Optimal. Leaf size=95 \[ \frac {a^2}{12 x^2}+\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{6 x^3}+\frac {a^3 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{3 x}-\frac {\cosh ^{-1}(a x)^2}{4 x^4}-\frac {1}{3} a^4 \log (x) \]
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Rubi [A]
time = 0.23, 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 = {5883, 5933,
5918, 29, 30} \begin {gather*} -\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} \end {gather*}
Antiderivative was successfully verified.
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Rule 29
Rule 30
Rule 5883
Rule 5918
Rule 5933
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*}
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Mathematica [A]
time = 0.06, size = 69, normalized size = 0.73 \begin {gather*} \frac {a^2 x^2+2 a x \sqrt {-1+a x} \sqrt {1+a x} \left (1+2 a^2 x^2\right ) \cosh ^{-1}(a x)-3 \cosh ^{-1}(a x)^2-4 a^4 x^4 \log (x)}{12 x^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.15, size = 123, normalized size = 1.29
method | result | size |
derivativedivides | \(a^{4} \left (\frac {2 \,\mathrm {arccosh}\left (a x \right )}{3}-\frac {-4 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+4 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4}-2 a x \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+3 \mathrm {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{12 a^{4} x^{4}}-\frac {\ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{3}\right )\) | \(123\) |
default | \(a^{4} \left (\frac {2 \,\mathrm {arccosh}\left (a x \right )}{3}-\frac {-4 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x +1}\, \sqrt {a x -1}\, a^{3} x^{3}+4 \,\mathrm {arccosh}\left (a x \right ) a^{4} x^{4}-2 a x \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+3 \mathrm {arccosh}\left (a x \right )^{2}-a^{2} x^{2}}{12 a^{4} x^{4}}-\frac {\ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right )}{3}\right )\) | \(123\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.47, size = 72, normalized size = 0.76 \begin {gather*} -\frac {1}{12} \, {\left (4 \, a^{2} \log \left (x\right ) - \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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 85, normalized size = 0.89 \begin {gather*} -\frac {4 \, a^{4} x^{4} \log \left (x\right ) - 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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 147, normalized size = 1.55 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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