Optimal. Leaf size=86 \[ -\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}-\frac{2}{3} a \sqrt{a \tanh ^3(x)}+\frac{a \tanh ^{-1}\left (\sqrt{\tanh (x)}\right ) \sqrt{a \tanh ^3(x)}}{\tanh ^{\frac{3}{2}}(x)}-\frac{a \sqrt{a \tanh ^3(x)} \tan ^{-1}\left (\sqrt{\tanh (x)}\right )}{\tanh ^{\frac{3}{2}}(x)} \]
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Rubi [A] time = 0.0366665, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.7, Rules used = {3658, 3473, 3476, 329, 298, 203, 206} \[ -\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}-\frac{2}{3} a \sqrt{a \tanh ^3(x)}+\frac{a \tanh ^{-1}\left (\sqrt{\tanh (x)}\right ) \sqrt{a \tanh ^3(x)}}{\tanh ^{\frac{3}{2}}(x)}-\frac{a \sqrt{a \tanh ^3(x)} \tan ^{-1}\left (\sqrt{\tanh (x)}\right )}{\tanh ^{\frac{3}{2}}(x)} \]
Antiderivative was successfully verified.
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Rule 3658
Rule 3473
Rule 3476
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \left (a \tanh ^3(x)\right )^{3/2} \, dx &=\frac{\left (a \sqrt{a \tanh ^3(x)}\right ) \int \tanh ^{\frac{9}{2}}(x) \, dx}{\tanh ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}+\frac{\left (a \sqrt{a \tanh ^3(x)}\right ) \int \tanh ^{\frac{5}{2}}(x) \, dx}{\tanh ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} a \sqrt{a \tanh ^3(x)}-\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}+\frac{\left (a \sqrt{a \tanh ^3(x)}\right ) \int \sqrt{\tanh (x)} \, dx}{\tanh ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} a \sqrt{a \tanh ^3(x)}-\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}-\frac{\left (a \sqrt{a \tanh ^3(x)}\right ) \operatorname{Subst}\left (\int \frac{\sqrt{x}}{-1+x^2} \, dx,x,\tanh (x)\right )}{\tanh ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} a \sqrt{a \tanh ^3(x)}-\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}-\frac{\left (2 a \sqrt{a \tanh ^3(x)}\right ) \operatorname{Subst}\left (\int \frac{x^2}{-1+x^4} \, dx,x,\sqrt{\tanh (x)}\right )}{\tanh ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} a \sqrt{a \tanh ^3(x)}-\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}+\frac{\left (a \sqrt{a \tanh ^3(x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1-x^2} \, dx,x,\sqrt{\tanh (x)}\right )}{\tanh ^{\frac{3}{2}}(x)}-\frac{\left (a \sqrt{a \tanh ^3(x)}\right ) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\tanh (x)}\right )}{\tanh ^{\frac{3}{2}}(x)}\\ &=-\frac{2}{3} a \sqrt{a \tanh ^3(x)}-\frac{a \tan ^{-1}\left (\sqrt{\tanh (x)}\right ) \sqrt{a \tanh ^3(x)}}{\tanh ^{\frac{3}{2}}(x)}+\frac{a \tanh ^{-1}\left (\sqrt{\tanh (x)}\right ) \sqrt{a \tanh ^3(x)}}{\tanh ^{\frac{3}{2}}(x)}-\frac{2}{7} a \tanh ^2(x) \sqrt{a \tanh ^3(x)}\\ \end{align*}
Mathematica [A] time = 0.0590554, size = 55, normalized size = 0.64 \[ -\frac{\left (a \tanh ^3(x)\right )^{3/2} \left (6 \tanh ^{\frac{7}{2}}(x)+14 \tanh ^{\frac{3}{2}}(x)-21 \tanh ^{-1}\left (\sqrt{\tanh (x)}\right )+21 \tan ^{-1}\left (\sqrt{\tanh (x)}\right )\right )}{21 \tanh ^{\frac{9}{2}}(x)} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 76, normalized size = 0.9 \begin{align*}{\frac{1}{21\, \left ( \tanh \left ( x \right ) \right ) ^{3}{a}^{2}} \left ( a \left ( \tanh \left ( x \right ) \right ) ^{3} \right ) ^{{\frac{3}{2}}} \left ( 21\,{a}^{7/2}{\it Artanh} \left ({\frac{\sqrt{a\tanh \left ( x \right ) }}{\sqrt{a}}} \right ) -21\,{a}^{7/2}\arctan \left ({\frac{\sqrt{a\tanh \left ( x \right ) }}{\sqrt{a}}} \right ) -6\, \left ( a\tanh \left ( x \right ) \right ) ^{7/2}-14\,{a}^{2} \left ( a\tanh \left ( x \right ) \right ) ^{3/2} \right ) \left ( a\tanh \left ( x \right ) \right ) ^{-{\frac{3}{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 \left (a \tanh \left (x\right )^{3}\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 = 2.7242, size = 4099, normalized size = 47.66 \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 \left (a \tanh ^{3}{\left (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 = 1.30995, size = 462, normalized size = 5.37 \begin{align*} -\frac{1}{42} \,{\left (42 \, \sqrt{a} \arctan \left (-\frac{\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}}{\sqrt{a}}\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 21 \, \sqrt{a} \log \left ({\left | -\sqrt{a} e^{\left (2 \, x\right )} + \sqrt{a e^{\left (4 \, x\right )} - a} \right |}\right ) \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + \frac{16 \,{\left (21 \,{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}\right )}^{6} a \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 42 \,{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}\right )}^{5} a^{\frac{3}{2}} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 119 \,{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}\right )}^{4} a^{2} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 56 \,{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}\right )}^{3} a^{\frac{5}{2}} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 63 \,{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}\right )}^{2} a^{3} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 14 \,{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a}\right )} a^{\frac{7}{2}} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right ) + 5 \, a^{4} \mathrm{sgn}\left (e^{\left (4 \, x\right )} - 1\right )\right )}}{{\left (\sqrt{a} e^{\left (2 \, x\right )} - \sqrt{a e^{\left (4 \, x\right )} - a} + \sqrt{a}\right )}^{7}}\right )} a \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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