt {a^2-x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cosh ^{-1}\left (\frac {x}{a}\right )^{3/2}}{\left (a^2-x^2\right )^{3/2}} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 4.10, size = 0, normalized size = 0.00 \[\int \frac {\mathrm {arccosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\left (a^{2}-x^{2}\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\left (- \left (- a + x\right ) \left (a + x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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]
[Out]
________________________________________________________________________________________
Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}}{{\left (a^2-x^2\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________