Optimal. Leaf size=46 \[ \frac{2 \sinh (a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}-\frac{2 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{3 b} \]
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Rubi [A] time = 0.0191879, antiderivative size = 46, 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, 2641} \[ \frac{2 \sinh (a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}-\frac{2 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2636
Rule 2641
Rubi steps
\begin{align*} \int \frac{1}{\cosh ^{\frac{5}{2}}(a+b x)} \, dx &=\frac{2 \sinh (a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}+\frac{1}{3} \int \frac{1}{\sqrt{\cosh (a+b x)}} \, dx\\ &=-\frac{2 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{3 b}+\frac{2 \sinh (a+b x)}{3 b \cosh ^{\frac{3}{2}}(a+b x)}\\ \end{align*}
Mathematica [C] time = 0.0636724, size = 84, normalized size = 1.83 \[ \frac{2 \left (\cosh (a+b x) \sqrt{\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (a+b x)\right )}{3 b \cosh ^{\frac{3}{2}}(a+b x)} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.06, size = 217, normalized size = 4.7 \begin{align*}{\frac{2}{3\,b} \left ( 2\,\sqrt{- \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}}\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cosh \left ( 1/2\,bx+a/2 \right ) ,\sqrt{2} \right ) \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}+\sqrt{- \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \sinh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1}{\it EllipticF} \left ( \cosh \left ({\frac{bx}{2}}+{\frac{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} \right ) \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}}{\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 ( 2\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{2}-1 \right ) ^{-{\frac{3}{2}}} \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{-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}{\cosh \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}{\cosh \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(-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{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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