Optimal. Leaf size=31 \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
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Rubi [A] time = 0.0469297, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {3885, 63, 207} \[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d} \]
Antiderivative was successfully verified.
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Rule 3885
Rule 63
Rule 207
Rubi steps
\begin{align*} \int \frac{\tanh (c+d x)}{\sqrt{a+b \text{sech}(c+d x)}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+x}} \, dx,x,b \text{sech}(c+d x)\right )}{d}\\ &=-\frac{2 \operatorname{Subst}\left (\int \frac{1}{-a+x^2} \, dx,x,\sqrt{a+b \text{sech}(c+d x)}\right )}{d}\\ &=\frac{2 \tanh ^{-1}\left (\frac{\sqrt{a+b \text{sech}(c+d x)}}{\sqrt{a}}\right )}{\sqrt{a} d}\\ \end{align*}
Mathematica [B] time = 0.129582, size = 73, normalized size = 2.35 \[ \frac{2 \sqrt{a \cosh (c+d x)+b} \tanh ^{-1}\left (\frac{\sqrt{a \cosh (c+d x)+b}}{\sqrt{a \cosh (c+d x)}}\right )}{d \sqrt{a \cosh (c+d x)} \sqrt{a+b \text{sech}(c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.034, size = 26, normalized size = 0.8 \begin{align*} 2\,{\frac{1}{d\sqrt{a}}{\it Artanh} \left ({\frac{\sqrt{a+b{\rm sech} \left (dx+c\right )}}{\sqrt{a}}} \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 \frac{\tanh \left (d x + c\right )}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 9.52898, size = 1492, normalized size = 48.13 \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 \frac{\tanh{\left (c + d x \right )}}{\sqrt{a + b \operatorname{sech}{\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 \frac{\tanh \left (d x + c\right )}{\sqrt{b \operatorname{sech}\left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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