Integrand size = 14, antiderivative size = 160 \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b} \]
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Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {5449, 5433, 5406, 2235, 2236} \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {3 \sqrt {\pi } e^{-a} \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}+\frac {\sqrt {\frac {\pi }{3}} e^{-3 a} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 \sqrt {\pi } e^a \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {\sqrt {\frac {\pi }{3}} e^{3 a} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b} \]
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Rule 2235
Rule 2236
Rule 5406
Rule 5433
Rule 5449
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {3}{4} x^2 \cosh \left (a+b x^2\right )+\frac {1}{4} x^2 \cosh \left (3 a+3 b x^2\right )\right ) \, dx \\ & = \frac {1}{4} \int x^2 \cosh \left (3 a+3 b x^2\right ) \, dx+\frac {3}{4} \int x^2 \cosh \left (a+b x^2\right ) \, dx \\ & = \frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b}-\frac {\int \sinh \left (3 a+3 b x^2\right ) \, dx}{24 b}-\frac {3 \int \sinh \left (a+b x^2\right ) \, dx}{8 b} \\ & = \frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b}+\frac {\int e^{-3 a-3 b x^2} \, dx}{48 b}-\frac {\int e^{3 a+3 b x^2} \, dx}{48 b}+\frac {3 \int e^{-a-b x^2} \, dx}{16 b}-\frac {3 \int e^{a+b x^2} \, dx}{16 b} \\ & = \frac {3 e^{-a} \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right )}{32 b^{3/2}}+\frac {e^{-3 a} \sqrt {\frac {\pi }{3}} \text {erf}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}-\frac {3 e^a \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right )}{32 b^{3/2}}-\frac {e^{3 a} \sqrt {\frac {\pi }{3}} \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )}{96 b^{3/2}}+\frac {3 x \sinh \left (a+b x^2\right )}{8 b}+\frac {x \sinh \left (3 a+3 b x^2\right )}{24 b} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.15 \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {-27 \sqrt {\pi } \cosh (a) \text {erfi}\left (\sqrt {b} x\right )-\sqrt {3 \pi } \cosh (3 a) \text {erfi}\left (\sqrt {3} \sqrt {b} x\right )+27 \sqrt {\pi } \text {erf}\left (\sqrt {b} x\right ) (\cosh (a)-\sinh (a))-27 \sqrt {\pi } \text {erfi}\left (\sqrt {b} x\right ) \sinh (a)+\sqrt {3 \pi } \text {erf}\left (\sqrt {3} \sqrt {b} x\right ) (\cosh (3 a)-\sinh (3 a))-\sqrt {3 \pi } \text {erfi}\left (\sqrt {3} \sqrt {b} x\right ) \sinh (3 a)+108 \sqrt {b} x \sinh \left (a+b x^2\right )+12 \sqrt {b} x \sinh \left (3 \left (a+b x^2\right )\right )}{288 b^{3/2}} \]
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Time = 0.19 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.98
method | result | size |
risch | \(-\frac {{\mathrm e}^{-3 a} x \,{\mathrm e}^{-3 b \,x^{2}}}{48 b}+\frac {{\mathrm e}^{-3 a} \sqrt {\pi }\, \sqrt {3}\, \operatorname {erf}\left (x \sqrt {3}\, \sqrt {b}\right )}{288 b^{\frac {3}{2}}}-\frac {3 \,{\mathrm e}^{-a} x \,{\mathrm e}^{-b \,x^{2}}}{16 b}+\frac {3 \,\operatorname {erf}\left (x \sqrt {b}\right ) \sqrt {\pi }\, {\mathrm e}^{-a}}{32 b^{\frac {3}{2}}}+\frac {{\mathrm e}^{3 a} x \,{\mathrm e}^{3 b \,x^{2}}}{48 b}-\frac {{\mathrm e}^{3 a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-3 b}\, x \right )}{96 b \sqrt {-3 b}}+\frac {3 \,{\mathrm e}^{a} {\mathrm e}^{b \,x^{2}} x}{16 b}-\frac {3 \,{\mathrm e}^{a} \sqrt {\pi }\, \operatorname {erf}\left (\sqrt {-b}\, x \right )}{32 b \sqrt {-b}}\) | \(157\) |
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Leaf count of result is larger than twice the leaf count of optimal. 903 vs. \(2 (114) = 228\).
Time = 0.26 (sec) , antiderivative size = 903, normalized size of antiderivative = 5.64 \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\text {Too large to display} \]
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\[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\int x^{2} \cosh ^{3}{\left (a + b x^{2} \right )}\, dx \]
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none
Time = 0.27 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.01 \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=-\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{288 \, \sqrt {-b} b} + \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{288 \, b^{\frac {3}{2}}} + \frac {x e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} + \frac {3 \, x e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-b x^{2} - a\right )}}{16 \, b} - \frac {x e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {b} x\right ) e^{\left (-a\right )}}{32 \, b^{\frac {3}{2}}} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (\sqrt {-b} x\right ) e^{a}}{32 \, \sqrt {-b} b} \]
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Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.04 \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {-b} x\right ) e^{\left (3 \, a\right )}}{288 \, \sqrt {-b} b} - \frac {\sqrt {3} \sqrt {\pi } \operatorname {erf}\left (-\sqrt {3} \sqrt {b} x\right ) e^{\left (-3 \, a\right )}}{288 \, b^{\frac {3}{2}}} + \frac {x e^{\left (3 \, b x^{2} + 3 \, a\right )}}{48 \, b} + \frac {3 \, x e^{\left (b x^{2} + a\right )}}{16 \, b} - \frac {3 \, x e^{\left (-b x^{2} - a\right )}}{16 \, b} - \frac {x e^{\left (-3 \, b x^{2} - 3 \, a\right )}}{48 \, b} - \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {b} x\right ) e^{\left (-a\right )}}{32 \, b^{\frac {3}{2}}} + \frac {3 \, \sqrt {\pi } \operatorname {erf}\left (-\sqrt {-b} x\right ) e^{a}}{32 \, \sqrt {-b} b} \]
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Timed out. \[ \int x^2 \cosh ^3\left (a+b x^2\right ) \, dx=\int x^2\,{\mathrm {cosh}\left (b\,x^2+a\right )}^3 \,d x \]
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