Integrand size = 23, antiderivative size = 40 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}} \]
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Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783} \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {a^2 x^2+1}}{a \sqrt {\text {arcsinh}(a x)} \sqrt {a^2 c x^2+c}} \]
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Rule 5783
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c+a^2 c x^2} \sqrt {\text {arcsinh}(a x)}} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{a \sqrt {c \left (1+a^2 x^2\right )} \sqrt {\text {arcsinh}(a x)}} \]
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Time = 0.26 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.90
method | result | size |
default | \(-\frac {2 \sqrt {a^{2} x^{2}+1}}{\sqrt {\operatorname {arcsinh}\left (a x \right )}\, a \sqrt {c \left (a^{2} x^{2}+1\right )}}\) | \(36\) |
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Time = 0.24 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.42 \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \, \sqrt {a^{2} c x^{2} + c} \sqrt {a^{2} x^{2} + 1}}{{\left (a^{3} c x^{2} + a c\right )} \sqrt {\log \left (a x + \sqrt {a^{2} x^{2} + 1}\right )}} \]
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\[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{\sqrt {c \left (a^{2} x^{2} + 1\right )} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {1}{\sqrt {a^{2} c x^{2} + c} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {c+a^2 c x^2} \text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}\,\sqrt {c\,a^2\,x^2+c}} \,d x \]
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