Optimal. Leaf size=32 \[ \frac {\sqrt {-1+a x} \cosh ^{-1}(a x)^2}{2 a \sqrt {1-a x}} \]
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Rubi [A]
time = 0.02, antiderivative size = 32, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {5892}
\begin {gather*} \frac {\sqrt {a x-1} \cosh ^{-1}(a x)^2}{2 a \sqrt {1-a x}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5892
Rubi steps
\begin {align*} \int \frac {\cosh ^{-1}(a x)}{\sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=\frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a \sqrt {1-a^2 x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 45, normalized size = 1.41 \begin {gather*} \frac {\sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{2 a \sqrt {1-a^2 x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.26, size = 51, normalized size = 1.59
method | result | size |
default | \(-\frac {\sqrt {-\left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2}}{2 a \left (a^{2} x^{2}-1\right )}\) | \(51\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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.
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Fricas [F]
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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {acosh}{\left (a x \right )}}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\mathrm {acosh}\left (a\,x\right )}{\sqrt {1-a^2\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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