Optimal. Leaf size=107 \[ -\frac {1}{2} \sqrt {\pi } \sqrt {c} e^{\frac {b^2}{4 c}-a} \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {c} e^{a-\frac {b^2}{4 c}} \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )-\frac {\cosh \left (a+b x+c x^2\right )}{x} \]
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Rubi [A] time = 0.09, antiderivative size = 107, normalized size of antiderivative = 1.00, 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 = {5391, 5374, 2234, 2204, 2205} \[ -\frac {1}{2} \sqrt {\pi } \sqrt {c} e^{\frac {b^2}{4 c}-a} \text {Erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )+\frac {1}{2} \sqrt {\pi } \sqrt {c} e^{a-\frac {b^2}{4 c}} \text {Erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right )-\frac {\cosh \left (a+b x+c x^2\right )}{x} \]
Antiderivative was successfully verified.
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Rule 2204
Rule 2205
Rule 2234
Rule 5374
Rule 5391
Rubi steps
\begin {align*} \int \left (\frac {\cosh \left (a+b x+c x^2\right )}{x^2}-\frac {b \sinh \left (a+b x+c x^2\right )}{x}\right ) \, dx &=-\left (b \int \frac {\sinh \left (a+b x+c x^2\right )}{x} \, dx\right )+\int \frac {\cosh \left (a+b x+c x^2\right )}{x^2} \, dx\\ &=-\frac {\cosh \left (a+b x+c x^2\right )}{x}+(2 c) \int \sinh \left (a+b x+c x^2\right ) \, dx\\ &=-\frac {\cosh \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 {\cosh \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 {\cosh \left (a+b x+c x^2\right )}{x}-\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 )\\ \end {align*}
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Mathematica [A] time = 8.27, size = 132, normalized size = 1.23 \[ \frac {\sqrt {\pi } \sqrt {c} x \text {erf}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\sinh \left (a-\frac {b^2}{4 c}\right )-\cosh \left (a-\frac {b^2}{4 c}\right )\right )+\sqrt {\pi } \sqrt {c} x \text {erfi}\left (\frac {b+2 c x}{2 \sqrt {c}}\right ) \left (\sinh \left (a-\frac {b^2}{4 c}\right )+\cosh \left (a-\frac {b^2}{4 c}\right )\right )-2 \cosh (a+x (b+c x))}{2 x} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.90, size = 330, normalized size = 3.08 \[ -\frac {\sqrt {\pi } {\left (x \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + x \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (x \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + x \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {-c} \operatorname {erf}\left (\frac {{\left (2 \, c x + b\right )} \sqrt {-c}}{2 \, c}\right ) + \sqrt {\pi } {\left (x \cosh \left (c x^{2} + b x + a\right ) \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - x \cosh \left (c x^{2} + b x + a\right ) \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) + {\left (x \cosh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right ) - x \sinh \left (-\frac {b^{2} - 4 \, a c}{4 \, c}\right )\right )} \sinh \left (c x^{2} + b x + a\right )\right )} \sqrt {c} \operatorname {erf}\left (\frac {2 \, c x + b}{2 \, \sqrt {c}}\right ) + \cosh \left (c x^{2} + b x + a\right )^{2} + 2 \, \cosh \left (c x^{2} + b x + a\right ) \sinh \left (c x^{2} + b x + a\right ) + \sinh \left (c x^{2} + b x + a\right )^{2} + 1}{2 \, {\left (x \cosh \left (c x^{2} + b x + a\right ) + x \sinh \left (c x^{2} + b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b \sinh \left (c x^{2} + b x + a\right )}{x} + \frac {\cosh \left (c x^{2} + b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.39, size = 0, normalized size = 0.00 \[ \int \frac {\cosh \left (c \,x^{2}+b x +a \right )}{x^{2}}-\frac {b \sinh \left (c \,x^{2}+b x +a \right )}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {b \sinh \left (c x^{2} + b x + a\right )}{x} + \frac {\cosh \left (c x^{2} + b x + a\right )}{x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\mathrm {cosh}\left (c\,x^2+b\,x+a\right )}{x^2}-\frac {b\,\mathrm {sinh}\left (c\,x^2+b\,x+a\right )}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \left (- \frac {\cosh {\left (a + b x + c x^{2} \right )}}{x^{2}}\right )\, dx - \int \frac {b \sinh {\left (a + b x + c x^{2} \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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