Integrand size = 20, antiderivative size = 26 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2 b \arctan \left (\sinh \left (c+d \sqrt {x}\right )\right )}{d} \]
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Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 5544, 3855} \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2 b \arctan \left (\sinh \left (c+d \sqrt {x}\right )\right )}{d} \]
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Rule 14
Rule 3855
Rule 5544
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a}{\sqrt {x}}+\frac {b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}}\right ) \, dx \\ & = 2 a \sqrt {x}+b \int \frac {\text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx \\ & = 2 a \sqrt {x}+(2 b) \text {Subst}\left (\int \text {sech}(c+d x) \, dx,x,\sqrt {x}\right ) \\ & = 2 a \sqrt {x}+\frac {2 b \arctan \left (\sinh \left (c+d \sqrt {x}\right )\right )}{d} \\ \end{align*}
Time = 0.06 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=2 a \sqrt {x}+\frac {2 b \arctan \left (\sinh \left (c+d \sqrt {x}\right )\right )}{d} \]
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Time = 0.26 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.88
method | result | size |
derivativedivides | \(\frac {2 b \arctan \left (\sinh \left (c +d \sqrt {x}\right )\right )}{d}+2 a \sqrt {x}\) | \(23\) |
default | \(\frac {2 b \arctan \left (\sinh \left (c +d \sqrt {x}\right )\right )}{d}+2 a \sqrt {x}\) | \(23\) |
parts | \(\frac {2 b \arctan \left (\sinh \left (c +d \sqrt {x}\right )\right )}{d}+2 a \sqrt {x}\) | \(23\) |
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Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.27 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left (a d \sqrt {x} + 2 \, b \arctan \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right )\right )\right )}}{d} \]
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Time = 0.45 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.42 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=2 a \sqrt {x} + 2 b \left (\begin {cases} \sqrt {x} \operatorname {sech}{\left (c \right )} & \text {for}\: d = 0 \\\frac {2 \operatorname {atan}{\left (\tanh {\left (\frac {c}{2} + \frac {d \sqrt {x}}{2} \right )} \right )}}{d} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.19 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.85 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=2 \, a \sqrt {x} + \frac {2 \, b \arctan \left (\sinh \left (d \sqrt {x} + c\right )\right )}{d} \]
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Time = 0.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.12 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=\frac {2 \, {\left (d \sqrt {x} + c\right )} a}{d} + \frac {4 \, b \arctan \left (e^{\left (d \sqrt {x} + c\right )}\right )}{d} \]
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Time = 1.98 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.65 \[ \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx=2\,a\,\sqrt {x}+\frac {4\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}} \]
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