Optimal. Leaf size=42 \[ \frac{2 \sinh (a+b x)}{b \sqrt{\cosh (a+b x)}}+\frac{2 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]
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Rubi [A] time = 0.0185497, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2636, 2639} \[ \frac{2 \sinh (a+b x)}{b \sqrt{\cosh (a+b x)}}+\frac{2 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2639
Rubi steps
\begin{align*} \int \frac{1}{\cosh ^{\frac{3}{2}}(a+b x)} \, dx &=\frac{2 \sinh (a+b x)}{b \sqrt{\cosh (a+b x)}}-\int \sqrt{\cosh (a+b x)} \, dx\\ &=\frac{2 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b}+\frac{2 \sinh (a+b x)}{b \sqrt{\cosh (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.0563266, size = 42, normalized size = 1. \[ \frac{2 \sinh (a+b x)}{b \sqrt{\cosh (a+b x)}}+\frac{2 i E\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.045, size = 103, normalized size = 2.5 \begin{align*} 2\,{\frac{\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\it EllipticE} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) +2\,\cosh \left ( 1/2\,bx+a/2 \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}{\sinh \left ( 1/2\,bx+a/2 \right ) \sqrt{2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}b}} \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}{\cosh \left (b x + a\right )^{\frac{3}{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}{\cosh \left (b x + a\right )^{\frac{3}{2}}}, x\right ) \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 \frac{1}{\cosh \left (b x + a\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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