Optimal. Leaf size=57 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d} \]
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Rubi [A] time = 0.0315493, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.417, Rules used = {3476, 329, 212, 206, 203} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{b \tanh (c+d x)}} \, dx &=-\frac{b \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (-b^2+x^2\right )} \, dx,x,b \tanh (c+d x)\right )}{d}\\ &=-\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{-b^2+x^4} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{d}+\frac{\operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{d}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{\sqrt{b} d}\\ \end{align*}
Mathematica [A] time = 0.0336064, size = 49, normalized size = 0.86 \[ \frac{\sqrt{\tanh (c+d x)} \left (\tanh ^{-1}\left (\sqrt{\tanh (c+d x)}\right )+\tan ^{-1}\left (\sqrt{\tanh (c+d x)}\right )\right )}{d \sqrt{b \tanh (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 46, normalized size = 0.8 \begin{align*}{\frac{1}{d}\arctan \left ({\sqrt{b\tanh \left ( dx+c \right ) }{\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{b}}}}+{\frac{1}{d}{\it Artanh} \left ({\sqrt{b\tanh \left ( dx+c \right ) }{\frac{1}{\sqrt{b}}}} \right ){\frac{1}{\sqrt{b}}}} \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{1}{\sqrt{b \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 = 2.68511, size = 1647, normalized size = 28.89 \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{1}{\sqrt{b \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 \frac{1}{\sqrt{b \tanh \left (d x + c\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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