Optimal. Leaf size=50 \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right )-\frac{1}{2} \tanh (x) \sqrt{\tanh ^2(x)+1}-\frac{5}{2} \sinh ^{-1}(\tanh (x)) \]
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Rubi [A] time = 0.0388703, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6, Rules used = {3661, 416, 523, 215, 377, 206} \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \tanh (x)}{\sqrt{\tanh ^2(x)+1}}\right )-\frac{1}{2} \tanh (x) \sqrt{\tanh ^2(x)+1}-\frac{5}{2} \sinh ^{-1}(\tanh (x)) \]
Antiderivative was successfully verified.
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Rule 3661
Rule 416
Rule 523
Rule 215
Rule 377
Rule 206
Rubi steps
\begin{align*} \int \left (1+\tanh ^2(x)\right )^{3/2} \, dx &=\operatorname{Subst}\left (\int \frac{\left (1+x^2\right )^{3/2}}{1-x^2} \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{2} \tanh (x) \sqrt{1+\tanh ^2(x)}-\frac{1}{2} \operatorname{Subst}\left (\int \frac{-3-5 x^2}{\left (1-x^2\right ) \sqrt{1+x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac{1}{2} \tanh (x) \sqrt{1+\tanh ^2(x)}-\frac{5}{2} \operatorname{Subst}\left (\int \frac{1}{\sqrt{1+x^2}} \, dx,x,\tanh (x)\right )+4 \operatorname{Subst}\left (\int \frac{1}{\left (1-x^2\right ) \sqrt{1+x^2}} \, dx,x,\tanh (x)\right )\\ &=-\frac{5}{2} \sinh ^{-1}(\tanh (x))-\frac{1}{2} \tanh (x) \sqrt{1+\tanh ^2(x)}+4 \operatorname{Subst}\left (\int \frac{1}{1-2 x^2} \, dx,x,\frac{\tanh (x)}{\sqrt{1+\tanh ^2(x)}}\right )\\ &=-\frac{5}{2} \sinh ^{-1}(\tanh (x))+2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{2} \tanh (x)}{\sqrt{1+\tanh ^2(x)}}\right )-\frac{1}{2} \tanh (x) \sqrt{1+\tanh ^2(x)}\\ \end{align*}
Mathematica [A] time = 0.133756, size = 74, normalized size = 1.48 \[ -\frac{\left (\tanh ^2(x)+1\right )^{3/2} \left (-4 \sqrt{2} \sinh ^{-1}\left (\sqrt{2} \sinh (x)\right ) \cosh ^3(x)+\sinh (x) \sqrt{\cosh (2 x)} \cosh (x)+5 \cosh ^3(x) \tanh ^{-1}\left (\frac{\sinh (x)}{\sqrt{\cosh (2 x)}}\right )\right )}{2 \cosh ^{\frac{3}{2}}(2 x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.025, size = 158, normalized size = 3.2 \begin{align*}{\frac{1}{6} \left ( \left ( 1+\tanh \left ( x \right ) \right ) ^{2}-2\,\tanh \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{\tanh \left ( x \right ) }{4}\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}-2\,\tanh \left ( x \right ) }}-{\frac{5\,{\it Arcsinh} \left ( \tanh \left ( x \right ) \right ) }{2}}+\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}-2\,\tanh \left ( x \right ) }-\sqrt{2}{\it Artanh} \left ({\frac{ \left ( 2-2\,\tanh \left ( x \right ) \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( 1+\tanh \left ( x \right ) \right ) ^{2}-2\,\tanh \left ( x \right ) }}}} \right ) -{\frac{1}{6} \left ( \left ( \tanh \left ( x \right ) -1 \right ) ^{2}+2\,\tanh \left ( x \right ) \right ) ^{{\frac{3}{2}}}}-{\frac{\tanh \left ( x \right ) }{4}\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}+2\,\tanh \left ( x \right ) }}-\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}+2\,\tanh \left ( x \right ) }+\sqrt{2}{\it Artanh} \left ({\frac{ \left ( 2\,\tanh \left ( x \right ) +2 \right ) \sqrt{2}}{4}{\frac{1}{\sqrt{ \left ( \tanh \left ( x \right ) -1 \right ) ^{2}+2\,\tanh \left ( x \right ) }}}} \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{\left (\tanh \left (x\right )^{2} + 1\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 = 2.0444, size = 3526, normalized size = 70.52 \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 (\tanh ^{2}{\left (x \right )} + 1\right )^{\frac{3}{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.25641, size = 273, normalized size = 5.46 \begin{align*} -\frac{1}{4} \, \sqrt{2}{\left (5 \, \sqrt{2} \log \left (\frac{\sqrt{2} - \sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1}{\sqrt{2} + \sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} - 1}\right ) - \frac{4 \,{\left (3 \,{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{3} -{\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - \sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} - 1\right )}}{{\left ({\left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right )}^{2} - 2 \, \sqrt{e^{\left (4 \, x\right )} + 1} + 2 \, e^{\left (2 \, x\right )} - 1\right )}^{2}} + 4 \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )} + 1\right ) + 4 \, \log \left (\sqrt{e^{\left (4 \, x\right )} + 1} - e^{\left (2 \, x\right )}\right ) - 4 \, \log \left (-\sqrt{e^{\left (4 \, x\right )} + 1} + e^{\left (2 \, x\right )} + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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