Integrand size = 15, antiderivative size = 136 \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {x^2}{4}-\frac {b e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}-\frac {b e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \]
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Time = 0.07 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {5503, 5491, 5483, 2266, 2235, 2236} \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\frac {\pi }{2}} b e^{\frac {b^2}{2 c}-2 a} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}-\frac {\sqrt {\frac {\pi }{2}} b e^{2 a-\frac {b^2}{2 c}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {x^2}{4} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5483
Rule 5491
Rule 5503
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {x}{2}+\frac {1}{2} x \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx \\ & = \frac {x^2}{4}+\frac {1}{2} \int x \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx \\ & = \frac {x^2}{4}+\frac {\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {b \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c} \\ & = \frac {x^2}{4}+\frac {\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {b \int e^{-2 a-2 b x-2 c x^2} \, dx}{8 c}-\frac {b \int e^{2 a+2 b x+2 c x^2} \, dx}{8 c} \\ & = \frac {x^2}{4}+\frac {\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac {\left (b e^{2 a-\frac {b^2}{2 c}}\right ) \int e^{\frac {(2 b+4 c x)^2}{8 c}} \, dx}{8 c}-\frac {\left (b e^{-2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(-2 b-4 c x)^2}{8 c}} \, dx}{8 c} \\ & = \frac {x^2}{4}-\frac {b 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 )}{16 c^{3/2}}-\frac {b 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 )}{16 c^{3/2}}+\frac {\sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \\ \end{align*}
Time = 0.27 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.14 \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {b \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (-\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )-b \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} \left (2 c x^2+\sinh (2 (a+x (b+c x)))\right )}{32 c^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.04
method | result | size |
risch | \(\frac {x^{2}}{4}-\frac {{\mathrm e}^{-2 c \,x^{2}-2 b x -2 a}}{16 c}-\frac {b \sqrt {\pi }\, {\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{32 c^{\frac {3}{2}}}+\frac {{\mathrm e}^{2 c \,x^{2}+2 b x +2 a}}{16 c}+\frac {b \sqrt {\pi }\, {\mathrm e}^{\frac {4 a c -b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{16 c \sqrt {-2 c}}\) | \(141\) |
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Leaf count of result is larger than twice the leaf count of optimal. 647 vs. \(2 (106) = 212\).
Time = 0.26 (sec) , antiderivative size = 647, normalized size of antiderivative = 4.76 \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {8 \, c^{2} x^{2} \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, c \cosh \left (c x^{2} + b x + a\right )^{4} + 8 \, c \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right )^{3} + 2 \, c \sinh \left (c x^{2} + b x + a\right )^{4} + \sqrt {2} \sqrt {\pi } {\left (b \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + b \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (b \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + b \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, {\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + b \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 ) - \sqrt {2} \sqrt {\pi } {\left (b \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - b \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (b \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - b \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 2 \, {\left (b \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - b \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 ) + 4 \, {\left (2 \, c^{2} x^{2} + 3 \, c \cosh \left (c x^{2} + b x + a\right )^{2}\right )} \sinh \left (c x^{2} + b x + a\right )^{2} + 8 \, {\left (2 \, c^{2} x^{2} \cosh \left (c x^{2} + b x + a\right ) + c \cosh \left (c x^{2} + b x + a\right )^{3}\right )} \sinh \left (c x^{2} + b x + a\right ) - 2 \, c}{32 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, c^{2} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )^{2}\right )}} \]
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\[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int x \cosh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]
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Time = 0.27 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.47 \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, x^{2} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{32 \, \sqrt {c}} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {\sqrt {2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{32 \, \sqrt {-c}} \]
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Time = 0.26 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.04 \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, x^{2} + \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} - 2 \, e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{32 \, c} + \frac {\frac {\sqrt {2} \sqrt {\pi } b \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} + 2 \, e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{32 \, c} \]
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Timed out. \[ \int x \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int x\,{\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}^2 \,d x \]
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