Optimal. Leaf size=107 \[ \frac{1}{2} \sqrt{\pi } \sqrt{c} e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )+\frac{1}{2} \sqrt{\pi } \sqrt{c} e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )-\frac{\sinh \left (a+b x+c x^2\right )}{x} \]
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Rubi [A] time = 0.0822187, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.152, Rules used = {5390, 5375, 2234, 2204, 2205} \[ \frac{1}{2} \sqrt{\pi } \sqrt{c} e^{\frac{b^2}{4 c}-a} \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )+\frac{1}{2} \sqrt{\pi } \sqrt{c} e^{a-\frac{b^2}{4 c}} \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )-\frac{\sinh \left (a+b x+c x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 5390
Rule 5375
Rule 2234
Rule 2204
Rule 2205
Rubi steps
\begin{align*} \int \left (-\frac{b \cosh \left (a+b x+c x^2\right )}{x}+\frac{\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx &=-\left (b \int \frac{\cosh \left (a+b x+c x^2\right )}{x} \, dx\right )+\int \frac{\sinh \left (a+b x+c x^2\right )}{x^2} \, dx\\ &=-\frac{\sinh \left (a+b x+c x^2\right )}{x}+(2 c) \int \cosh \left (a+b x+c x^2\right ) \, dx\\ &=-\frac{\sinh \left (a+b x+c x^2\right )}{x}+c \int e^{-a-b x-c x^2} \, dx+c \int e^{a+b x+c x^2} \, dx\\ &=-\frac{\sinh \left (a+b x+c x^2\right )}{x}+\left (c e^{a-\frac{b^2}{4 c}}\right ) \int e^{\frac{(b+2 c x)^2}{4 c}} \, dx+\left (c e^{-a+\frac{b^2}{4 c}}\right ) \int e^{-\frac{(-b-2 c x)^2}{4 c}} \, dx\\ &=\frac{1}{2} \sqrt{c} e^{-a+\frac{b^2}{4 c}} \sqrt{\pi } \text{erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )+\frac{1}{2} \sqrt{c} e^{a-\frac{b^2}{4 c}} \sqrt{\pi } \text{erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right )-\frac{\sinh \left (a+b x+c x^2\right )}{x}\\ \end{align*}
Mathematica [A] time = 0.599733, size = 132, normalized size = 1.23 \[ \frac{\sqrt{\pi } \sqrt{c} x \text{Erf}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\cosh \left (a-\frac{b^2}{4 c}\right )-\sinh \left (a-\frac{b^2}{4 c}\right )\right )+\sqrt{\pi } \sqrt{c} x \text{Erfi}\left (\frac{b+2 c x}{2 \sqrt{c}}\right ) \left (\sinh \left (a-\frac{b^2}{4 c}\right )+\cosh \left (a-\frac{b^2}{4 c}\right )\right )-2 \sinh (a+x (b+c x))}{2 x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.112, size = 0, normalized size = 0. \begin{align*} \int -{\frac{b\cosh \left ( c{x}^{2}+bx+a \right ) }{x}}+{\frac{\sinh \left ( c{x}^{2}+bx+a \right ) }{{x}^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \cosh \left (c x^{2} + b x + a\right )}{x} + \frac{\sinh \left (c x^{2} + b x + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.09119, size = 872, normalized size = 8.15 \begin{align*} -\frac{\sqrt{\pi }{\left (x \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + x \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (x \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) + x \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 (x \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - x \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) +{\left (x \cosh \left (-\frac{b^{2} - 4 \, a c}{4 \, c}\right ) - x \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 ) + \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + \sinh \left (c x^{2} + b x + a\right )^{2} - 1}{2 \,{\left (x \cosh \left (c x^{2} + b x + a\right ) + x \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 - \frac{\sinh{\left (a + b x + c x^{2} \right )}}{x^{2}}\, dx - \int \frac{b \cosh{\left (a + b x + c x^{2} \right )}}{x}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{b \cosh \left (c x^{2} + b x + a\right )}{x} + \frac{\sinh \left (c x^{2} + b x + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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