Integrand size = 19, antiderivative size = 261 \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\frac {e (2 c d-b e) \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {e^2 e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}-\frac {(2 c d-b e)^2 e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}-\frac {e^2 e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {(2 c d-b e)^2 e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}} \]
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Time = 0.14 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {5494, 5490, 5482, 2266, 2235, 2236, 5483} \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=-\frac {\sqrt {\pi } e^{\frac {b^2}{4 c}-a} (2 c d-b e)^2 \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}+\frac {\sqrt {\pi } e^{a-\frac {b^2}{4 c}} (2 c d-b e)^2 \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}-\frac {\sqrt {\pi } e^2 e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}-\frac {\sqrt {\pi } e^2 e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{8 c^{3/2}}+\frac {e (2 c d-b e) \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5482
Rule 5483
Rule 5490
Rule 5494
Rubi steps \begin{align*} \text {integral}& = \frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {e^2 \int \cosh \left (a+b x+c x^2\right ) \, dx}{2 c}-\frac {(-2 c d+b e) \int (d+e x) \sinh \left (a+b x+c x^2\right ) \, dx}{2 c} \\ & = \frac {e (2 c d-b e) \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {e^2 \int e^{-a-b x-c x^2} \, dx}{4 c}-\frac {e^2 \int e^{a+b x+c x^2} \, dx}{4 c}+\frac {(2 c d-b e)^2 \int \sinh \left (a+b x+c x^2\right ) \, dx}{4 c^2} \\ & = \frac {e (2 c d-b e) \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {(2 c d-b e)^2 \int e^{-a-b x-c x^2} \, dx}{8 c^2}+\frac {(2 c d-b e)^2 \int e^{a+b x+c x^2} \, dx}{8 c^2}-\frac {\left (e^2 e^{a-\frac {b^2}{4 c}}\right ) \int e^{\frac {(b+2 c x)^2}{4 c}} \, dx}{4 c}-\frac {\left (e^2 e^{-a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(-b-2 c x)^2}{4 c}} \, dx}{4 c} \\ & = \frac {e (2 c d-b e) \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {e^2 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 {e^2 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 {\left ((2 c d-b e)^2 e^{a-\frac {b^2}{4 c}}\right ) \int e^{\frac {(b+2 c x)^2}{4 c}} \, dx}{8 c^2}-\frac {\left ((2 c d-b e)^2 e^{-a+\frac {b^2}{4 c}}\right ) \int e^{-\frac {(-b-2 c x)^2}{4 c}} \, dx}{8 c^2} \\ & = \frac {e (2 c d-b e) \cosh \left (a+b x+c x^2\right )}{4 c^2}+\frac {e (d+e x) \cosh \left (a+b x+c x^2\right )}{2 c}-\frac {e^2 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)^2 e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}}-\frac {e^2 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 {(2 c d-b e)^2 e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )}{16 c^{5/2}} \\ \end{align*}
Time = 0.40 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.74 \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\frac {4 \sqrt {c} e (4 c d-b e+2 c e x) \cosh (a+x (b+c x))+\left (4 c^2 d^2+b^2 e^2+2 c e (-2 b d+e)\right ) \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (-\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )+\left (4 c^2 d^2+b^2 e^2-2 c e (2 b d+e)\right ) \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )}{16 c^{5/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(459\) vs. \(2(205)=410\).
Time = 0.64 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.76
method | result | size |
risch | \(-\frac {\operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{-\frac {4 a c -b^{2}}{4 c}}}{4 \sqrt {c}}+\frac {{\mathrm e}^{-a} e^{2} x \,{\mathrm e}^{-x \left (c x +b \right )}}{4 c}-\frac {{\mathrm e}^{-a} e^{2} b \,{\mathrm e}^{-x \left (c x +b \right )}}{8 c^{2}}-\frac {{\mathrm e}^{-a} e^{2} b^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{16 c^{\frac {5}{2}}}-\frac {{\mathrm e}^{-a} e^{2} \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{8 c^{\frac {3}{2}}}+\frac {{\mathrm e}^{-a} d e \,{\mathrm e}^{-x \left (c x +b \right )}}{2 c}+\frac {{\mathrm e}^{-a} d e b \sqrt {\pi }\, {\mathrm e}^{\frac {b^{2}}{4 c}} \operatorname {erf}\left (\sqrt {c}\, x +\frac {b}{2 \sqrt {c}}\right )}{4 c^{\frac {3}{2}}}-\frac {\operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right ) \sqrt {\pi }\, d^{2} {\mathrm e}^{\frac {4 a c -b^{2}}{4 c}}}{4 \sqrt {-c}}+\frac {{\mathrm e}^{a} e^{2} x \,{\mathrm e}^{x \left (c x +b \right )}}{4 c}-\frac {{\mathrm e}^{a} e^{2} b \,{\mathrm e}^{x \left (c x +b \right )}}{8 c^{2}}-\frac {{\mathrm e}^{a} e^{2} b^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{16 c^{2} \sqrt {-c}}+\frac {{\mathrm e}^{a} e^{2} \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{8 c \sqrt {-c}}+\frac {{\mathrm e}^{a} d e \,{\mathrm e}^{x \left (c x +b \right )}}{2 c}+\frac {{\mathrm e}^{a} d e b \sqrt {\pi }\, {\mathrm e}^{-\frac {b^{2}}{4 c}} \operatorname {erf}\left (-\sqrt {-c}\, x +\frac {b}{2 \sqrt {-c}}\right )}{4 c \sqrt {-c}}\) | \(460\) |
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Leaf count of result is larger than twice the leaf count of optimal. 634 vs. \(2 (205) = 410\).
