Optimal. Leaf size=39 \[ 2 x-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{a}+x \cosh ^{-1}(a x)^2 \]
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Rubi [A]
time = 0.08, antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5879, 5915, 8}
\begin {gather*} x \cosh ^{-1}(a x)^2-\frac {2 \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)}{a}+2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5879
Rule 5915
Rubi steps
\begin {align*} \int \cosh ^{-1}(a x)^2 \, dx &=x \cosh ^{-1}(a x)^2-(2 a) \int \frac {x \cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{a}+x \cosh ^{-1}(a x)^2+2 \int 1 \, dx\\ &=2 x-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{a}+x \cosh ^{-1}(a x)^2\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 39, normalized size = 1.00 \begin {gather*} 2 x-\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)}{a}+x \cosh ^{-1}(a x)^2 \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.77, size = 39, normalized size = 1.00
method | result | size |
derivativedivides | \(\frac {\mathrm {arccosh}\left (a x \right )^{2} a x -2 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+2 a x}{a}\) | \(39\) |
default | \(\frac {\mathrm {arccosh}\left (a x \right )^{2} a x -2 \,\mathrm {arccosh}\left (a x \right ) \sqrt {a x -1}\, \sqrt {a x +1}+2 a x}{a}\) | \(39\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 32, normalized size = 0.82 \begin {gather*} x \operatorname {arcosh}\left (a x\right )^{2} + 2 \, x - \frac {2 \, \sqrt {a^{2} x^{2} - 1} \operatorname {arcosh}\left (a x\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 59, normalized size = 1.51 \begin {gather*} \frac {a x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 2 \, a x - 2 \, \sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.08, size = 39, normalized size = 1.00 \begin {gather*} \begin {cases} x \operatorname {acosh}^{2}{\left (a x \right )} + 2 x - \frac {2 \sqrt {a^{2} x^{2} - 1} \operatorname {acosh}{\left (a x \right )}}{a} & \text {for}\: a \neq 0 \\- \frac {\pi ^{2} x}{4} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.40, size = 62, normalized size = 1.59 \begin {gather*} x \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )^{2} + 2 \, a {\left (\frac {x}{a} - \frac {\sqrt {a^{2} x^{2} - 1} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )}{a^{2}}\right )} \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 = -0.03 \begin {gather*} \int {\mathrm {acosh}\left (a\,x\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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