Optimal. Leaf size=95 \[ -\frac {1}{3} a^4 \log (x)+\frac {a^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{3 x}+\frac {a^2}{12 x^2}-\frac {\cosh ^{-1}(a x)^2}{4 x^4}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{6 x^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.36, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5662, 5748, 5724, 29, 30} \[ \frac {a^2}{12 x^2}-\frac {1}{3} a^4 \log (x)+\frac {a^3 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{3 x}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{6 x^3}-\frac {\cosh ^{-1}(a x)^2}{4 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 29
Rule 30
Rule 5662
Rule 5724
Rule 5748
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^5} \, dx &=-\frac {\cosh ^{-1}(a x)^2}{4 x^4}+\frac {1}{2} a \int \frac {\cosh ^{-1}(a x)}{x^4 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{6 x^3}-\frac {\cosh ^{-1}(a x)^2}{4 x^4}-\frac {1}{6} a^2 \int \frac {1}{x^3} \, dx+\frac {1}{3} a^3 \int \frac {\cosh ^{-1}(a x)}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=\frac {a^2}{12 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{6 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x}-\frac {\cosh ^{-1}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \int \frac {1}{x} \, dx\\ &=\frac {a^2}{12 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{6 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x}-\frac {\cosh ^{-1}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \log (x)\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.09, size = 69, normalized size = 0.73 \[ \frac {-4 a^4 x^4 \log (x)+a^2 x^2+2 a x \sqrt {a x-1} \sqrt {a x+1} \left (2 a^2 x^2+1\right ) \cosh ^{-1}(a x)-3 \cosh ^{-1}(a x)^2}{12 x^4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.61, size = 85, normalized size = 0.89 \[ -\frac {4 \, a^{4} x^{4} \log \relax (x) - a^{2} x^{2} - 2 \, {\left (2 \, a^{3} x^{3} + a x\right )} \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) + 3 \, \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{12 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.49, size = 147, normalized size = 1.55 \[ -\frac {1}{12} \, {\left (2 \, a^{3} \log \left (x^{2}\right ) - 4 \, a^{3} \log \left ({\left | -x {\left | a \right |} + \sqrt {a^{2} x^{2} - 1} \right |}\right ) - \frac {8 \, {\left (3 \, {\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )} a^{2} {\left | a \right |} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{{\left ({\left (x {\left | a \right |} - \sqrt {a^{2} x^{2} - 1}\right )}^{2} + 1\right )}^{3}} - \frac {2 \, a^{3} x^{2} + a}{x^{2}}\right )} a - \frac {\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.29, size = 109, normalized size = 1.15 \[ \frac {a^{4} \mathrm {arccosh}\left (a x \right )}{3}+\frac {a^{3} \mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{3 x}+\frac {a \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}}{6 x^{3}}+\frac {a^{2}}{12 x^{2}}-\frac {\mathrm {arccosh}\left (a x \right )^{2}}{4 x^{4}}-\frac {a^{4} \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{3} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.62, size = 72, normalized size = 0.76 \[ -\frac {1}{12} \, {\left (4 \, a^{2} \log \relax (x) - \frac {1}{x^{2}}\right )} a^{2} + \frac {1}{6} \, {\left (\frac {2 \, \sqrt {a^{2} x^{2} - 1} a^{2}}{x} + \frac {\sqrt {a^{2} x^{2} - 1}}{x^{3}}\right )} a \operatorname {arcosh}\left (a x\right ) - \frac {\operatorname {arcosh}\left (a x\right )^{2}}{4 \, x^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^5} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{5}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________