Optimal. Leaf size=15 \[ \frac {\sinh \left (a+b x^2\right )}{2 b} \]
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Rubi [A] time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {5321, 2637} \[ \frac {\sinh \left (a+b x^2\right )}{2 b} \]
Antiderivative was successfully verified.
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Rule 2637
Rule 5321
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*}
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Mathematica [B] time = 0.01, 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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fricas [A] time = 0.45, size = 13, normalized size = 0.87 \[ \frac {\sinh \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.11, size = 27, normalized size = 1.80 \[ \frac {e^{\left (b x^{2} + a\right )} - e^{\left (-b x^{2} - a\right )}}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 14, normalized size = 0.93 \[ \frac {\sinh \left (b \,x^{2}+a \right )}{2 b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.32, size = 13, normalized size = 0.87 \[ \frac {\sinh \left (b x^{2} + a\right )}{2 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.05, size = 13, normalized size = 0.87 \[ \frac {\mathrm {sinh}\left (b\,x^2+a\right )}{2\,b} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.21, size = 19, normalized size = 1.27 \[ \begin {cases} \frac {\sinh {\left (a + b x^{2} \right )}}{2 b} & \text {for}\: b \neq 0 \\\frac {x^{2} \cosh {\relax (a )}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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