Integrand size = 12, antiderivative size = 69 \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{8 b^{3/2}}-\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{8 b^{3/2}}+\frac {x \sinh \left (a+b x^2\right )}{2 b} \]
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Time = 0.03 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5433, 5406, 2235, 2236} \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\frac {\sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{8 b^{3/2}}-\frac {\sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{8 b^{3/2}}+\frac {x \sinh \left (a+b x^2\right )}{2 b} \]
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Rule 2235
Rule 2236
Rule 5406
Rule 5433
Rubi steps \begin{align*} \text {integral}& = \frac {x \sinh \left (a+b x^2\right )}{2 b}-\frac {\int \sinh \left (a+b x^2\right ) \, dx}{2 b} \\ & = \frac {x \sinh \left (a+b x^2\right )}{2 b}+\frac {\int e^{-a-b x^2} \, dx}{4 b}-\frac {\int e^{a+b x^2} \, dx}{4 b} \\ & = \frac {e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{8 b^{3/2}}-\frac {e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{8 b^{3/2}}+\frac {x \sinh \left (a+b x^2\right )}{2 b} \\ \end{align*}
Time = 0.05 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.97 \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\frac {\sqrt {\pi } \text {erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))-\sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right ) (\cosh (a)+\sinh (a))+4 \sqrt {b} x \sinh \left (a+b x^2\right )}{8 b^{3/2}} \]
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Time = 0.05 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07
method | result | size |
risch | \(-\frac {{\mathrm e}^{-a} x \,{\mathrm e}^{-b \,x^{2}}}{4 b}+\frac {\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{8 b^{\frac {3}{2}}}+\frac {{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}} x}{4 b}-\frac {{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{8 b \sqrt {-b}}\) | \(74\) |
meijerg | \(-\frac {i \cosh \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (\frac {x \sqrt {2}\, \left (i b \right )^{\frac {3}{2}} {\mathrm e}^{b \,x^{2}}}{4 \sqrt {\pi }\, b}-\frac {x \sqrt {2}\, \left (i b \right )^{\frac {3}{2}} {\mathrm e}^{-b \,x^{2}}}{4 \sqrt {\pi }\, b}+\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erf}\left (x \sqrt {b}\right )}{8 b^{\frac {3}{2}}}-\frac {\left (i b \right )^{\frac {3}{2}} \sqrt {2}\, \operatorname {erfi}\left (x \sqrt {b}\right )}{8 b^{\frac {3}{2}}}\right )}{2 b \sqrt {i b}}-\frac {\sinh \left (a \right ) \sqrt {\pi }\, \sqrt {2}\, \left (\frac {x \sqrt {2}\, \left (i b \right )^{\frac {5}{2}} {\mathrm e}^{-b \,x^{2}}}{4 \sqrt {\pi }\, b^{2}}+\frac {x \sqrt {2}\, \left (i b \right )^{\frac {5}{2}} {\mathrm e}^{b \,x^{2}}}{4 \sqrt {\pi }\, b^{2}}-\frac {\left (i b \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erf}\left (x \sqrt {b}\right )}{8 b^{\frac {5}{2}}}-\frac {\left (i b \right )^{\frac {5}{2}} \sqrt {2}\, \operatorname {erfi}\left (x \sqrt {b}\right )}{8 b^{\frac {5}{2}}}\right )}{2 b \sqrt {i b}}\) | \(221\) |
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Leaf count of result is larger than twice the leaf count of optimal. 189 vs. \(2 (49) = 98\).
Time = 0.25 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.74 \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\frac {2 \, b x \cosh \left (b x^{2} + a\right )^{2} + 4 \, b x \cosh \left (b x^{2} + a\right ) \sinh \left (b x^{2} + a\right ) + 2 \, b x \sinh \left (b x^{2} + a\right )^{2} + \sqrt {\pi } {\left (\cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) + {\left (\cosh \left (a\right ) + \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right ) + \cosh \left (b x^{2} + a\right ) \sinh \left (a\right )\right )} \sqrt {-b} \operatorname {erf}\left (\sqrt {-b} x\right ) + \sqrt {\pi } {\left (\cosh \left (b x^{2} + a\right ) \cosh \left (a\right ) + {\left (\cosh \left (a\right ) - \sinh \left (a\right )\right )} \sinh \left (b x^{2} + a\right ) - \cosh \left (b x^{2} + a\right ) \sinh \left (a\right )\right )} \sqrt {b} \operatorname {erf}\left (\sqrt {b} x\right ) - 2 \, b x}{8 \, {\left (b^{2} \cosh \left (b x^{2} + a\right ) + b^{2} \sinh \left (b x^{2} + a\right )\right )}} \]
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\[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\int x^{2} \cosh {\left (a + b x^{2} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 110 vs. \(2 (49) = 98\).
Time = 0.18 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.59 \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\frac {1}{3} \, x^{3} \cosh \left (b x^{2} + a\right ) - \frac {1}{24} \, b {\left (\frac {2 \, {\left (2 \, b x^{3} e^{a} - 3 \, x e^{a}\right )} e^{\left (b x^{2}\right )}}{b^{2}} + \frac {2 \, {\left (2 \, b x^{3} + 3 \, x\right )} e^{\left (-b x^{2} - a\right )}}{b^{2}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{b^{\frac {5}{2}}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{\sqrt {-b} b^{2}}\right )} \]
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none
Time = 0.26 (sec) , antiderivative size = 75, normalized size of antiderivative = 1.09 \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\frac {x e^{\left (b x^{2} + a\right )}}{4 \, b} - \frac {x e^{\left (-b x^{2} - a\right )}}{4 \, b} - \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{8 \, b^{\frac {3}{2}}} + \frac {\sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{8 \, \sqrt {-b} b} \]
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Timed out. \[ \int x^2 \cosh \left (a+b x^2\right ) \, dx=\int x^2\,\mathrm {cosh}\left (b\,x^2+a\right ) \,d x \]
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