Optimal. Leaf size=49 \[ -\frac{2}{3} \sin ^{-1}\left (\frac{\cosh (x)}{\sqrt{2}}\right )+\frac{1}{6} \sqrt{1-\sinh ^2(x)} \text{sech}(x)-\frac{1}{3} \tanh ^3(x) \sin ^{-1}(\sinh (x))+\tanh (x) \sin ^{-1}(\sinh (x)) \]
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Rubi [A] time = 0.140496, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3767, 4844, 12, 4357, 451, 216} \[ \frac{1}{6} \sqrt{2-\cosh ^2(x)} \text{sech}(x)-\frac{2}{3} \sin ^{-1}\left (\frac{\cosh (x)}{\sqrt{2}}\right )-\frac{1}{3} \tanh ^3(x) \sin ^{-1}(\sinh (x))+\tanh (x) \sin ^{-1}(\sinh (x)) \]
Antiderivative was successfully verified.
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Rule 3767
Rule 4844
Rule 12
Rule 4357
Rule 451
Rule 216
Rubi steps
\begin{align*} \int \sin ^{-1}(\sinh (x)) \text{sech}^4(x) \, dx &=\sin ^{-1}(\sinh (x)) \tanh (x)-\frac{1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\int \frac{(2+\cosh (2 x)) \text{sech}(x) \tanh (x)}{3 \sqrt{1-\sinh ^2(x)}} \, dx\\ &=\sin ^{-1}(\sinh (x)) \tanh (x)-\frac{1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\frac{1}{3} \int \frac{(2+\cosh (2 x)) \text{sech}(x) \tanh (x)}{\sqrt{1-\sinh ^2(x)}} \, dx\\ &=\sin ^{-1}(\sinh (x)) \tanh (x)-\frac{1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\frac{1}{3} \operatorname{Subst}\left (\int \frac{1+2 x^2}{x^2 \sqrt{2-x^2}} \, dx,x,\cosh (x)\right )\\ &=\frac{1}{6} \sqrt{2-\cosh ^2(x)} \text{sech}(x)+\sin ^{-1}(\sinh (x)) \tanh (x)-\frac{1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)-\frac{2}{3} \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-x^2}} \, dx,x,\cosh (x)\right )\\ &=-\frac{2}{3} \sin ^{-1}\left (\frac{\cosh (x)}{\sqrt{2}}\right )+\frac{1}{6} \sqrt{2-\cosh ^2(x)} \text{sech}(x)+\sin ^{-1}(\sinh (x)) \tanh (x)-\frac{1}{3} \sin ^{-1}(\sinh (x)) \tanh ^3(x)\\ \end{align*}
Mathematica [C] time = 0.228016, size = 66, normalized size = 1.35 \[ \frac{1}{12} \left (8 i \log \left (\sqrt{3-\cosh (2 x)}+i \sqrt{2} \cosh (x)\right )+\sqrt{6-2 \cosh (2 x)} \text{sech}(x)+4 (\cosh (2 x)+2) \tanh (x) \text{sech}^2(x) \sin ^{-1}(\sinh (x))\right ) \]
Antiderivative was successfully verified.
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Maple [F] time = 0.046, size = 0, normalized size = 0. \begin{align*} \int \arcsin \left ( \sinh \left ( x \right ) \right ) \left ({\rm sech} \left (x\right ) \right ) ^{4}\, 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*} -\frac{4 \,{\left (3 \, e^{\left (2 \, x\right )} + 1\right )} \arctan \left (e^{\left (2 \, x\right )} - 1, \sqrt{e^{\left (2 \, x\right )} + 2 \, e^{x} - 1} \sqrt{-e^{\left (2 \, x\right )} + 2 \, e^{x} + 1}\right ) + 16 \,{\left (e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 1\right )} \int -\frac{{\left (3 \, e^{\left (4 \, x\right )} + e^{\left (2 \, x\right )}\right )} e^{\left (\frac{1}{2} \, \log \left (e^{\left (2 \, x\right )} + 2 \, e^{x} - 1\right ) + \frac{1}{2} \, \log \left (-e^{\left (2 \, x\right )} + 2 \, e^{x} + 1\right )\right )}}{{\left (e^{\left (8 \, x\right )} - 4 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} - 4 \, e^{\left (2 \, x\right )} + 1\right )}{\left (e^{\left (2 \, x\right )} + 2 \, e^{x} - 1\right )}{\left (e^{\left (2 \, x\right )} - 2 \, e^{x} - 1\right )} - e^{\left (12 \, x\right )} + 6 \, e^{\left (10 \, x\right )} + e^{\left (8 \, x\right )} - 12 \, e^{\left (6 \, x\right )} + e^{\left (4 \, x\right )} + 6 \, e^{\left (2 \, x\right )} - 1}\,{d x}}{3 \,{\left (e^{\left (6 \, x\right )} + 3 \, e^{\left (4 \, x\right )} + 3 \, e^{\left (2 \, x\right )} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.55435, size = 1785, normalized size = 36.43 \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(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \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 \arcsin \left (\sinh \left (x\right )\right ) \operatorname{sech}\left (x\right )^{4}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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