Integrand size = 24, antiderivative size = 48 \[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {c-a^2 c x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 48, 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} \[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {c-a^2 c x^2}} \]
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Rule 5892
Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\frac {2 \sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^{5/2}}{5 a \sqrt {c-a^2 c x^2}} \]
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Time = 0.55 (sec) , antiderivative size = 41, normalized size of antiderivative = 0.85
method | result | size |
default | \(\frac {2 \operatorname {arccosh}\left (a x \right )^{\frac {5}{2}} \sqrt {a x -1}\, \sqrt {a x +1}}{5 \sqrt {-c \left (a x -1\right ) \left (a x +1\right )}\, a}\) | \(41\) |
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Exception generated. \[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {\operatorname {acosh}^{\frac {3}{2}}{\left (a x \right )}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{\frac {3}{2}}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)^{3/2}}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^{3/2}}{\sqrt {c-a^2\,c\,x^2}} \,d x \]
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