Optimal. Leaf size=66 \[ \frac{2 \sinh (a+b x)}{5 b \text{sech}^{\frac{3}{2}}(a+b x)}-\frac{6 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{5 b} \]
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Rubi [A] time = 0.0329784, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {3769, 3771, 2639} \[ \frac{2 \sinh (a+b x)}{5 b \text{sech}^{\frac{3}{2}}(a+b x)}-\frac{6 i \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{5 b} \]
Antiderivative was successfully verified.
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Rule 3769
Rule 3771
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\text{sech}^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 \sinh (a+b x)}{5 b \text{sech}^{\frac{3}{2}}(a+b x)}+\frac{3}{5} \int \frac{1}{\sqrt{\text{sech}(a+b x)}} \, dx\\ &=\frac{2 \sinh (a+b x)}{5 b \text{sech}^{\frac{3}{2}}(a+b x)}+\frac{1}{5} \left (3 \sqrt{\cosh (a+b x)} \sqrt{\text{sech}(a+b x)}\right ) \int \sqrt{\cosh (a+b x)} \, dx\\ &=-\frac{6 i \sqrt{\cosh (a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right ) \sqrt{\text{sech}(a+b x)}}{5 b}+\frac{2 \sinh (a+b x)}{5 b \text{sech}^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [A] time = 0.0697306, size = 59, normalized size = 0.89 \[ \frac{\sqrt{\text{sech}(a+b x)} \left (\sinh (a+b x)+\sinh (3 (a+b x))-12 i \sqrt{\cosh (a+b x)} E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )\right )}{10 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.309, size = 188, normalized size = 2.9 \begin{align*}{\frac{2}{5\,b}\sqrt{ \left ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}} \left ( 8\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{7}-16\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}+10\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}-3\,\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+1}{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) -2\,\cosh \left ( 1/2\,bx+a/2 \right ) \right ){\frac{1}{\sqrt{2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{4}+ \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}}} \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-1}{\frac{1}{\sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}}}} \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 \frac{1}{\operatorname{sech}\left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{\operatorname{sech}\left (b x + a\right )^{\frac{5}{2}}}, x\right ) \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 \frac{1}{\operatorname{sech}^{\frac{5}{2}}{\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 \frac{1}{\operatorname{sech}\left (b x + a\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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