Optimal. Leaf size=25 \[ \frac{2}{7} (\text{sech}(x)+1)^{7/2}-\frac{4}{5} (\text{sech}(x)+1)^{5/2} \]
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Rubi [A] time = 0.100077, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {4373, 1570, 1469, 627, 43} \[ \frac{2}{7} (\text{sech}(x)+1)^{7/2}-\frac{4}{5} (\text{sech}(x)+1)^{5/2} \]
Antiderivative was successfully verified.
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Rule 4373
Rule 1570
Rule 1469
Rule 627
Rule 43
Rubi steps
\begin{align*} \int \text{sech}(x) \sqrt{1+\text{sech}(x)} \tanh ^3(x) \, dx &=-\operatorname{Subst}\left (\int \frac{\sqrt{1+\frac{1}{x}} \left (1-x^2\right )}{x^4} \, dx,x,\cosh (x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{\left (-1+\frac{1}{x^2}\right ) \sqrt{1+\frac{1}{x}}}{x^2} \, dx,x,\cosh (x)\right )\\ &=\operatorname{Subst}\left (\int \sqrt{1+x} \left (-1+x^2\right ) \, dx,x,\text{sech}(x)\right )\\ &=\operatorname{Subst}\left (\int (-1+x) (1+x)^{3/2} \, dx,x,\text{sech}(x)\right )\\ &=\operatorname{Subst}\left (\int \left (-2 (1+x)^{3/2}+(1+x)^{5/2}\right ) \, dx,x,\text{sech}(x)\right )\\ &=-\frac{4}{5} (1+\text{sech}(x))^{5/2}+\frac{2}{7} (1+\text{sech}(x))^{7/2}\\ \end{align*}
Mathematica [A] time = 0.200845, size = 30, normalized size = 1.2 \[ -\frac{8}{35} \cosh ^4\left (\frac{x}{2}\right ) (9 \cosh (x)-5) \text{sech}^3(x) \sqrt{\text{sech}(x)+1} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\rm sech} \left (x\right )\sqrt{1+{\rm sech} \left (x\right )} \left ( \tanh \left ( x \right ) \right ) ^{3}\, dx \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 \sqrt{\operatorname{sech}\left (x\right ) + 1} \operatorname{sech}\left (x\right ) \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 = 2.07246, size = 1465, normalized size = 58.6 \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{\operatorname{sech}{\left (x \right )} + 1} \tanh ^{3}{\left (x \right )} \operatorname{sech}{\left (x \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.1923, size = 62, normalized size = 2.48 \begin{align*} -\frac{2 \,{\left ({\left ({\left ({\left ({\left ({\left ({\left (9 \, e^{x} + 35\right )} e^{x} + 49\right )} e^{x} + 35\right )} e^{x} + 35\right )} e^{x} + 49\right )} e^{x} + 35\right )} e^{x} + 9\right )}}{35 \,{\left (e^{\left (2 \, x\right )} + 1\right )}^{\frac{7}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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