Optimal. Leaf size=78 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{\tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{2}{b d \sqrt{b \tanh (c+d x)}} \]
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Rubi [A] time = 0.048383, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {3474, 3476, 329, 298, 203, 206} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{\tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{2}{b d \sqrt{b \tanh (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 3474
Rule 3476
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{(b \tanh (c+d x))^{3/2}} \, dx &=-\frac{2}{b d \sqrt{b \tanh (c+d x)}}+\frac{\int \sqrt{b \tanh (c+d x)} \, dx}{b^2}\\ &=-\frac{2}{b d \sqrt{b \tanh (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{-b^2+x^2} \, dx,x,b \tanh (c+d x)\right )}{b d}\\ &=-\frac{2}{b d \sqrt{b \tanh (c+d x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{-b^2+x^4} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{b d}\\ &=-\frac{2}{b d \sqrt{b \tanh (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{b d}-\frac{\operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \tanh (c+d x)}\right )}{b d}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \tanh (c+d x)}}{\sqrt{b}}\right )}{b^{3/2} d}-\frac{2}{b d \sqrt{b \tanh (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.0736574, size = 36, normalized size = 0.46 \[ -\frac{2 \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};\tanh ^2(c+d x)\right )}{b d \sqrt{b \tanh (c+d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 65, normalized size = 0.8 \begin{align*} -{\frac{1}{d}\arctan \left ({\sqrt{b\tanh \left ( dx+c \right ) }{\frac{1}{\sqrt{b}}}} \right ){b}^{-{\frac{3}{2}}}}+{\frac{1}{d}{\it Artanh} \left ({\sqrt{b\tanh \left ( dx+c \right ) }{\frac{1}{\sqrt{b}}}} \right ){b}^{-{\frac{3}{2}}}}-2\,{\frac{1}{bd\sqrt{b\tanh \left ( dx+c \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{1}{\left (b \tanh \left (d x + c\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.70914, size = 2574, normalized size = 33. \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}{\left (b \tanh{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28834, size = 207, normalized size = 2.65 \begin{align*} \frac{\frac{\pi + \log \left ({\left | b \right |}\right ) + 8}{\sqrt{b} d} - \frac{4 \, \arctan \left (-\frac{\sqrt{b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt{b e^{\left (4 \, d x + 4 \, c\right )} - b}}{\sqrt{b}}\right )}{\sqrt{b} d} - \frac{2 \, \log \left ({\left | -\sqrt{b} e^{\left (2 \, d x + 2 \, c\right )} + \sqrt{b e^{\left (4 \, d x + 4 \, c\right )} - b} \right |}\right )}{\sqrt{b} d} + \frac{16}{{\left (\sqrt{b} e^{\left (2 \, d x + 2 \, c\right )} - \sqrt{b e^{\left (4 \, d x + 4 \, c\right )} - b} - \sqrt{b}\right )} d}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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