Optimal. Leaf size=58 \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{d} \]
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Rubi [A] time = 0.0319265, antiderivative size = 58, 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, 298, 203, 206} \[ \frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{d}-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{d} \]
Antiderivative was successfully verified.
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Rule 3476
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \sqrt{b \tanh (c+d x)} \, dx &=-\frac{b \operatorname{Subst}\left (\int \frac{\sqrt{x}}{-b^2+x^2} \, dx,x,b \tanh (c+d x)\right )}{d}\\ &=-\frac{(2 b) \operatorname{Subst}\left (\int \frac{x^2}{-b^2+x^4} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{d}\\ &=\frac{b \operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{d}-\frac{b \operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{d}\\ &=-\frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{d}\\ \end{align*}
Mathematica [A] time = 0.0380801, size = 51, normalized size = 0.88 \[ \frac{\sqrt{b \tanh (c+d x)} \left (\tanh ^{-1}\left (\sqrt{\tanh (c+d x)}\right )-\tan ^{-1}\left (\sqrt{\tanh (c+d x)}\right )\right )}{d \sqrt{\tanh (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.025, size = 47, normalized size = 0.8 \begin{align*} -{\frac{1}{d}\arctan \left ({\sqrt{b\tanh \left ( dx+c \right ) }{\frac{1}{\sqrt{b}}}} \right ) \sqrt{b}}+{\frac{1}{d}{\it Artanh} \left ({\sqrt{b\tanh \left ( dx+c \right ) }{\frac{1}{\sqrt{b}}}} \right ) \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 \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 = 2.82232, size = 1635, normalized size = 28.19 \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{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 \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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