Optimal. Leaf size=42 \[ -\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5885, 3382}
\begin {gather*} \frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}-\frac {x \sqrt {a x-1} \sqrt {a x+1}}{a \cosh ^{-1}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 5885
Rubi steps
\begin {align*} \int \frac {x}{\cosh ^{-1}(a x)^2} \, dx &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Subst}\left (\int \frac {\cosh (2 x)}{x} \, dx,x,\cosh ^{-1}(a x)\right )}{a^2}\\ &=-\frac {x \sqrt {-1+a x} \sqrt {1+a x}}{a \cosh ^{-1}(a x)}+\frac {\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2}\\ \end {align*}
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Mathematica [A]
time = 0.18, size = 44, normalized size = 1.05 \begin {gather*} \frac {-\frac {a x \sqrt {\frac {-1+a x}{1+a x}} (1+a x)}{\cosh ^{-1}(a x)}+\text {Chi}\left (2 \cosh ^{-1}(a x)\right )}{a^2} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 2.45, size = 28, normalized size = 0.67
method | result | size |
derivativedivides | \(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2 \,\mathrm {arccosh}\left (a x \right )}+\hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{2}}\) | \(28\) |
default | \(\frac {-\frac {\sinh \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{2 \,\mathrm {arccosh}\left (a x \right )}+\hyperbolicCosineIntegral \left (2 \,\mathrm {arccosh}\left (a x \right )\right )}{a^{2}}\) | \(28\) |
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 {x}{\operatorname {acosh}^{2}{\left (a x \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 = -0.02 \begin {gather*} \int \frac {x}{{\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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