Optimal. Leaf size=125 \[ \frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]
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Rubi [A] time = 0.07, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5301, 5299, 2204, 2205} \[ \frac {3 \sqrt {\pi } e^{-a} \text {Erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {Erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 \sqrt {\pi } e^a \text {Erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {Erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 5299
Rule 5301
Rubi steps
\begin {align*} \int \cosh ^3\left (a+b x^2\right ) \, dx &=\int \left (\frac {3}{4} \cosh \left (a+b x^2\right )+\frac {1}{4} \cosh \left (3 a+3 b x^2\right )\right ) \, dx\\ &=\frac {1}{4} \int \cosh \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int \cosh \left (a+b x^2\right ) \, dx\\ &=\frac {1}{8} \int e^{-3 a-3 b x^2} \, dx+\frac {1}{8} \int e^{3 a+3 b x^2} \, dx+\frac {3}{8} \int e^{-a-b x^2} \, dx+\frac {3}{8} \int e^{a+b x^2} \, dx\\ &=\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.13, size = 136, normalized size = 1.09 \[ \frac {\sqrt {\frac {\pi }{3}} \left (3 \sqrt {3} (\cosh (a)-\sinh (a)) \text {erf}\left (\sqrt {b} x\right )+(\cosh (3 a)-\sinh (3 a)) \text {erf}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \sinh (a) \text {erfi}\left (\sqrt {b} x\right )+\sinh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \cosh (a) \text {erfi}\left (\sqrt {b} x\right )+\cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )\right )}{16 \sqrt {b}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.55, size = 113, normalized size = 0.90 \[ -\frac {\sqrt {3} \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) - \sqrt {3} \sqrt {\pi } \sqrt {b} {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) + 9 \, \sqrt {\pi } \sqrt {-b} {\left (\cosh \relax (a) + \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) - 9 \, \sqrt {\pi } \sqrt {b} {\left (\cosh \relax (a) - \sinh \relax (a)\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{48 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 95, normalized size = 0.76 \[ -\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 86, normalized size = 0.69 \[ \frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \erf \left (x \sqrt {3}\, \sqrt {b}\right )}{48 \sqrt {b}}+\frac {3 \erf \left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{16 \sqrt {b}}+\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \erf \left (\sqrt {-3 b}\, x \right )}{16 \sqrt {-3 b}}+\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \erf \left (\sqrt {-b}\, x \right )}{16 \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.40, size = 91, normalized size = 0.73 \[ \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\mathrm {cosh}\left (b\,x^2+a\right )}^3 \,d x \]
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 \[ \int \cosh ^{3}{\left (a + b x^{2} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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