Optimal. Leaf size=19 \[ \frac{\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
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Rubi [A] time = 0.0622082, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {3198, 261} \[ \frac{\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 3198
Rule 261
Rubi steps
\begin{align*} \int \cosh (x) \sinh (x) \sqrt{a+b \sinh ^2(x)} \, dx &=\operatorname{Subst}\left (\int x \sqrt{a+b x^2} \, dx,x,\sinh (x)\right )\\ &=\frac{\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b}\\ \end{align*}
Mathematica [A] time = 0.0120327, size = 19, normalized size = 1. \[ \frac{\left (a+b \sinh ^2(x)\right )^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 16, normalized size = 0.8 \begin{align*}{\frac{1}{3\,b} \left ( a+b \left ( \sinh \left ( x \right ) \right ) ^{2} \right ) ^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06767, size = 20, normalized size = 1.05 \begin{align*} \frac{{\left (b \sinh \left (x\right )^{2} + a\right )}^{\frac{3}{2}}}{3 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19973, size = 464, normalized size = 24.42 \begin{align*} \frac{\sqrt{2}{\left (b \cosh \left (x\right )^{4} + 4 \, b \cosh \left (x\right ) \sinh \left (x\right )^{3} + b \sinh \left (x\right )^{4} + 2 \,{\left (2 \, a - b\right )} \cosh \left (x\right )^{2} + 2 \,{\left (3 \, b \cosh \left (x\right )^{2} + 2 \, a - b\right )} \sinh \left (x\right )^{2} + 4 \,{\left (b \cosh \left (x\right )^{3} +{\left (2 \, a - b\right )} \cosh \left (x\right )\right )} \sinh \left (x\right ) + b\right )} \sqrt{\frac{b \cosh \left (x\right )^{2} + b \sinh \left (x\right )^{2} + 2 \, a - b}{\cosh \left (x\right )^{2} - 2 \, \cosh \left (x\right ) \sinh \left (x\right ) + \sinh \left (x\right )^{2}}}}{24 \,{\left (b \cosh \left (x\right )^{3} + 3 \, b \cosh \left (x\right )^{2} \sinh \left (x\right ) + 3 \, b \cosh \left (x\right ) \sinh \left (x\right )^{2} + b \sinh \left (x\right )^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.90936, size = 46, normalized size = 2.42 \begin{align*} \begin{cases} \frac{a \sqrt{a + b \sinh ^{2}{\left (x \right )}}}{3 b} + \frac{\sqrt{a + b \sinh ^{2}{\left (x \right )}} \sinh ^{2}{\left (x \right )}}{3} & \text{for}\: b \neq 0 \\\frac{\sqrt{a} \sinh ^{2}{\left (x \right )}}{2} & \text{otherwise} \end{cases} \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 \sqrt{b \sinh \left (x\right )^{2} + a} \cosh \left (x\right ) \sinh \left (x\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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