Optimal. Leaf size=35 \[ \frac{2 \tanh ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \tanh ^{\frac{7}{2}}(a+b x)}{7 b} \]
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Rubi [A] time = 0.0366913, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {2607, 14} \[ \frac{2 \tanh ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \tanh ^{\frac{7}{2}}(a+b x)}{7 b} \]
Antiderivative was successfully verified.
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Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \text{sech}^4(a+b x) \sqrt{\tanh (a+b x)} \, dx &=-\frac{i \operatorname{Subst}\left (\int \sqrt{-i x} \left (1+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (\sqrt{-i x}-(-i x)^{5/2}\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac{2 \tanh ^{\frac{3}{2}}(a+b x)}{3 b}-\frac{2 \tanh ^{\frac{7}{2}}(a+b x)}{7 b}\\ \end{align*}
Mathematica [A] time = 0.105289, size = 29, normalized size = 0.83 \[ \frac{2 \tanh ^{\frac{3}{2}}(a+b x) \left (3 \text{sech}^2(a+b x)+4\right )}{21 b} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.194, size = 0, normalized size = 0. \begin{align*} \int \left ({\rm sech} \left (bx+a\right ) \right ) ^{4}\sqrt{\tanh \left ( bx+a \right ) }\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.80421, size = 475, normalized size = 13.57 \begin{align*} \frac{32 \, \sqrt{e^{\left (-b x - a\right )} + 1} \sqrt{-e^{\left (-b x - a\right )} + 1} e^{\left (-2 \, b x - 2 \, a\right )}}{21 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt{e^{\left (-2 \, b x - 2 \, a\right )} + 1}} - \frac{32 \, \sqrt{e^{\left (-b x - a\right )} + 1} \sqrt{-e^{\left (-b x - a\right )} + 1} e^{\left (-4 \, b x - 4 \, a\right )}}{21 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt{e^{\left (-2 \, b x - 2 \, a\right )} + 1}} - \frac{8 \, \sqrt{e^{\left (-b x - a\right )} + 1} \sqrt{-e^{\left (-b x - a\right )} + 1} e^{\left (-6 \, b x - 6 \, a\right )}}{21 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt{e^{\left (-2 \, b x - 2 \, a\right )} + 1}} + \frac{8 \, \sqrt{e^{\left (-b x - a\right )} + 1} \sqrt{-e^{\left (-b x - a\right )} + 1}}{21 \, b{\left (3 \, e^{\left (-2 \, b x - 2 \, a\right )} + 3 \, e^{\left (-4 \, b x - 4 \, a\right )} + e^{\left (-6 \, b x - 6 \, a\right )} + 1\right )} \sqrt{e^{\left (-2 \, b x - 2 \, a\right )} + 1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.8987, size = 1521, normalized size = 43.46 \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 \sqrt{\tanh{\left (a + b x \right )}} \operatorname{sech}^{4}{\left (a + b x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \operatorname{sech}\left (b x + a\right )^{4} \sqrt{\tanh \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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