Integrand size = 10, antiderivative size = 125 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]
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Time = 0.06 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5409, 5407, 2235, 2236} \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}}+\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{16 \sqrt {b}}+\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{16 \sqrt {b}} \]
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Rule 2235
Rule 2236
Rule 5407
Rule 5409
Rubi steps \begin{align*} \text {integral}& = \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*}
Time = 0.10 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.09 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{3}} \left (3 \sqrt {3} \cosh (a) \text {erfi}\left (\sqrt {b} x\right )+\cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+3 \sqrt {3} \text {erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))+3 \sqrt {3} \text {erfi}\left (\sqrt {b} x\right ) \sinh (a)+\text {erf}\left (\sqrt {3} \sqrt {b} x\right ) (\cosh (3 a)-\sinh (3 a))+\text {erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)\right )}{16 \sqrt {b}} \]
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Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (x \sqrt {3}\, \sqrt {b}\right )}{48 \sqrt {b}}+\frac {3 \,\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{16 \sqrt {b}}+\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 b}\, x \right )}{16 \sqrt {-3 b}}+\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{16 \sqrt {-b}}\) | \(86\) |
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Time = 0.26 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.90 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (3 \, a\right ) + \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) - \sqrt {3} \sqrt {\pi } \sqrt {b} {\left (\cosh \left (3 \, a\right ) - \sinh \left (3 \, a\right )\right )} \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) + 9 \, \sqrt {\pi } \sqrt {-b} {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {-b} x\right ) - 9 \, \sqrt {\pi } \sqrt {b} {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \operatorname {erf}\left (\sqrt {b} x\right )}{48 \, b} \]
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\[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\int \cosh ^{3}{\left (a + b x^{2} \right )}\, dx \]
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Time = 0.28 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.73 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]
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Time = 0.25 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.76 \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{48 \, \sqrt {-b}} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{48 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{16 \, \sqrt {b}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{16 \, \sqrt {-b}} \]
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Timed out. \[ \int \cosh ^3\left (a+b x^2\right ) \, dx=\int {\mathrm {cosh}\left (b\,x^2+a\right )}^3 \,d x \]
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