Optimal. Leaf size=46 \[ \frac{2 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{3 b}-\frac{2 i \text{EllipticF}\left (\frac{1}{2} i (a+b x),2\right )}{3 b} \]
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Rubi [A] time = 0.0194128, 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 = {2635, 2641} \[ \frac{2 \sinh (a+b x) \sqrt{\cosh (a+b x)}}{3 b}-\frac{2 i F\left (\left .\frac{1}{2} i (a+b x)\right |2\right )}{3 b} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2641
Rubi steps
\begin{align*} \int \cosh ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{3 b}+\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 \sqrt{\cosh (a+b x)} \sinh (a+b x)}{3 b}\\ \end{align*}
Mathematica [C] time = 0.0939686, size = 81, normalized size = 1.76 \[ \frac{2 \sqrt{\sinh (2 (a+b x))+\cosh (2 (a+b x))+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};-\cosh (2 (a+b x))-\sinh (2 (a+b x))\right )+\sinh (2 (a+b x))}{3 b \sqrt{\cosh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.033, size = 174, normalized size = 3.8 \begin{align*}{\frac{2}{3\,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 ( 4\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{5}-6\, \left ( \cosh \left ( 1/2\,bx+a/2 \right ) \right ) ^{3}+\sqrt{- \left ( \sinh \left ({\frac{bx}{2}}+{\frac{a}{2}} \right ) \right ) ^{2}}\sqrt{-2\, \left ( \cosh \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 ) \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 \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 (\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 \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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