Integrand size = 23, antiderivative size = 23 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\text {Int}\left (\frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}},x\right ) \]
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Not integrable
Time = 0.07 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx \\ \end{align*}
Not integrable
Time = 5.71 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx \]
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Not integrable
Time = 0.11 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {\operatorname {arcsinh}\left (a x \right )^{n}}{x \sqrt {a^{2} x^{2}+1}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]
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Not integrable
Time = 0.68 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {\operatorname {asinh}^{n}{\left (a x \right )}}{x \sqrt {a^{2} x^{2} + 1}}\, dx \]
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Not integrable
Time = 0.36 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]
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Not integrable
Time = 0.43 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int { \frac {\operatorname {arsinh}\left (a x\right )^{n}}{\sqrt {a^{2} x^{2} + 1} x} \,d x } \]
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Not integrable
Time = 2.68 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\text {arcsinh}(a x)^n}{x \sqrt {1+a^2 x^2}} \, dx=\int \frac {{\mathrm {asinh}\left (a\,x\right )}^n}{x\,\sqrt {a^2\,x^2+1}} \,d x \]
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