Optimal. Leaf size=19 \[ -\frac{\log \left (\cos \left (a+b x+c x^2\right )\right )}{2 c} \]
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Rubi [A] time = 0.0176489, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {3763} \[ -\frac{\log \left (\cos \left (a+b x+c x^2\right )\right )}{2 c} \]
Antiderivative was successfully verified.
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Rule 3763
Rubi steps
\begin{align*} \int \left (\frac{b \tan \left (a+b x+c x^2\right )}{2 c}+x \tan \left (a+b x+c x^2\right )\right ) \, dx &=\frac{b \int \tan \left (a+b x+c x^2\right ) \, dx}{2 c}+\int x \tan \left (a+b x+c x^2\right ) \, dx\\ &=-\frac{\log \left (\cos \left (a+b x+c x^2\right )\right )}{2 c}\\ \end{align*}
Mathematica [A] time = 0.676132, size = 18, normalized size = 0.95 \[ -\frac{\log (\cos (a+x (b+c x)))}{2 c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.03, size = 18, normalized size = 1. \begin{align*} -{\frac{\ln \left ( \cos \left ( c{x}^{2}+bx+a \right ) \right ) }{2\,c}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 1.20718, size = 112, normalized size = 5.89 \begin{align*} -\frac{\log \left (\cos \left (2 \, c x^{2}\right )^{2} + 2 \, \cos \left (2 \, c x^{2}\right ) \cos \left (2 \, b x + 2 \, a\right ) + \cos \left (2 \, b x + 2 \, a\right )^{2} + \sin \left (2 \, c x^{2}\right )^{2} - 2 \, \sin \left (2 \, c x^{2}\right ) \sin \left (2 \, b x + 2 \, a\right ) + \sin \left (2 \, b x + 2 \, a\right )^{2}\right )}{4 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.37484, size = 59, normalized size = 3.11 \begin{align*} -\frac{\log \left (\frac{1}{\tan \left (c x^{2} + b x + a\right )^{2} + 1}\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*} \frac{\int b \tan{\left (a + b x + c x^{2} \right )}\, dx + \int 2 c x \tan{\left (a + b x + c x^{2} \right )}\, dx}{2 c} \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 x \tan \left (c x^{2} + b x + a\right ) + \frac{b \tan \left (c x^{2} + b x + a\right )}{2 \, c}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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