Integrand size = 27, antiderivative size = 27 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\frac {c \log (a+b \text {arcsinh}(c x))}{b}+\text {Int}\left (\frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))},x\right ) \]
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Not integrable
Time = 0.20 (sec) , antiderivative size = 27, 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 = {} \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {c^2}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}+\frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))}\right ) \, dx \\ & = c^2 \int \frac {1}{\sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ & = \frac {c \log (a+b \text {arcsinh}(c x))}{b}+\int \frac {1}{x^2 \sqrt {1+c^2 x^2} (a+b \text {arcsinh}(c x))} \, dx \\ \end{align*}
Not integrable
Time = 2.09 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx \]
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Not integrable
Time = 0.11 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.93
\[\int \frac {\sqrt {c^{2} x^{2}+1}}{x^{2} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.15 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.62 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {c^{2} x^{2} + 1}}{x^{2} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )}\, dx \]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
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Not integrable
Time = 0.29 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int { \frac {\sqrt {c^{2} x^{2} + 1}}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )} x^{2}} \,d x } \]
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Not integrable
Time = 2.56 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1+c^2 x^2}}{x^2 (a+b \text {arcsinh}(c x))} \, dx=\int \frac {\sqrt {c^2\,x^2+1}}{x^2\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )} \,d x \]
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