Integrand size = 19, antiderivative size = 160 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {(d+e x)^2}{4 e}+\frac {(2 c d-b e) e^{-2 a+\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {(2 c d-b e) e^{2 a-\frac {b^2}{2 c}} \sqrt {\frac {\pi }{2}} \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \]
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Time = 0.11 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {5503, 5491, 5483, 2266, 2235, 2236} \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\frac {\pi }{2}} e^{\frac {b^2}{2 c}-2 a} (2 c d-b e) \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} e^{2 a-\frac {b^2}{2 c}} (2 c d-b e) \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right )}{16 c^{3/2}}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(d+e x)^2}{4 e} \]
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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 {1}{2} (d+e x)+\frac {1}{2} (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx \\ & = \frac {(d+e x)^2}{4 e}+\frac {1}{2} \int (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx \\ & = \frac {(d+e x)^2}{4 e}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(2 c d-b e) \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c} \\ & = \frac {(d+e x)^2}{4 e}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {(2 c d-b e) \int e^{-2 a-2 b x-2 c x^2} \, dx}{8 c}+\frac {(2 c d-b e) \int e^{2 a+2 b x+2 c x^2} \, dx}{8 c} \\ & = \frac {(d+e x)^2}{4 e}+\frac {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac {\left ((2 c d-b e) 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 ((2 c d-b e) e^{-2 a+\frac {b^2}{2 c}}\right ) \int e^{-\frac {(-2 b-4 c x)^2}{8 c}} \, dx}{8 c} \\ & = \frac {(d+e x)^2}{4 e}+\frac {(2 c d-b e) 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 {(2 c d-b e) 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 {e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.11 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) \sqrt {2 \pi } \text {erf}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )-\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+(2 c d-b e) \sqrt {2 \pi } \text {erfi}\left (\frac {b+2 c x}{\sqrt {2} \sqrt {c}}\right ) \left (\cosh \left (2 a-\frac {b^2}{2 c}\right )+\sinh \left (2 a-\frac {b^2}{2 c}\right )\right )+4 \sqrt {c} (2 c x (2 d+e x)+e \sinh (2 (a+x (b+c x))))}{32 c^{3/2}} \]
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Time = 0.16 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.44
method | result | size |
risch | \(\frac {e \,x^{2}}{4}+\frac {d x}{2}+\frac {\operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right ) \sqrt {2}\, \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {4 a c -b^{2}}{2 c}}}{16 \sqrt {c}}-\frac {{\mathrm e}^{-2 a} e \,{\mathrm e}^{-2 x \left (c x +b \right )}}{16 c}-\frac {{\mathrm e}^{-2 a} e b \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{2 c}} \sqrt {2}\, \operatorname {erf}\left (\sqrt {2}\, \sqrt {c}\, x +\frac {b \sqrt {2}}{2 \sqrt {c}}\right )}{32 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {4 a c -b^{2}}{2 c}}}{8 \sqrt {-2 c}}+\frac {{\mathrm e}^{2 a} e \,{\mathrm e}^{2 x \left (c x +b \right )}}{16 c}+\frac {{\mathrm e}^{2 a} e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{2 c}} \operatorname {erf}\left (-\sqrt {-2 c}\, x +\frac {b}{\sqrt {-2 c}}\right )}{16 c \sqrt {-2 c}}\) | \(231\) |
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Leaf count of result is larger than twice the leaf count of optimal. 777 vs. \(2 (130) = 260\).
Time = 0.27 (sec) , antiderivative size = 777, normalized size of antiderivative = 4.86 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {2 \, c e \cosh \left (c x^{2} + b x + a\right )^{4} + 8 \, c e \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right )^{3} + 2 \, c e \sinh \left (c x^{2} + b x + a\right )^{4} - \sqrt {2} \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\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 (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left (2 \, c d - b e\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 ) + \sqrt {2} \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right )^{2} \sinh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\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 (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right ) - {\left (2 \, c d - b e\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 ) + 8 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \cosh \left (c x^{2} + b x + a\right )^{2} + 4 \, {\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x + 3 \, c e \cosh \left (c x^{2} + b x + a\right )^{2}\right )} \sinh \left (c x^{2} + b x + a\right )^{2} - 2 \, c e + 8 \, {\left (c e \cosh \left (c x^{2} + b x + a\right )^{3} + 2 \, {\left (c^{2} e x^{2} + 2 \, c^{2} d x\right )} \cosh \left (c x^{2} + b x + a\right )\right )} \sinh \left (c x^{2} + b x + a\right )}{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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\[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \cosh ^{2}{\left (a + b x + c x^{2} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 301 vs. \(2 (130) = 260\).
Time = 0.37 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.88 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{16} \, {\left (\frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {-c} x - \frac {\sqrt {2} b}{2 \, \sqrt {-c}}\right ) e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}} + \frac {\sqrt {2} \sqrt {\pi } \operatorname {erf}\left (\sqrt {2} \sqrt {c} x + \frac {\sqrt {2} b}{2 \, \sqrt {c}}\right ) e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} + 8 \, x\right )} d + \frac {1}{32} \, {\left (8 \, x^{2} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {\sqrt {2} e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\sqrt {c}}\right )} e^{\left (2 \, a - \frac {b^{2}}{2 \, c}\right )}}{\sqrt {c}} - \frac {\sqrt {2} {\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\sqrt {\frac {1}{2}} \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {\sqrt {2} c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{2 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e^{\left (-2 \, a + \frac {b^{2}}{2 \, c}\right )}}{\sqrt {-c}}\right )} e \]
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Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.03 \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\frac {1}{4} \, e x^{2} + \frac {1}{2} \, d x - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {c}} + 2 \, e e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a\right )}}{32 \, c} - \frac {\frac {\sqrt {2} \sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {2} \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{2 \, c}\right )}}{\sqrt {-c}} - 2 \, e e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a\right )}}{32 \, c} \]
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Timed out. \[ \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx=\int {\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}^2\,\left (d+e\,x\right ) \,d x \]
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