Integrand size = 18, antiderivative size = 268 \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {x^3}{6}-\frac {b^2 e^{2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {e^{2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {b^2 e^{-2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {e^{-2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c} \]
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Time = 0.17 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.444, Rules used = {5503, 5495, 5491, 5483, 2266, 2236, 2235, 5482} \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=-\frac {\sqrt {\frac {\pi }{2}} e^{2 a+\frac {b^2}{2 c}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} b^2 e^{2 a+\frac {b^2}{2 c}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{-2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} b^2 e^{-2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {x^3}{6} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5482
Rule 5483
Rule 5491
Rule 5495
Rule 5503
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x^2}{2}+\frac {1}{2} x^2 \cosh \left (2 a+2 b x-2 c x^2\right )\right ) \, dx \\ & = \frac {x^3}{6}+\frac {1}{2} \int x^2 \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx \\ & = \frac {x^3}{6}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {\int \sinh \left (2 a+2 b x-2 c x^2\right ) \, dx}{8 c}+\frac {b \int x \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx}{4 c} \\ & = \frac {x^3}{6}-\frac {b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {b^2 \int \cosh \left (2 a+2 b x-2 c x^2\right ) \, dx}{8 c^2}+\frac {\int e^{2 a+2 b x-2 c x^2} \, dx}{16 c}-\frac {\int e^{-2 a-2 b x+2 c x^2} \, dx}{16 c} \\ & = \frac {x^3}{6}-\frac {b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {b^2 \int e^{2 a+2 b x-2 c x^2} \, dx}{16 c^2}+\frac {b^2 \int e^{-2 a-2 b x+2 c x^2} \, dx}{16 c^2}-\frac {e^{-2 a-\frac {b^2}{2 c}} \int e^{\frac {(-2 b+4 c x)^2}{8 c}} \, dx}{16 c}+\frac {e^{2 a+\frac {b^2}{2 c}} \int e^{-\frac {(2 b-4 c x)^2}{8 c}} \, dx}{16 c} \\ & = \frac {x^3}{6}-\frac {e^{2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}+\frac {e^{-2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c}+\frac {\left (b^2 e^{-2 a-\frac {b^2}{2 c}}\right ) \int e^{\frac {(-2 b+4 c x)^2}{8 c}} \, dx}{16 c^2}+\frac {\left (b^2 e^{2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(2 b-4 c x)^2}{8 c}} \, dx}{16 c^2} \\ & = \frac {x^3}{6}-\frac {b^2 e^{2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}-\frac {e^{2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {b^2 e^{-2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{5/2}}+\frac {e^{-2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b-2 c x}{\sqrt {2} \sqrt {c}}\right )}{32 c^{3/2}}-\frac {b \sinh \left (2 a+2 b x-2 c x^2\right )}{16 c^2}-\frac {x \sinh \left (2 a+2 b x-2 c x^2\right )}{8 c} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.68 \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {3 \left (b^2-c\right ) \sqrt {2 \pi } \text {erfi}\left (\frac {-b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a+\frac {b^2}{2 c}\right )-\sinh \left (2 a+\frac {b^2}{2 c}\right )\right )+3 \left (b^2+c\right ) \sqrt {2 \pi } \text {erf}\left (\frac {-b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a+\frac {b^2}{2 c}\right )+\sinh \left (2 a+\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} \left (8 c^2 x^3-3 (b+2 c x) \sinh (2 (a+x (b-c x)))\right )}{192 c^{5/2}} \]
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Time = 0.16 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.02
method | result | size |
risch | \(\frac {x^{3}}{6}+\frac {x \,{\mathrm e}^{2 c \,x^{2}-2 b x -2 a}}{16 c}+\frac {b \,{\mathrm e}^{2 c \,x^{2}-2 b x -2 a}}{32 c^{2}}+\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{2 c}} \operatorname {erf}\left (\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{32 c^{2} \sqrt {-2 c}}-\frac {\sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c +b^{2}}{2 c}} \operatorname {erf}\left (\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{32 c \sqrt {-2 c}}-\frac {x \,{\mathrm e}^{-2 c \,x^{2}+2 b x +2 a}}{16 c}-\frac {b \,{\mathrm e}^{-2 c \,x^{2}+2 b x +2 a}}{32 c^{2}}-\frac {b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (-\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{64 c^{\frac {5}{2}}}-\frac {\sqrt {\pi }\, {\mathrm e}^{\frac {4 a c +b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (-\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{64 c^{\frac {3}{2}}}\) | \(273\) |
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Leaf count of result is larger than twice the leaf count of optimal. 843 vs. \(2 (218) = 436\).
