Integrand size = 10, antiderivative size = 137 \[ \int x^m \text {arcsinh}(a x)^2 \, dx=\frac {x^{1+m} \text {arcsinh}(a x)^2}{1+m}-\frac {2 a x^{2+m} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2}+\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )}{6+11 m+6 m^2+m^3} \]
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Time = 0.07 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5776, 5817} \[ \int x^m \text {arcsinh}(a x)^2 \, dx=\frac {2 a^2 x^{m+3} \, _3F_2\left (1,\frac {m}{2}+\frac {3}{2},\frac {m}{2}+\frac {3}{2};\frac {m}{2}+2,\frac {m}{2}+\frac {5}{2};-a^2 x^2\right )}{m^3+6 m^2+11 m+6}-\frac {2 a x^{m+2} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {m+2}{2},\frac {m+4}{2},-a^2 x^2\right )}{m^2+3 m+2}+\frac {x^{m+1} \text {arcsinh}(a x)^2}{m+1} \]
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Rule 5776
Rule 5817
Rubi steps \begin{align*} \text {integral}& = \frac {x^{1+m} \text {arcsinh}(a x)^2}{1+m}-\frac {(2 a) \int \frac {x^{1+m} \text {arcsinh}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{1+m} \\ & = \frac {x^{1+m} \text {arcsinh}(a x)^2}{1+m}-\frac {2 a x^{2+m} \text {arcsinh}(a x) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )}{2+3 m+m^2}+\frac {2 a^2 x^{3+m} \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )}{6+11 m+6 m^2+m^3} \\ \end{align*}
Time = 0.03 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.90 \[ \int x^m \text {arcsinh}(a x)^2 \, dx=\frac {x^{1+m} \left ((3+m) \text {arcsinh}(a x) \left ((2+m) \text {arcsinh}(a x)-2 a x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {2+m}{2},\frac {4+m}{2},-a^2 x^2\right )\right )+2 a^2 x^2 \, _3F_2\left (1,\frac {3}{2}+\frac {m}{2},\frac {3}{2}+\frac {m}{2};2+\frac {m}{2},\frac {5}{2}+\frac {m}{2};-a^2 x^2\right )\right )}{(1+m) (2+m) (3+m)} \]
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\[\int x^{m} \operatorname {arcsinh}\left (a x \right )^{2}d x\]
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\[ \int x^m \text {arcsinh}(a x)^2 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{2} \,d x } \]
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\[ \int x^m \text {arcsinh}(a x)^2 \, dx=\int x^{m} \operatorname {asinh}^{2}{\left (a x \right )}\, dx \]
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\[ \int x^m \text {arcsinh}(a x)^2 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{2} \,d x } \]
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\[ \int x^m \text {arcsinh}(a x)^2 \, dx=\int { x^{m} \operatorname {arsinh}\left (a x\right )^{2} \,d x } \]
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Timed out. \[ \int x^m \text {arcsinh}(a x)^2 \, dx=\int x^m\,{\mathrm {asinh}\left (a\,x\right )}^2 \,d x \]
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