Integrand size = 17, antiderivative size = 128 \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e) e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c} \]
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Time = 0.05 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {5491, 5483, 2266, 2235, 2236} \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} (2 c d-b e) \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} (2 c d-b e) \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5483
Rule 5491
Rubi steps \begin{align*} \text {integral}& = \frac {e \sinh \left (a+b x+c x^2\right )}{2 c}-\frac {(-2 c d+b e) \int \cosh \left (a+b x+c x^2\right ) \, dx}{2 c} \\ & = \frac {e \sinh \left (a+b x+c x^2\right )}{2 c}+\frac {(2 c d-b e) \int e^{-a-b x-c x^2} \, dx}{4 c}+\frac {(2 c d-b e) \int e^{a+b x+c x^2} \, dx}{4 c} \\ & = \frac {e \sinh \left (a+b x+c x^2\right )}{2 c}+\frac {\left ((2 c d-b e) e^{a-\frac {b^2}{4 c}}\right ) \int e^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c}+\frac {\left ((2 c d-b e) e^{-a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(-b-2 c x)^2}{4 c}} \, dx}{4 c} \\ & = \frac {(2 c d-b e) e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e) e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {e \sinh \left (a+b x+c x^2\right )}{2 c} \\ \end{align*}
Time = 0.16 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.14 \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\frac {(2 c d-b e) \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )-\sinh \left (a-\frac {b^2}{4 c}\right )\right )+(2 c d-b e) \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )+4 \sqrt {c} e \sinh (a+x (b+c x))}{8 c^{3/2}} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.56
method | result | size |
risch | \(\frac {\operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}}}{4 \sqrt {c}}-\frac {{\mathrm e}^{-a} e \,{\mathrm e}^{-x \left (c x +b \right )}}{4 c}-\frac {{\mathrm e}^{-a} e b \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, d \,{\mathrm e}^{\frac {4 a c -b^{2}}{4 c}}}{4 \sqrt {-c}}+\frac {{\mathrm e}^{a} e \,{\mathrm e}^{x \left (c x +b \right )}}{4 c}+\frac {{\mathrm e}^{a} e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}}\) | \(200\) |
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Leaf count of result is larger than twice the leaf count of optimal. 412 vs. \(2 (100) = 200\).
Time = 0.26 (sec) , antiderivative size = 412, normalized size of antiderivative = 3.22 \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\frac {2 \, c e \cosh \left (c x^{2} + b x + a\right )^{2} + 4 \, c e \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, c e \sinh \left (c x^{2} + b x + a\right )^{2} - \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (2 \, c d - b e\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {\pi } {\left ({\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (2 \, c d - b e\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (2 \, c d - b e\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (2 \, c d - b e\right )} \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right ) - 2 \, c e}{8 \, {\left (c^{2} \cosh \left (c x^{2} + b x + a\right ) + c^{2} \sinh \left (c x^{2} + b x + a\right )\right )}} \]
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\[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right ) \cosh {\left (a + b x + c x^{2} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 254 vs. \(2 (100) = 200\).
Time = 0.28 (sec) , antiderivative size = 254, normalized size of antiderivative = 1.98 \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {-c} x - \frac {b}{2 \, \sqrt {-c}}\right ) e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} + \frac {\sqrt {\pi } d \operatorname {erf}\left (\sqrt {c} x + \frac {b}{2 \, \sqrt {c}}\right ) e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {-\frac {{\left (2 \, c x + b\right )}^{2}}{c}} c^{\frac {3}{2}}} - \frac {2 \, e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\sqrt {c}}\right )} e e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {c}} - \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b {\left (\operatorname {erf}\left (\frac {1}{2} \, \sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}}\right ) - 1\right )}}{\sqrt {\frac {{\left (2 \, c x + b\right )}^{2}}{c}} \left (-c\right )^{\frac {3}{2}}} + \frac {2 \, c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {3}{2}}}\right )} e e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{8 \, \sqrt {-c}} \]
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Time = 0.27 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09 \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=-\frac {\frac {\sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, e e^{\left (-c x^{2} - b x - a\right )}}{8 \, c} - \frac {\frac {\sqrt {\pi } {\left (2 \, c d - b e\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} - 2 \, e e^{\left (c x^{2} + b x + a\right )}}{8 \, c} \]
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Timed out. \[ \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx=\int \mathrm {cosh}\left (c\,x^2+b\,x+a\right )\,\left (d+e\,x\right ) \,d x \]
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