Integrand size = 23, antiderivative size = 23 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\text {Int}\left (\frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3},x\right ) \]
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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 = {} \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx \\ \end{align*}
Not integrable
Time = 4.88 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx \]
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Not integrable
Time = 0.74 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91
\[\int \frac {\left (a +b \,\operatorname {arcsec}\left (c x \right )\right ) \sqrt {e \,x^{2}+d}}{x^{3}}d x\]
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Not integrable
Time = 0.27 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 14.24 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.96 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\left (a + b \operatorname {asec}{\left (c x \right )}\right ) \sqrt {d + e x^{2}}}{x^{3}}\, dx \]
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Exception generated. \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\text {Exception raised: ValueError} \]
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Not integrable
Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\int { \frac {\sqrt {e x^{2} + d} {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}}{x^{3}} \,d x } \]
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Not integrable
Time = 1.31 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {d+e x^2} \left (a+b \sec ^{-1}(c x)\right )}{x^3} \, dx=\int \frac {\sqrt {e\,x^2+d}\,\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}{x^3} \,d x \]
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