Optimal. Leaf size=15 \[ \frac{\sinh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.0149193, antiderivative size = 15, 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 = {5321, 2637} \[ \frac{\sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 5321
Rule 2637
Rubi steps
\begin{align*} \int x \cosh \left (a+b x^2\right ) \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \cosh (a+b x) \, dx,x,x^2\right )\\ &=\frac{\sinh \left (a+b x^2\right )}{2 b}\\ \end{align*}
Mathematica [B] time = 0.0097035, size = 31, normalized size = 2.07 \[ \frac{\sinh (a) \cosh \left (b x^2\right )}{2 b}+\frac{\cosh (a) \sinh \left (b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 14, normalized size = 0.9 \begin{align*}{\frac{\sinh \left ( b{x}^{2}+a \right ) }{2\,b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.10586, size = 18, normalized size = 1.2 \begin{align*} \frac{\sinh \left (b x^{2} + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.75891, size = 31, normalized size = 2.07 \begin{align*} \frac{\sinh \left (b x^{2} + a\right )}{2 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.233205, size = 19, normalized size = 1.27 \begin{align*} \begin{cases} \frac{\sinh{\left (a + b x^{2} \right )}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{2} \cosh{\left (a \right )}}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.31239, size = 36, normalized size = 2.4 \begin{align*} \frac{e^{\left (b x^{2} + a\right )} - e^{\left (-b x^{2} - a\right )}}{4 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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