Integrand size = 10, antiderivative size = 10 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\frac {x^{1+m} \text {arcsinh}(a x)^3}{1+m}-\frac {3 a \text {Int}\left (\frac {x^{1+m} \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}},x\right )}{1+m} \]
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Not integrable
Time = 0.08 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^m \text {arcsinh}(a x)^3 \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \text {arcsinh}(a x)^3}{1+m}-\frac {(3 a) \int \frac {x^{1+m} \text {arcsinh}(a x)^2}{\sqrt {1+a^2 x^2}} \, dx}{1+m} \\ \end{align*}
Not integrable
Time = 0.38 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^m \text {arcsinh}(a x)^3 \, dx \]
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Not integrable
Time = 0.34 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00
\[\int x^{m} \operatorname {arcsinh}\left (a x \right )^{3}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
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Not integrable
Time = 3.05 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^{m} \operatorname {asinh}^{3}{\left (a x \right )}\, dx \]
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Not integrable
Time = 0.48 (sec) , antiderivative size = 134, normalized size of antiderivative = 13.40 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{3} \,d x } \]
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Not integrable
Time = 2.88 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int x^m \text {arcsinh}(a x)^3 \, dx=\int x^m\,{\mathrm {asinh}\left (a\,x\right )}^3 \,d x \]
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