Optimal. Leaf size=53 \[ \frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {5299, 2204, 2205} \[ \frac {\sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}}+\frac {\sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5299
Rubi steps
\begin {align*} \int \cosh \left (a+b x^2\right ) \, dx &=\frac {1}{2} \int e^{-a-b x^2} \, dx+\frac {1}{2} \int e^{a+b x^2} \, dx\\ &=\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{4 \sqrt {b}}+\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{4 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.85 \[ \frac {\sqrt {\pi } \left ((\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x\right )+(\sinh (a)+\cosh (a)) \text {erfi}\left (\sqrt {b} x\right )\right )}{4 \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 49, normalized size = 0.92 \[ -\frac {\sqrt {\pi } \sqrt {-b} {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) - \sqrt {\pi } \sqrt {b} {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{4 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.11, size = 41, normalized size = 0.77 \[ -\frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{4 \, \sqrt {b}} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{4 \, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 40, normalized size = 0.75 \[ \frac {\erf \left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{4 \sqrt {b}}+\frac {{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{4 \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 87, normalized size = 1.64 \[ -\frac {1}{4} \, b {\left (\frac {2 \, x e^{\left (b x^{2} + a\right )}}{b} + \frac {2 \, x e^{\left (-b x^{2} - a\right )}}{b} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{b^{\frac {3}{2}}} - \frac {\sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b} b}\right )} + x \cosh \left (b x^{2} + a\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \mathrm {cosh}\left (b\,x^2+a\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh {\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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