Optimal. Leaf size=80 \[ \frac{2 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{3 b}+\frac{2 i \sqrt{i \sinh (a+b x)} \text{EllipticF}\left (\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right ),2\right )}{3 b \sqrt{\sinh (a+b x)}} \]
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Rubi [A] time = 0.0318454, antiderivative size = 80, 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 = {2635, 2642, 2641} \[ \frac{2 \sqrt{\sinh (a+b x)} \cosh (a+b x)}{3 b}+\frac{2 i \sqrt{i \sinh (a+b x)} F\left (\left .\frac{1}{2} \left (i a+i b x-\frac{\pi }{2}\right )\right |2\right )}{3 b \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2635
Rule 2642
Rule 2641
Rubi steps
\begin{align*} \int \sinh ^{\frac{3}{2}}(a+b x) \, dx &=\frac{2 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{3 b}-\frac{1}{3} \int \frac{1}{\sqrt{\sinh (a+b x)}} \, dx\\ &=\frac{2 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{3 b}-\frac{\sqrt{i \sinh (a+b x)} \int \frac{1}{\sqrt{i \sinh (a+b x)}} \, dx}{3 \sqrt{\sinh (a+b x)}}\\ &=\frac{2 i F\left (\left .\frac{1}{2} \left (i a-\frac{\pi }{2}+i b x\right )\right |2\right ) \sqrt{i \sinh (a+b x)}}{3 b \sqrt{\sinh (a+b x)}}+\frac{2 \cosh (a+b x) \sqrt{\sinh (a+b x)}}{3 b}\\ \end{align*}
Mathematica [C] time = 0.0872205, size = 83, normalized size = 1.04 \[ \frac{\sinh (2 (a+b x))-2 \sqrt{-\sinh (2 a+2 b x)-\cosh (2 a+2 b x)+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{5}{4};\cosh (2 (a+b x))+\sinh (2 (a+b x))\right )}{3 b \sqrt{\sinh (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.042, size = 100, normalized size = 1.3 \begin{align*}{\frac{1}{b\cosh \left ( bx+a \right ) } \left ( -{\frac{i}{3}}\sqrt{1-i\sinh \left ( bx+a \right ) }\sqrt{2}\sqrt{1+i\sinh \left ( bx+a \right ) }\sqrt{i\sinh \left ( bx+a \right ) }{\it EllipticF} \left ( \sqrt{1-i\sinh \left ( bx+a \right ) },{\frac{\sqrt{2}}{2}} \right ) +{\frac{2\,\sinh \left ( bx+a \right ) \left ( \cosh \left ( bx+a \right ) \right ) ^{2}}{3}} \right ){\frac{1}{\sqrt{\sinh \left ( bx+a \right ) }}}} \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 \sinh \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 (\sinh \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sinh ^{\frac{3}{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 \sinh \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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