Optimal. Leaf size=64 \[ \frac{\tanh ^{-1}\left (\sqrt{\tanh (x)}\right ) \tanh ^{\frac{3}{2}}(x)}{\sqrt{a \tanh ^3(x)}}-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}-\frac{\tanh ^{\frac{3}{2}}(x) \tan ^{-1}\left (\sqrt{\tanh (x)}\right )}{\sqrt{a \tanh ^3(x)}} \]
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Rubi [A] time = 0.030095, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {3658, 3474, 3476, 329, 298, 203, 206} \[ \frac{\tanh ^{-1}\left (\sqrt{\tanh (x)}\right ) \tanh ^{\frac{3}{2}}(x)}{\sqrt{a \tanh ^3(x)}}-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}-\frac{\tanh ^{\frac{3}{2}}(x) \tan ^{-1}\left (\sqrt{\tanh (x)}\right )}{\sqrt{a \tanh ^3(x)}} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3474
Rule 3476
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{a \tanh ^3(x)}} \, dx &=\frac{\tanh ^{\frac{3}{2}}(x) \int \frac{1}{\tanh ^{\frac{3}{2}}(x)} \, dx}{\sqrt{a \tanh ^3(x)}}\\ &=-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}+\frac{\tanh ^{\frac{3}{2}}(x) \int \sqrt{\tanh (x)} \, dx}{\sqrt{a \tanh ^3(x)}}\\ &=-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}-\frac{\tanh ^{\frac{3}{2}}(x) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{-1+x^2} \, dx,x,\tanh (x)\right )}{\sqrt{a \tanh ^3(x)}}\\ &=-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}-\frac{\left (2 \tanh ^{\frac{3}{2}}(x)\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt{\tanh (x)}\right )}{\sqrt{a \tanh ^3(x)}}\\ &=-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}+\frac{\tanh ^{\frac{3}{2}}(x) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\tanh (x)}\right )}{\sqrt{a \tanh ^3(x)}}-\frac{\tanh ^{\frac{3}{2}}(x) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\tanh (x)}\right )}{\sqrt{a \tanh ^3(x)}}\\ &=-\frac{2 \tanh (x)}{\sqrt{a \tanh ^3(x)}}-\frac{\tan ^{-1}\left (\sqrt{\tanh (x)}\right ) \tanh ^{\frac{3}{2}}(x)}{\sqrt{a \tanh ^3(x)}}+\frac{\tanh ^{-1}\left (\sqrt{\tanh (x)}\right ) \tanh ^{\frac{3}{2}}(x)}{\sqrt{a \tanh ^3(x)}}\\ \end{align*}
Mathematica [C] time = 0.0169666, size = 26, normalized size = 0.41 \[ -\frac{2 \tanh (x) \, _2F_1\left (-\frac{1}{4},1;\frac{3}{4};\tanh ^2(x)\right )}{\sqrt{a \tanh ^3(x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.04, size = 65, normalized size = 1. \begin{align*} -{\tanh \left ( x \right ) \left ( 2\,{a}^{5/2}-{\it Artanh} \left ({\sqrt{a\tanh \left ( x \right ) }{\frac{1}{\sqrt{a}}}} \right ){a}^{2}\sqrt{a\tanh \left ( x \right ) }+\arctan \left ({\sqrt{a\tanh \left ( x \right ) }{\frac{1}{\sqrt{a}}}} \right ){a}^{2}\sqrt{a\tanh \left ( x \right ) } \right ){\frac{1}{\sqrt{a \left ( \tanh \left ( x \right ) \right ) ^{3}}}}{a}^{-{\frac{5}{2}}}} \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{a \tanh \left (x\right )^{3}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.45095, size = 1763, normalized size = 27.55 \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{a \tanh ^{3}{\left (x \right )}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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