Optimal. Leaf size=311 \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^2 e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^2 e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^3}{6 e} \]
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Rubi [A] time = 0.385622, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 8, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.381, Rules used = {5395, 5387, 5374, 2234, 2204, 2205, 5383, 5375} \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e)^2 \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e)^2 \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{5/2}}+\frac{\sqrt{\frac{\pi }{2}} e^2 e^{\frac{b^2}{2 c}-2 a} \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}-\frac{\sqrt{\frac{\pi }{2}} e^2 e^{2 a-\frac{b^2}{2 c}} \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{32 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^3}{6 e} \]
Antiderivative was successfully verified.
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Rule 5395
Rule 5387
Rule 5374
Rule 2234
Rule 2204
Rule 2205
Rule 5383
Rule 5375
Rubi steps
\begin{align*} \int (d+e x)^2 \cosh ^2\left (a+b x+c x^2\right ) \, dx &=\int \left (\frac{1}{2} (d+e x)^2+\frac{1}{2} (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right )\right ) \, dx\\ &=\frac{(d+e x)^3}{6 e}+\frac{1}{2} \int (d+e x)^2 \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}-\frac{e^2 \int \sinh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c}+\frac{(2 c d-b e) \int (d+e x) \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{4 c}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{e^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c}-\frac{e^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c}+\frac{(2 c d-b e)^2 \int \cosh \left (2 a+2 b x+2 c x^2\right ) \, dx}{8 c^2}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(2 c d-b e)^2 \int e^{-2 a-2 b x-2 c x^2} \, dx}{16 c^2}+\frac{(2 c d-b e)^2 \int e^{2 a+2 b x+2 c x^2} \, dx}{16 c^2}-\frac{\left (e^2 e^{2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{16 c}+\frac{\left (e^2 e^{-2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{16 c}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e^2 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 )}{32 c^{3/2}}-\frac{e^2 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 )}{32 c^{3/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{\left ((2 c d-b e)^2 e^{2 a-\frac{b^2}{2 c}}\right ) \int e^{\frac{(2 b+4 c x)^2}{8 c}} \, dx}{16 c^2}+\frac{\left ((2 c d-b e)^2 e^{-2 a+\frac{b^2}{2 c}}\right ) \int e^{-\frac{(-2 b-4 c x)^2}{8 c}} \, dx}{16 c^2}\\ &=\frac{(d+e x)^3}{6 e}+\frac{e^2 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 )}{32 c^{3/2}}+\frac{(2 c d-b e)^2 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 )}{32 c^{5/2}}-\frac{e^2 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 )}{32 c^{3/2}}+\frac{(2 c d-b e)^2 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 )}{32 c^{5/2}}+\frac{e (2 c d-b e) \sinh \left (2 a+2 b x+2 c x^2\right )}{16 c^2}+\frac{e (d+e x) \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}\\ \end{align*}
Mathematica [A] time = 1.33651, size = 240, normalized size = 0.77 \[ \frac{3 \sqrt{2 \pi } \left (b^2 e^2+c e (e-4 b d)+4 c^2 d^2\right ) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\cosh \left (2 a-\frac{b^2}{2 c}\right )-\sinh \left (2 a-\frac{b^2}{2 c}\right )\right )+3 \sqrt{2 \pi } \left (b^2 e^2-c e (4 b d+e)+4 c^2 d^2\right ) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )+\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )+4 \sqrt{c} \left (3 e \sinh (2 (a+x (b+c x))) (-b e+4 c d+2 c e x)+8 c^2 x \left (3 d^2+3 d e x+e^2 x^2\right )\right )}{192 c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.08, size = 558, normalized size = 1.8 \begin{align*}{\frac{{e}^{2}{x}^{3}}{6}}+{\frac{de{x}^{2}}{2}}+{\frac{{d}^{2}x}{2}}+{\frac{{d}^{2}\sqrt{\pi }\sqrt{2}}{16}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{e}^{2}x{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}+{\frac{{e}^{2}b{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{32\,{c}^{2}}}+{\frac{{b}^{2}{e}^{2}\sqrt{\pi }\sqrt{2}}{64}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{e}^{2}\sqrt{\pi }\sqrt{2}}{64}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-{\frac{de{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{8\,c}}-{\frac{bde\sqrt{\pi }\sqrt{2}}{16}{{\rm e}^{-{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( \sqrt{2}\sqrt{c}x+{\frac{b\sqrt{2}}{2}{\frac{1}{\sqrt{c}}}} \right ){c}^{-{\frac{3}{2}}}}-{\frac{{d}^{2}\sqrt{\pi }}{8}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}+{\frac{{e}^{2}x{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}-{\frac{{e}^{2}b{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{32\,{c}^{2}}}-{\frac{{b}^{2}{e}^{2}\sqrt{\pi }}{32\,{c}^{2}}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}+{\frac{{e}^{2}\sqrt{\pi }}{32\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}}+{\frac{de{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{8\,c}}+{\frac{bde\sqrt{\pi }}{8\,c}{{\rm e}^{{\frac{4\,ac-{b}^{2}}{2\,c}}}}{\it Erf} \left ( -\sqrt{-2\,c}x+{b{\frac{1}{\sqrt{-2\,c}}}} \right ){\frac{1}{\sqrt{-2\,c}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.10776, size = 814, normalized size = 2.62 \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 = 2.10577, size = 2700, normalized size = 8.68 \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 ^{2}{\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 = 1.30703, size = 608, normalized size = 1.95 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi } d^{2} \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 )}}{16 \, \sqrt{c}} - \frac{\sqrt{2} \sqrt{\pi } d^{2} \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 )}}{16 \, \sqrt{-c}} + \frac{1}{6} \, x^{3} e^{2} + \frac{1}{2} \, d x^{2} e + \frac{1}{2} \, d^{2} x + \frac{\frac{\sqrt{2} \sqrt{\pi } b d \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}{2 \, c}\right )}}{\sqrt{c}} - 2 \, d e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a + 1\right )}}{16 \, c} + \frac{\frac{\sqrt{2} \sqrt{\pi } b d \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}{2 \, c}\right )}}{\sqrt{-c}} + 2 \, d e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a + 1\right )}}{16 \, c} - \frac{\frac{\sqrt{2} \sqrt{\pi }{\left (b^{2} + c\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 + 4 \, c}{2 \, c}\right )}}{\sqrt{c}} + 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a + 2\right )}}{64 \, c^{2}} - \frac{\frac{\sqrt{2} \sqrt{\pi }{\left (b^{2} - c\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 - 4 \, c}{2 \, c}\right )}}{\sqrt{-c}} - 2 \,{\left (c{\left (2 \, x + \frac{b}{c}\right )} - 2 \, b\right )} e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a + 2\right )}}{64 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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