Optimal. Leaf size=26 \[ \frac{a x^2}{2}-\frac{b \log \left (\cos \left (c+d x^2\right )\right )}{2 d} \]
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Rubi [A] time = 0.0258382, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {14, 3747, 3475} \[ \frac{a x^2}{2}-\frac{b \log \left (\cos \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 3747
Rule 3475
Rubi steps
\begin{align*} \int x \left (a+b \tan \left (c+d x^2\right )\right ) \, dx &=\int \left (a x+b x \tan \left (c+d x^2\right )\right ) \, dx\\ &=\frac{a x^2}{2}+b \int x \tan \left (c+d x^2\right ) \, dx\\ &=\frac{a x^2}{2}+\frac{1}{2} b \operatorname{Subst}\left (\int \tan (c+d x) \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}-\frac{b \log \left (\cos \left (c+d x^2\right )\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0180102, size = 26, normalized size = 1. \[ \frac{a x^2}{2}-\frac{b \log \left (\cos \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 30, normalized size = 1.2 \begin{align*}{\frac{a{x}^{2}}{2}}-{\frac{b\ln \left ( \cos \left ( d{x}^{2}+c \right ) \right ) }{2\,d}}+{\frac{ac}{2\,d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.04563, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{b \log \left (\sec \left (d x^{2} + c\right )\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.6185, size = 72, normalized size = 2.77 \begin{align*} \frac{2 \, a d x^{2} - b \log \left (\frac{1}{\tan \left (d x^{2} + c\right )^{2} + 1}\right )}{4 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.292229, size = 36, normalized size = 1.38 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b \log{\left (\tan ^{2}{\left (c + d x^{2} \right )} + 1 \right )}}{4 d} & \text{for}\: d \neq 0 \\\frac{x^{2} \left (a + b \tan{\left (c \right )}\right )}{2} & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18515, size = 38, normalized size = 1.46 \begin{align*} \frac{{\left (d x^{2} + c\right )} a - b \log \left ({\left | \cos \left (d x^{2} + c\right ) \right |}\right )}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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