Optimal. Leaf size=51 \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d} \]
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Rubi [A] time = 0.0535916, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {3885, 50, 63, 207} \[ \frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 50
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \sqrt{a+b \text{sech}(c+d x)} \tanh (c+d x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{a+x}}{x} \, dx,x,b \text{sech}(c+d x)\right )}{d}\\ &=-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}-\frac{a \operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+x}} \, dx,x,b \text{sech}(c+d x)\right )}{d}\\ &=-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}-\frac{(2 a) \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{d}\\ &=\frac{2 \sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{d}-\frac{2 \sqrt{a+b \text{sech}(c+d x)}}{d}\\ \end{align*}
Mathematica [A] time = 0.141392, size = 90, normalized size = 1.76 \[ -\frac{2 \sqrt{a+b \text{sech}(c+d x)} \left (\sqrt{a \cosh (c+d x)+b}-\sqrt{a \cosh (c+d x)} \tanh ^{-1}\left (\frac{\sqrt{a \cosh (c+d x)+b}}{\sqrt{a \cosh (c+d x)}}\right )\right )}{d \sqrt{a \cosh (c+d x)+b}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 43, normalized size = 0.8 \begin{align*} -{\frac{1}{d} \left ( 2\,\sqrt{a+b{\rm sech} \left (dx+c\right )}-2\,\sqrt{a}{\it Artanh} \left ({\frac{\sqrt{a+b{\rm sech} \left (dx+c\right )}}{\sqrt{a}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 8.72209, size = 1613, normalized size = 31.63 \begin{align*} \text{result too large to display} \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 \sqrt{a + b \operatorname{sech}{\left (c + d x \right )}} \tanh{\left (c + d x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \operatorname{sech}\left (d x + c\right ) + a} \tanh \left (d x + c\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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