Optimal. Leaf size=26 \[ \frac{a x^2}{2}+\frac{b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]
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Rubi [A] time = 0.024407, 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, 5436, 3770} \[ \frac{a x^2}{2}+\frac{b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 5436
Rule 3770
Rubi steps
\begin{align*} \int x \left (a+b \text{sech}\left (c+d x^2\right )\right ) \, dx &=\int \left (a x+b x \text{sech}\left (c+d x^2\right )\right ) \, dx\\ &=\frac{a x^2}{2}+b \int x \text{sech}\left (c+d x^2\right ) \, dx\\ &=\frac{a x^2}{2}+\frac{1}{2} b \operatorname{Subst}\left (\int \text{sech}(c+d x) \, dx,x,x^2\right )\\ &=\frac{a x^2}{2}+\frac{b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d}\\ \end{align*}
Mathematica [A] time = 0.0138505, size = 26, normalized size = 1. \[ \frac{a x^2}{2}+\frac{b \tan ^{-1}\left (\sinh \left (c+d x^2\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 30, normalized size = 1.2 \begin{align*}{\frac{a{x}^{2}}{2}}+{\frac{b\arctan \left ( \sinh \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 = 1.17881, size = 30, normalized size = 1.15 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{b \arctan \left (\sinh \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 = 2.40488, size = 88, normalized size = 3.38 \begin{align*} \frac{a d x^{2} + 2 \, b \arctan \left (\cosh \left (d x^{2} + c\right ) + \sinh \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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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x \left (a + b \operatorname{sech}{\left (c + d x^{2} \right )}\right )\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13095, size = 38, normalized size = 1.46 \begin{align*} \frac{{\left (d x^{2} + c\right )} a}{2 \, d} + \frac{b \arctan \left (e^{\left (d x^{2} + c\right )}\right )}{d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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