Optimal. Leaf size=91 \[ \frac{\sqrt{\pi } e^{-a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}-\frac{\sqrt{\pi } e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]
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Rubi [A] time = 0.0323885, antiderivative size = 91, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {5374, 2234, 2205, 2204} \[ \frac{\sqrt{\pi } e^{-a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}-\frac{\sqrt{\pi } e^{a+\frac{b^2}{4 c}} \text{Erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
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Rule 5374
Rule 2234
Rule 2205
Rule 2204
Rubi steps
\begin{align*} \int \sinh \left (a+b x-c x^2\right ) \, dx &=\frac{1}{2} \int e^{a+b x-c x^2} \, dx-\frac{1}{2} \int e^{-a-b x+c x^2} \, dx\\ &=-\left (\frac{1}{2} e^{-a-\frac{b^2}{4 c}} \int e^{\frac{(-b+2 c x)^2}{4 c}} \, dx\right )+\frac{1}{2} e^{a+\frac{b^2}{4 c}} \int e^{-\frac{(b-2 c x)^2}{4 c}} \, dx\\ &=-\frac{e^{a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}+\frac{e^{-a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b-2 c x}{2 \sqrt{c}}\right )}{4 \sqrt{c}}\\ \end{align*}
Mathematica [A] time = 0.0693669, size = 109, normalized size = 1.2 \[ \frac{\sqrt{\pi } \left (\text{Erf}\left (\frac{2 c x-b}{2 \sqrt{c}}\right ) \left (\sinh \left (a+\frac{b^2}{4 c}\right )+\cosh \left (a+\frac{b^2}{4 c}\right )\right )+\text{Erfi}\left (\frac{2 c x-b}{2 \sqrt{c}}\right ) \left (\sinh \left (a+\frac{b^2}{4 c}\right )-\cosh \left (a+\frac{b^2}{4 c}\right )\right )\right )}{4 \sqrt{c}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.032, size = 79, normalized size = 0.9 \begin{align*} -{\frac{\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{\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}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.52258, size = 691, normalized size = 7.59 \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 [A] time = 2.06003, size = 296, normalized size = 3.25 \begin{align*} \frac{\sqrt{\pi } \sqrt{-c}{\left (\cosh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) - \sinh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{{\left (2 \, c x - b\right )} \sqrt{-c}}{2 \, c}\right ) + \sqrt{\pi } \sqrt{c}{\left (\cosh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right ) + \sinh \left (\frac{b^{2} + 4 \, a c}{4 \, c}\right )\right )} \operatorname{erf}\left (\frac{2 \, c x - b}{2 \, \sqrt{c}}\right )}{4 \, c} \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 \sinh{\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 = 1.42254, size = 109, normalized size = 1.2 \begin{align*} -\frac{\sqrt{\pi } \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 } \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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