Optimal. Leaf size=128 \[ \frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{e \sinh \left (a+b x+c x^2\right )}{2 c} \]
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Rubi [A] time = 0.0599674, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.294, Rules used = {5383, 5375, 2234, 2204, 2205} \[ \frac{\sqrt{\pi } e^{\frac{b^2}{4 c}-a} (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{\sqrt{\pi } e^{a-\frac{b^2}{4 c}} (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )}{8 c^{3/2}}+\frac{e \sinh \left (a+b x+c x^2\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 5383
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int (d+e x) \cosh \left (a+b x+c x^2\right ) \, dx &=\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*}
Mathematica [A] time = 0.232393, size = 146, normalized size = 1.14 \[ \frac{\sqrt{\pi } (2 c d-b e) \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\cosh \left (a-\frac{b^2}{4 c}\right )-\sinh \left (a-\frac{b^2}{4 c}\right )\right )+\sqrt{\pi } (2 c d-b e) \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )+4 \sqrt{c} e \sinh (a+x (b+c x))}{8 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.036, size = 211, normalized size = 1.7 \begin{align*}{\frac{d\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{{\rm e}^{-c{x}^{2}-bx-a}}}{4\,c}}-{\frac{be\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{d\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{{\rm e}^{c{x}^{2}+bx+a}}}{4\,c}}+{\frac{be\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.37208, size = 343, normalized size = 2.68 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.24789, size = 1033, normalized size = 8.07 \begin{align*} \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 )}} \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{\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 [B] time = 1.23379, size = 282, normalized size = 2.2 \begin{align*} -\frac{\sqrt{\pi } 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}\right )}}{4 \, \sqrt{c}} - \frac{\sqrt{\pi } 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}\right )}}{4 \, \sqrt{-c}} + \frac{\frac{\sqrt{\pi } b \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 \, e^{\left (-c x^{2} - b x - a + 1\right )}}{8 \, c} + \frac{\frac{\sqrt{\pi } b \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 \, e^{\left (c x^{2} + b x + a + 1\right )}}{8 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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