Optimal. Leaf size=45 \[ \frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b \sqrt{1-c x}}{2 c^2 \sqrt{\frac{1}{c x+1}}} \]
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Rubi [A] time = 0.0136379, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {6283, 74} \[ \frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )-\frac{b \sqrt{1-c x}}{2 c^2 \sqrt{\frac{1}{c x+1}}} \]
Antiderivative was successfully verified.
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Rule 6283
Rule 74
Rubi steps
\begin{align*} \int x \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )+\frac{1}{2} \left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{x}{\sqrt{1-c x} \sqrt{1+c x}} \, dx\\ &=-\frac{b \sqrt{1-c x}}{2 c^2 \sqrt{\frac{1}{1+c x}}}+\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )\\ \end{align*}
Mathematica [A] time = 0.0515405, size = 57, normalized size = 1.27 \[ \frac{a x^2}{2}+b \left (-\frac{1}{2 c^2}-\frac{x}{2 c}\right ) \sqrt{\frac{1-c x}{c x+1}}+\frac{1}{2} b x^2 \text{sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.178, size = 63, normalized size = 1.4 \begin{align*}{\frac{1}{{c}^{2}} \left ({\frac{{c}^{2}{x}^{2}a}{2}}+b \left ({\frac{{\rm arcsech} \left (cx\right ){c}^{2}{x}^{2}}{2}}-{\frac{cx}{2}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.979394, size = 49, normalized size = 1.09 \begin{align*} \frac{1}{2} \, a x^{2} + \frac{1}{2} \,{\left (x^{2} \operatorname{arsech}\left (c x\right ) - \frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c}\right )} b \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90979, size = 157, normalized size = 3.49 \begin{align*} \frac{b c x^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) + a c x^{2} - b x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}}}{2 \, c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.31947, size = 46, normalized size = 1.02 \begin{align*} \begin{cases} \frac{a x^{2}}{2} + \frac{b x^{2} \operatorname{asech}{\left (c x \right )}}{2} - \frac{b \sqrt{- c^{2} x^{2} + 1}}{2 c^{2}} & \text{for}\: c \neq 0 \\\frac{x^{2} \left (a + \infty b\right )}{2} & \text{otherwise} \end{cases} \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{\left (b \operatorname{arsech}\left (c x\right ) + a\right )} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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