Integrand size = 24, antiderivative size = 50 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Time = 0.03 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {5892} \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {\frac {x}{a}-1} \sqrt {\frac {x}{a}+1} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Rule 5892
Rubi steps \begin{align*} \text {integral}& = \frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \\ \end{align*}
Time = 0.04 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\frac {2 a \sqrt {-1+\frac {x}{a}} \sqrt {1+\frac {x}{a}} \text {arccosh}\left (\frac {x}{a}\right )^{5/2}}{5 \sqrt {a^2-x^2}} \]
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Time = 0.56 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {2 \operatorname {arccosh}\left (\frac {x}{a}\right )^{\frac {5}{2}} \sqrt {-\frac {a -x}{a}}\, \sqrt {\frac {a +x}{a}}\, a}{5 \sqrt {\left (a -x \right ) \left (a +x \right )}}\) | \(44\) |
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Exception generated. \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {\operatorname {acosh}^{\frac {3}{2}}{\left (\frac {x}{a} \right )}}{\sqrt {- \left (- a + x\right ) \left (a + x\right )}}\, dx \]
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\[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\sqrt {a^{2} - x^{2}}} \,d x } \]
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\[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (\frac {x}{a}\right )^{\frac {3}{2}}}{\sqrt {a^{2} - x^{2}}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}\left (\frac {x}{a}\right )^{3/2}}{\sqrt {a^2-x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (\frac {x}{a}\right )}^{3/2}}{\sqrt {a^2-x^2}} \,d x \]
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