Optimal. Leaf size=261 \[ \frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^2 e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } e^2 e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.172052, antiderivative size = 261, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 7, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.368, Rules used = {5387, 5374, 2234, 2204, 2205, 5383, 5375} \[ \frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{16 c^{5/2}}+\frac{\sqrt{\pi } e^2 e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}-\frac{\sqrt{\pi } e^2 e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 5387
Rule 5374
Rule 2234
Rule 2204
Rule 2205
Rule 5383
Rule 5375
Rubi steps
\begin{align*} \int (d+e x)^2 \cosh \left (a+b x+c x^2\right ) \, dx &=\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}-\frac{e^2 \int \sinh \left (a+b x+c x^2\right ) \, dx}{2 c}-\frac{(-2 c d+b e) \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx}{2 c}\\ &=\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \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 \cosh \left (a+b x+c x^2\right ) \, dx}{4 c^2}\\ &=\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \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 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{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}+\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 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}}+\frac{e (2 c d-b e) \sinh \left (a+b x+c x^2\right )}{4 c^2}+\frac{e (d+e x) \sinh \left (a+b x+c x^2\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.527869, size = 194, normalized size = 0.74 \[ \frac{\sqrt{\pi } \left (b^2 e^2+2 c e (e-2 b d)+4 c^2 d^2\right ) \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\cosh \left (a-\frac{b^2}{4 c}\right )-\sinh \left (a-\frac{b^2}{4 c}\right )\right )+\sqrt{\pi } \left (b^2 e^2-2 c e (2 b d+e)+4 c^2 d^2\right ) \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )+4 \sqrt{c} e \sinh (a+x (b+c x)) (-b e+4 c d+2 c e x)}{16 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 493, normalized size = 1.9 \begin{align*}{\frac{{d}^{2}\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{e}^{2}x{{\rm e}^{-c{x}^{2}-bx-a}}}{4\,c}}+{\frac{{e}^{2}b{{\rm e}^{-c{x}^{2}-bx-a}}}{8\,{c}^{2}}}+{\frac{{b}^{2}{e}^{2}\sqrt{\pi }}{16}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{e}^{2}\sqrt{\pi }}{8}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-{\frac{de{{\rm e}^{-c{x}^{2}-bx-a}}}{2\,c}}-{\frac{bde\sqrt{\pi }}{4}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( \sqrt{c}x+{\frac{b}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{d}^{2}\sqrt{\pi }}{4}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}}+{\frac{{e}^{2}x{{\rm e}^{c{x}^{2}+bx+a}}}{4\,c}}-{\frac{{e}^{2}b{{\rm e}^{c{x}^{2}+bx+a}}}{8\,{c}^{2}}}-{\frac{{b}^{2}{e}^{2}\sqrt{\pi }}{16\,{c}^{2}}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}}+{\frac{{e}^{2}\sqrt{\pi }}{8\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}}+{\frac{de{{\rm e}^{c{x}^{2}+bx+a}}}{2\,c}}+{\frac{bde\sqrt{\pi }}{4\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{4\,c}}}}{\it Erf} \left ( -\sqrt{-c}x+{\frac{b}{2}{\frac{1}{\sqrt{-c}}}} \right ){\frac{1}{\sqrt{-c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.50754, size = 724, normalized size = 2.77 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.23463, size = 1508, normalized size = 5.78 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d + e x\right )^{2} \cosh{\left (a + b x + c x^{2} \right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.27384, size = 522, normalized size = 2. \begin{align*} -\frac{\sqrt{\pi } d^{2} \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 )}}{4 \, \sqrt{c}} - \frac{\sqrt{\pi } d^{2} \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 )}}{4 \, \sqrt{-c}} + \frac{\frac{\sqrt{\pi } b d \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}{4 \, c}\right )}}{\sqrt{c}} - 2 \, d e^{\left (-c x^{2} - b x - a + 1\right )}}{4 \, c} + \frac{\frac{\sqrt{\pi } b d \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}{4 \, c}\right )}}{\sqrt{-c}} + 2 \, d e^{\left (c x^{2} + b x + a + 1\right )}}{4 \, c} - \frac{\frac{\sqrt{\pi }{\left (b^{2} + 2 \, c\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 + 8 \, c}{4 \, c}\right )}}{\sqrt{c}} + 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (-c x^{2} - b x - a + 2\right )}}{16 \, c^{2}} - \frac{\frac{\sqrt{\pi }{\left (b^{2} - 2 \, c\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 - 8 \, c}{4 \, c}\right )}}{\sqrt{-c}} - 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (c x^{2} + b x + a + 2\right )}}{16 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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