Integrand size = 33, antiderivative size = 107 \[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=\frac {1}{2} \sqrt {c} e^{-a+\frac {b^2}{4 c}} \sqrt {\pi } \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )+\frac {1}{2} \sqrt {c} e^{a-\frac {b^2}{4 c}} \sqrt {\pi } \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )-\frac {\sinh \left (a+b x+c x^2\right )}{x} \]
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Time = 0.06 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {5498, 5483, 2266, 2235, 2236} \[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=\frac {1}{2} \sqrt {\pi } \sqrt {c} e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {c} e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )-\frac {\sinh \left (a+b x+c x^2\right )}{x} \]
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Rule 2235
Rule 2236
Rule 2266
Rule 5483
Rule 5498
Rubi steps \begin{align*} \text {integral}& = -\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*}
Time = 0.44 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.23 \[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=\frac {\sqrt {c} \sqrt {\pi } x \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )-\sinh \left (a-\frac {b^2}{4 c}\right )\right )+\sqrt {c} \sqrt {\pi } x \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\cosh \left (a-\frac {b^2}{4 c}\right )+\sinh \left (a-\frac {b^2}{4 c}\right )\right )-2 \sinh (a+x (b+c x))}{2 x} \]
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\[\int \left (-\frac {b \cosh \left (c \,x^{2}+b x +a \right )}{x}+\frac {\sinh \left (c \,x^{2}+b x +a \right )}{x^{2}}\right )d x\]
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Leaf count of result is larger than twice the leaf count of optimal. 331 vs. \(2 (81) = 162\).
Time = 0.25 (sec) , antiderivative size = 331, normalized size of antiderivative = 3.09 \[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=-\frac {\sqrt {\pi } {\left (x \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + x \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (x \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + x \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) - \sqrt {\pi } {\left (x \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - x \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (x \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - x \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right ) + \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + \sinh \left (c x^{2} + b x + a\right )^{2} - 1}{2 \, {\left (x \cosh \left (c x^{2} + b x + a\right ) + x \sinh \left (c x^{2} + b x + a\right )\right )}} \]
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\[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=- \int \left (- \frac {\sinh {\left (a + b x + c x^{2} \right )}}{x^{2}}\right )\, dx - \int \frac {b \cosh {\left (a + b x + c x^{2} \right )}}{x}\, dx \]
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\[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=\int { -\frac {b \cosh \left (c x^{2} + b x + a\right )}{x} + \frac {\sinh \left (c x^{2} + b x + a\right )}{x^{2}} \,d x } \]
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\[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=\int { -\frac {b \cosh \left (c x^{2} + b x + a\right )}{x} + \frac {\sinh \left (c x^{2} + b x + a\right )}{x^{2}} \,d x } \]
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Timed out. \[ \int \left (-\frac {b \cosh \left (a+b x+c x^2\right )}{x}+\frac {\sinh \left (a+b x+c x^2\right )}{x^2}\right ) \, dx=\int \frac {\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}{x^2}-\frac {b\,\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}{x} \,d x \]
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