Optimal. Leaf size=47 \[ 2 a^2 \sqrt{x}+\frac{4 a b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )}{d}+\frac{2 b^2 \tanh \left (c+d \sqrt{x}\right )}{d} \]
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Rubi [A] time = 0.0567438, antiderivative size = 47, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {5436, 3773, 3770, 3767, 8} \[ 2 a^2 \sqrt{x}+\frac{4 a b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )}{d}+\frac{2 b^2 \tanh \left (c+d \sqrt{x}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 5436
Rule 3773
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int \frac{\left (a+b \text{sech}\left (c+d \sqrt{x}\right )\right )^2}{\sqrt{x}} \, dx &=2 \operatorname{Subst}\left (\int (a+b \text{sech}(c+d x))^2 \, dx,x,\sqrt{x}\right )\\ &=2 a^2 \sqrt{x}+(4 a b) \operatorname{Subst}\left (\int \text{sech}(c+d x) \, dx,x,\sqrt{x}\right )+\left (2 b^2\right ) \operatorname{Subst}\left (\int \text{sech}^2(c+d x) \, dx,x,\sqrt{x}\right )\\ &=2 a^2 \sqrt{x}+\frac{4 a b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )}{d}+\frac{\left (2 i b^2\right ) \operatorname{Subst}\left (\int 1 \, dx,x,-i \tanh \left (c+d \sqrt{x}\right )\right )}{d}\\ &=2 a^2 \sqrt{x}+\frac{4 a b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )}{d}+\frac{2 b^2 \tanh \left (c+d \sqrt{x}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.113166, size = 48, normalized size = 1.02 \[ \frac{2 \left (a \left (a \left (c+d \sqrt{x}\right )+2 b \tan ^{-1}\left (\sinh \left (c+d \sqrt{x}\right )\right )\right )+b^2 \tanh \left (c+d \sqrt{x}\right )\right )}{d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 51, normalized size = 1.1 \begin{align*} 2\,{a}^{2}\sqrt{x}+2\,{\frac{{b}^{2}\tanh \left ( c+d\sqrt{x} \right ) }{d}}+8\,{\frac{ba\arctan \left ({{\rm e}^{c+d\sqrt{x}}} \right ) }{d}}+2\,{\frac{{a}^{2}c}{d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.13577, size = 65, normalized size = 1.38 \begin{align*} 2 \, a^{2} \sqrt{x} + \frac{4 \, a b \arctan \left (\sinh \left (d \sqrt{x} + c\right )\right )}{d} + \frac{4 \, b^{2}}{d{\left (e^{\left (-2 \, d \sqrt{x} - 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.20083, size = 576, normalized size = 12.26 \begin{align*} \frac{2 \,{\left (a^{2} d \sqrt{x} \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, a^{2} d \sqrt{x} \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + a^{2} d \sqrt{x} \sinh \left (d \sqrt{x} + c\right )^{2} + a^{2} d \sqrt{x} - 2 \, b^{2} + 4 \,{\left (a b \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, a b \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + a b \sinh \left (d \sqrt{x} + c\right )^{2} + a b\right )} \arctan \left (\cosh \left (d \sqrt{x} + c\right ) + \sinh \left (d \sqrt{x} + c\right )\right )\right )}}{d \cosh \left (d \sqrt{x} + c\right )^{2} + 2 \, d \cosh \left (d \sqrt{x} + c\right ) \sinh \left (d \sqrt{x} + c\right ) + d \sinh \left (d \sqrt{x} + c\right )^{2} + d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (a + b \operatorname{sech}{\left (c + d \sqrt{x} \right )}\right )^{2}}{\sqrt{x}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19042, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (d \sqrt{x} + c\right )} a^{2}}{d} + \frac{8 \, a b \arctan \left (e^{\left (d \sqrt{x} + c\right )}\right )}{d} - \frac{4 \, b^{2}}{d{\left (e^{\left (2 \, d \sqrt{x} + 2 \, c\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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