Integrand size = 22, antiderivative size = 22 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\text {Int}\left (\frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}},x\right ) \]
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Not integrable
Time = 0.02 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]
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Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \\ \end{align*}
Not integrable
Time = 36.17 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx \]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 18, normalized size of antiderivative = 0.82
\[\int \frac {\left (a +b \,\operatorname {sech}\left (c +d \sqrt {x}\right )\right )^{2}}{x^{\frac {3}{2}}}d x\]
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Not integrable
Time = 0.26 (sec) , antiderivative size = 46, normalized size of antiderivative = 2.09 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 2.05 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}\right )^{2}}{x^{\frac {3}{2}}}\, dx \]
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Not integrable
Time = 0.55 (sec) , antiderivative size = 106, normalized size of antiderivative = 4.82 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int { \frac {{\left (b \operatorname {sech}\left (d \sqrt {x} + c\right ) + a\right )}^{2}}{x^{\frac {3}{2}}} \,d x } \]
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Not integrable
Time = 2.38 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+b \text {sech}\left (c+d \sqrt {x}\right )\right )^2}{x^{3/2}} \, dx=\int \frac {{\left (a+\frac {b}{\mathrm {cosh}\left (c+d\,\sqrt {x}\right )}\right )}^2}{x^{3/2}} \,d x \]
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