Optimal. Leaf size=26 \[ 2 a \sqrt {x}+\frac {2 b \tan ^{-1}\left (\sinh \left (c+d \sqrt {x}\right )\right )}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {14, 5436, 3770} \[ 2 a \sqrt {x}+\frac {2 b \tan ^{-1}\left (\sinh \left (c+d \sqrt {x}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3770
Rule 5436
Rubi steps
\begin {align*} \int \frac {a+b \text {sech}\left (c+d \sqrt {x}\right )}{\sqrt {x}} \, dx &=\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) \operatorname {Subst}\left (\int \text {sech}(c+d x) \, dx,x,\sqrt {x}\right )\\ &=2 a \sqrt {x}+\frac {2 b \tan ^{-1}\left (\sinh \left (c+d \sqrt {x}\right )\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 30, normalized size = 1.15 \[ \frac {2 \left (a \left (c+d \sqrt {x}\right )+b \tan ^{-1}\left (\sinh \left (c+d \sqrt {x}\right )\right )\right )}{d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.52, size = 33, normalized size = 1.27 \[ \frac {2 \, {\left (a d \sqrt {x} + 2 \, b \arctan \left (\cosh \left (d \sqrt {x} + c\right ) + \sinh \left (d \sqrt {x} + c\right )\right )\right )}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.14, size = 29, normalized size = 1.12 \[ \frac {2 \, {\left (d \sqrt {x} + c\right )} a}{d} + \frac {4 \, b \arctan \left (e^{\left (d \sqrt {x} + c\right )}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.13, size = 23, normalized size = 0.88 \[ \frac {2 b \arctan \left (\sinh \left (c +d \sqrt {x}\right )\right )}{d}+2 a \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.34, size = 22, normalized size = 0.85 \[ 2 \, a \sqrt {x} + \frac {2 \, b \arctan \left (\sinh \left (d \sqrt {x} + c\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.34, size = 43, normalized size = 1.65 \[ 2\,a\,\sqrt {x}+\frac {4\,\mathrm {atan}\left (\frac {b\,{\mathrm {e}}^{d\,\sqrt {x}}\,{\mathrm {e}}^c\,\sqrt {d^2}}{d\,\sqrt {b^2}}\right )\,\sqrt {b^2}}{\sqrt {d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {a + b \operatorname {sech}{\left (c + d \sqrt {x} \right )}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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