Integrand size = 21, antiderivative size = 40 \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^4}{4 a \sqrt {c+a^2 c x^2}} \]
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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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {5783} \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {a^2 x^2+1} \text {arcsinh}(a x)^4}{4 a \sqrt {a^2 c x^2+c}} \]
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Rule 5783
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^4}{4 a \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.02 \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\sqrt {1+a^2 x^2} \text {arcsinh}(a x)^4}{4 a \sqrt {c \left (1+a^2 x^2\right )}} \]
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Time = 0.24 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.98
method | result | size |
default | \(\frac {\sqrt {c \left (a^{2} x^{2}+1\right )}\, \operatorname {arcsinh}\left (a x \right )^{4}}{4 \sqrt {a^{2} x^{2}+1}\, a c}\) | \(39\) |
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\[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {\operatorname {asinh}^{3}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \]
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none
Time = 0.20 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.35 \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\frac {\operatorname {arsinh}\left (a x\right )^{4}}{4 \, a \sqrt {c}} \]
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\[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{3}}{\sqrt {a^{2} c x^{2} + c}} \,d x } \]
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Timed out. \[ \int \frac {\text {arcsinh}(a x)^3}{\sqrt {c+a^2 c x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^3}{\sqrt {c\,a^2\,x^2+c}} \,d x \]
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