Optimal. Leaf size=160 \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^2}{4 e} \]
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Rubi [A] time = 0.145921, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {5395, 5383, 5375, 2234, 2204, 2205} \[ \frac{\sqrt{\frac{\pi }{2}} e^{\frac{b^2}{2 c}-2 a} (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{\sqrt{\frac{\pi }{2}} e^{2 a-\frac{b^2}{2 c}} (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right )}{16 c^{3/2}}+\frac{e \sinh \left (2 a+2 b x+2 c x^2\right )}{8 c}+\frac{(d+e x)^2}{4 e} \]
Antiderivative was successfully verified.
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Rule 5395
Rule 5383
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (d+e x) \cosh ^2\left (a+b x+c x^2\right ) \, dx &=\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*}
Mathematica [A] time = 0.639251, size = 177, normalized size = 1.11 \[ \frac{\sqrt{2 \pi } (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\cosh \left (2 a-\frac{b^2}{2 c}\right )-\sinh \left (2 a-\frac{b^2}{2 c}\right )\right )+\sqrt{2 \pi } (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{\sqrt{2} \sqrt{c}}\right ) \left (\sinh \left (2 a-\frac{b^2}{2 c}\right )+\cosh \left (2 a-\frac{b^2}{2 c}\right )\right )+4 \sqrt{c} (e \sinh (2 (a+x (b+c x)))+2 c x (2 d+e x))}{32 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.057, size = 241, normalized size = 1.5 \begin{align*}{\frac{e{x}^{2}}{4}}+{\frac{dx}{2}}+{\frac{d\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{{\rm e}^{-2\,c{x}^{2}-2\,bx-2\,a}}}{16\,c}}-{\frac{be\sqrt{\pi }\sqrt{2}}{32}{{\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\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{{\rm e}^{2\,c{x}^{2}+2\,bx+2\,a}}}{16\,c}}+{\frac{be\sqrt{\pi }}{16\,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 = 1.93148, size = 406, normalized size = 2.54 \begin{align*} \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 \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.92095, size = 1922, normalized size = 12.01 \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 ) \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 = 1.33314, size = 335, normalized size = 2.09 \begin{align*} -\frac{\sqrt{2} \sqrt{\pi } 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}\right )}}{16 \, \sqrt{c}} - \frac{\sqrt{2} \sqrt{\pi } 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}\right )}}{16 \, \sqrt{-c}} + \frac{1}{4} \, x^{2} e + \frac{1}{2} \, d x + \frac{\frac{\sqrt{2} \sqrt{\pi } b \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 \, e^{\left (-2 \, c x^{2} - 2 \, b x - 2 \, a + 1\right )}}{32 \, c} + \frac{\frac{\sqrt{2} \sqrt{\pi } b \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 \, e^{\left (2 \, c x^{2} + 2 \, b x + 2 \, a + 1\right )}}{32 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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