Integrand size = 21, antiderivative size = 28 \[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {-1+a x} \log (\text {arccosh}(a x))}{a \sqrt {1-a x}} \]
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Time = 0.03 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5890} \[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {a x-1} \log (\text {arccosh}(a x))}{a \sqrt {1-a x}} \]
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Rule 5890
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+a x} \log (\text {arccosh}(a x))}{a \sqrt {1-a x}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.68 \[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\frac {\sqrt {\frac {-1+a x}{1+a x}} (1+a x) \log (\text {arccosh}(a x))}{a \sqrt {-((-1+a x) (1+a x))}} \]
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Time = 0.66 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.71
method | result | size |
default | \(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \ln \left (\operatorname {arccosh}\left (a x \right )\right )}{a \left (a^{2} x^{2}-1\right )}\) | \(48\) |
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Leaf count of result is larger than twice the leaf count of optimal. 55 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.96 \[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=-\frac {\sqrt {a^{2} x^{2} - 1} \sqrt {-a^{2} x^{2} + 1} \log \left (\log \left (a x + \sqrt {a^{2} x^{2} - 1}\right )\right )}{a^{3} x^{2} - a} \]
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\[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {1}{\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )} \operatorname {acosh}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {1}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
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\[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int { \frac {1}{\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {1-a^2 x^2} \text {arccosh}(a x)} \, dx=\int \frac {1}{\mathrm {acosh}\left (a\,x\right )\,\sqrt {1-a^2\,x^2}} \,d x \]
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