Time = 0.27 (sec) , antiderivative size = 843, normalized size of antiderivative = 3.15 \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {32 \, c^{3} x^{3} \cosh \left (c x^{2} - b x - a\right )^{2} + 6 \, {\left (2 \, c^{2} x + b c\right )} \cosh \left (c x^{2} - b x - a\right )^{4} + 24 \, {\left (2 \, c^{2} x + b c\right )} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (c x^{2} - b x - a\right )^{3} + 6 \, {\left (2 \, c^{2} x + b c\right )} \sinh \left (c x^{2} - b x - a\right )^{4} - 3 \, \sqrt {2} \sqrt {\pi } {\left ({\left (b^{2} - c\right )} \cosh \left (c x^{2} - b x - a\right )^{2} \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - {\left (b^{2} - c\right )} \cosh \left (c x^{2} - b x - a\right )^{2} \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left ({\left (b^{2} - c\right )} \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - {\left (b^{2} - c\right )} \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )^{2} + 2 \, {\left ({\left (b^{2} - c\right )} \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) - {\left (b^{2} - c\right )} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )} \sqrt {-c}}{2 \, c}\right ) + 3 \, \sqrt {2} \sqrt {\pi } {\left ({\left (b^{2} + c\right )} \cosh \left (c x^{2} - b x - a\right )^{2} \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (b^{2} + c\right )} \cosh \left (c x^{2} - b x - a\right )^{2} \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left ({\left (b^{2} + c\right )} \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (b^{2} + c\right )} \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )^{2} + 2 \, {\left ({\left (b^{2} + c\right )} \cosh \left (c x^{2} - b x - a\right ) \cosh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right ) + {\left (b^{2} + c\right )} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} - b x - a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {\sqrt {2} {\left (2 \, c x - b\right )}}{2 \, \sqrt {c}}\right ) - 12 \, c^{2} x + 4 \, {\left (8 \, c^{3} x^{3} + 9 \, {\left (2 \, c^{2} x + b c\right )} \cosh \left (c x^{2} - b x - a\right )^{2}\right )} \sinh \left (c x^{2} - b x - a\right )^{2} - 6 \, b c + 8 \, {\left (8 \, c^{3} x^{3} \cosh \left (c x^{2} - b x - a\right ) + 3 \, {\left (2 \, c^{2} x + b c\right )} \cosh \left (c x^{2} - b x - a\right )^{3}\right )} \sinh \left (c x^{2} - b x - a\right )}{192 \, {\left (c^{3} \cosh \left (c x^{2} - b x - a\right )^{2} + 2 \, c^{3} \cosh \left (c x^{2} - b x - a\right ) \sinh \left (c x^{2} - b x - a\right ) + c^{3} \sinh \left (c x^{2} - b x - a\right )^{2}\right )}} \]
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\[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\int x^{2} \cosh ^{2}{\left (a + b x - c x^{2} \right )}\, dx \]
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Time = 0.29 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.20 \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {1}{6} \, x^{3} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b^{2} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x - b\right )}^{2}}{c}} \left (-c\right )^{\frac {5}{2}}} - \frac {2 \, \sqrt {2} b c e^{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {2 \, {\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} e^{\left (2 \, a + \frac {b^{2}}{2 \, c}\right )}}{64 \, \sqrt {-c}} + \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x - b\right )} b^{2} {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x - b\right )}^{2}}{c}} c^{\frac {5}{2}}} + \frac {2 \, \sqrt {2} b e^{\left (\frac {{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {2 \, {\left (2 \, c x - b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x - b\right )}^{2}}{2 \, c}\right )}{\left (-\frac {{\left (2 \, c x - b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} e^{\left (-2 \, a - \frac {b^{2}}{2 \, c}\right )}}{64 \, \sqrt {c}} \]
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Time = 0.27 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.69 \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\frac {1}{6} \, x^{3} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} + c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} + 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c {\left (2 \, x - \frac {b}{c}\right )} + 2 \, b\right )} e^{\left (-2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{64 \, c^{2}} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (b^{2} - c\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x - \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} + 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c {\left (2 \, x - \frac {b}{c}\right )} + 2 \, b\right )} e^{\left (2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{64 \, c^{2}} \]
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Timed out. \[ \int x^2 \cosh ^2\left (a+b x-c x^2\right ) \, dx=\int x^2\,{\mathrm {cosh}\left (-c\,x^2+b\,x+a\right )}^2 \,d x \]
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