Integrand size = 22, antiderivative size = 46 \[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \]
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Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {5892} \[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {a x-1} \sqrt {a x+1} \text {arccosh}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \]
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Rule 5892
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\frac {\sqrt {-1+a x} \sqrt {1+a x} \text {arccosh}(a x)^4}{4 a \sqrt {c-a^2 c x^2}} \]
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Time = 0.56 (sec) , antiderivative size = 55, normalized size of antiderivative = 1.20
method | result | size |
default | \(-\frac {\sqrt {-c \left (a x -1\right ) \left (a x +1\right )}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \operatorname {arccosh}\left (a x \right )^{4}}{4 c \left (a^{2} x^{2}-1\right ) a}\) | \(55\) |
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\[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {\operatorname {acosh}^{3}{\left (a x \right )}}{\sqrt {- c \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]
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\[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\int { \frac {\operatorname {arcosh}\left (a x\right )^{3}}{\sqrt {-a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {\text {arccosh}(a x)^3}{\sqrt {c-a^2 c x^2}} \, dx=\int \frac {{\mathrm {acosh}\left (a\,x\right )}^3}{\sqrt {c-a^2\,c\,x^2}} \,d x \]
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