Time = 0.26 (sec) , antiderivative size = 634, normalized size of antiderivative = 2.43 \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\frac {4 \, c^{2} e^{2} x + 8 \, c^{2} d e - 2 \, b c e^{2} + 2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cosh \left (c x^{2} + b x + a\right )^{2} - \sqrt {\pi } {\left ({\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} - 2 \, c\right )} e^{2}\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 (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left ({\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\right )} \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - {\left (4 \, c^{2} d^{2} - 4 \, b c d e + {\left (b^{2} + 2 \, c\right )} e^{2}\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 ) + 4 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + 2 \, {\left (2 \, c^{2} e^{2} x + 4 \, c^{2} d e - b c e^{2}\right )} \sinh \left (c x^{2} + b x + a\right )^{2}}{16 \, {\left (c^{3} \cosh \left (c x^{2} + b x + a\right ) + c^{3} \sinh \left (c x^{2} + b x + a\right )\right )}} \]
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\[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\int \left (d + e x\right )^{2} \sinh {\left (a + b x + c x^{2} \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 536 vs. \(2 (205) = 410\).
Time = 0.37 (sec) , antiderivative size = 536, normalized size of antiderivative = 2.05 \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\frac {\sqrt {\pi } d^{2} \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^{2} \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 )} d e e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {c}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\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 {5}{2}}} - \frac {4 \, b e^{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{c^{\frac {3}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, -\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} c^{\frac {5}{2}}}\right )} e^{2} e^{\left (a - \frac {b^{2}}{4 \, c}\right )}}{16 \, \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 )} d e e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{4 \, \sqrt {-c}} + \frac {{\left (\frac {\sqrt {\pi } {\left (2 \, c x + b\right )} b^{2} {\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 {5}{2}}} + \frac {4 \, b c e^{\left (-\frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}}{\left (-c\right )^{\frac {5}{2}}} - \frac {4 \, {\left (2 \, c x + b\right )}^{3} \Gamma \left (\frac {3}{2}, \frac {{\left (2 \, c x + b\right )}^{2}}{4 \, c}\right )}{\left (\frac {{\left (2 \, c x + b\right )}^{2}}{c}\right )^{\frac {3}{2}} \left (-c\right )^{\frac {5}{2}}}\right )} e^{2} e^{\left (-a + \frac {b^{2}}{4 \, c}\right )}}{16 \, \sqrt {-c}} \]
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Time = 0.29 (sec) , antiderivative size = 227, normalized size of antiderivative = 0.87 \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\frac {\frac {\sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} + 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {c}} + 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (-c x^{2} - b x - a\right )}}{16 \, c^{2}} - \frac {\frac {\sqrt {\pi } {\left (4 \, c^{2} d^{2} - 4 \, b c d e + b^{2} e^{2} - 2 \, c e^{2}\right )} \operatorname {erf}\left (-\frac {1}{2} \, \sqrt {-c} {\left (2 \, x + \frac {b}{c}\right )}\right ) e^{\left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )}}{\sqrt {-c}} - 2 \, {\left (c e^{2} {\left (2 \, x + \frac {b}{c}\right )} + 4 \, c d e - 2 \, b e^{2}\right )} e^{\left (c x^{2} + b x + a\right )}}{16 \, c^{2}} \]
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Timed out. \[ \int (d+e x)^2 \sinh \left (a+b x+c x^2\right ) \, dx=\int \mathrm {sinh}\left (c\,x^2+b\,x+a\right )\,{\left (d+e\,x\right )}^2 \,d x \]
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