Optimal. Leaf size=45 \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )-\frac{2}{5} (\tanh (x)+1)^{5/2}-2 \sqrt{\tanh (x)+1} \]
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Rubi [A] time = 0.0581242, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {3543, 3478, 3480, 206} \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )-\frac{2}{5} (\tanh (x)+1)^{5/2}-2 \sqrt{\tanh (x)+1} \]
Antiderivative was successfully verified.
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Rule 3543
Rule 3478
Rule 3480
Rule 206
Rubi steps
\begin{align*} \int \tanh ^2(x) (1+\tanh (x))^{3/2} \, dx &=-\frac{2}{5} (1+\tanh (x))^{5/2}+\int (1+\tanh (x))^{3/2} \, dx\\ &=-2 \sqrt{1+\tanh (x)}-\frac{2}{5} (1+\tanh (x))^{5/2}+2 \int \sqrt{1+\tanh (x)} \, dx\\ &=-2 \sqrt{1+\tanh (x)}-\frac{2}{5} (1+\tanh (x))^{5/2}+4 \operatorname{Subst}\left (\int \frac{1}{2-x^2} \, dx,x,\sqrt{1+\tanh (x)}\right )\\ &=2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{1+\tanh (x)}}{\sqrt{2}}\right )-2 \sqrt{1+\tanh (x)}-\frac{2}{5} (1+\tanh (x))^{5/2}\\ \end{align*}
Mathematica [A] time = 0.197637, size = 53, normalized size = 1.18 \[ 2 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{\tanh (x)+1}}{\sqrt{2}}\right )-\frac{1}{5} \sqrt{\tanh (x)+1} \text{sech}^2(x) (2 \sinh (2 x)+7 \cosh (2 x)+5) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.019, size = 35, normalized size = 0.8 \begin{align*} 2\,{\it Artanh} \left ( 1/2\,\sqrt{1+\tanh \left ( x \right ) }\sqrt{2} \right ) \sqrt{2}-2\,\sqrt{1+\tanh \left ( x \right ) }-{\frac{2}{5} \left ( 1+\tanh \left ( x \right ) \right ) ^{{\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{\left (\tanh \left (x\right ) + 1\right )}^{\frac{3}{2}} \tanh \left (x\right )^{2}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.34753, size = 1450, normalized size = 32.22 \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 [A] time = 17.3534, size = 82, normalized size = 1.82 \begin{align*} - \frac{2 \left (\tanh{\left (x \right )} + 1\right )^{\frac{5}{2}}}{5} - 2 \sqrt{\tanh{\left (x \right )} + 1} - 4 \left (\begin{cases} - \frac{\sqrt{2} \operatorname{acoth}{\left (\frac{\sqrt{2} \sqrt{\tanh{\left (x \right )} + 1}}{2} \right )}}{2} & \text{for}\: \tanh{\left (x \right )} + 1 > 2 \\- \frac{\sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{\tanh{\left (x \right )} + 1}}{2} \right )}}{2} & \text{for}\: \tanh{\left (x \right )} + 1 < 2 \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.29067, size = 189, normalized size = 4.2 \begin{align*} \frac{1}{5} \, \sqrt{2}{\left (\frac{2 \,{\left (25 \,{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{4} - 60 \,{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{3} + 70 \,{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )}\right )}^{2} - 40 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 40 \, e^{\left (2 \, x\right )} + 9\right )}}{{\left (\sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} - e^{\left (2 \, x\right )} - 1\right )}^{5}} - 5 \, \log \left (-2 \, \sqrt{e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}} + 2 \, e^{\left (2 \, x\right )} + 1\right )\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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