Optimal. Leaf size=65 \[ -\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}+\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )^2-\frac{b^2 \log (x)}{c^2} \]
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Rubi [A] time = 0.0752577, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {6285, 5451, 4184, 3475} \[ -\frac{b \sqrt{\frac{1-c x}{c x+1}} (c x+1) \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}+\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )^2-\frac{b^2 \log (x)}{c^2} \]
Antiderivative was successfully verified.
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Rule 6285
Rule 5451
Rule 4184
Rule 3475
Rubi steps
\begin{align*} \int x \left (a+b \text{sech}^{-1}(c x)\right )^2 \, dx &=-\frac{\operatorname{Subst}\left (\int (a+b x)^2 \text{sech}^2(x) \tanh (x) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^2}\\ &=\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )^2-\frac{b \operatorname{Subst}\left (\int (a+b x) \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^2}\\ &=-\frac{b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}+\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )^2+\frac{b^2 \operatorname{Subst}\left (\int \tanh (x) \, dx,x,\text{sech}^{-1}(c x)\right )}{c^2}\\ &=-\frac{b \sqrt{\frac{1-c x}{1+c x}} (1+c x) \left (a+b \text{sech}^{-1}(c x)\right )}{c^2}+\frac{1}{2} x^2 \left (a+b \text{sech}^{-1}(c x)\right )^2-\frac{b^2 \log (x)}{c^2}\\ \end{align*}
Mathematica [A] time = 0.208577, size = 112, normalized size = 1.72 \[ \frac{a \left (a c^2 x^2-2 b \sqrt{\frac{1-c x}{c x+1}} (c x+1)\right )-2 b \text{sech}^{-1}(c x) \left (b \sqrt{\frac{1-c x}{c x+1}} (c x+1)-a c^2 x^2\right )+b^2 c^2 x^2 \text{sech}^{-1}(c x)^2-2 b^2 \log (c x)}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.253, size = 168, normalized size = 2.6 \begin{align*}{\frac{{a}^{2}{x}^{2}}{2}}-{\frac{{b}^{2}{\rm arcsech} \left (cx\right )}{{c}^{2}}}-{\frac{{b}^{2}{\rm arcsech} \left (cx\right )x}{c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}}+{\frac{{x}^{2}{b}^{2} \left ({\rm arcsech} \left (cx\right ) \right ) ^{2}}{2}}+{\frac{{b}^{2}}{{c}^{2}}\ln \left ( 1+ \left ({\frac{1}{cx}}+\sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) ^{2} \right ) }+ab{\rm arcsech} \left (cx\right ){x}^{2}-{\frac{xab}{c}\sqrt{-{\frac{cx-1}{cx}}}\sqrt{{\frac{cx+1}{cx}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.01994, size = 113, normalized size = 1.74 \begin{align*} \frac{1}{2} \, b^{2} x^{2} \operatorname{arsech}\left (c x\right )^{2} + \frac{1}{2} \, a^{2} x^{2} +{\left (x^{2} \operatorname{arsech}\left (c x\right ) - \frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1}}{c}\right )} a b -{\left (\frac{x \sqrt{\frac{1}{c^{2} x^{2}} - 1} \operatorname{arsech}\left (c x\right )}{c} + \frac{\log \left (x\right )}{c^{2}}\right )} b^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.06572, size = 446, normalized size = 6.86 \begin{align*} \frac{b^{2} c^{2} x^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )^{2} + a^{2} c^{2} x^{2} - 2 \, a b c^{2} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, a b c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 2 \, b^{2} \log \left (x\right ) + 2 \,{\left (a b c^{2} x^{2} - b^{2} c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - a b c^{2}\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.94712, size = 99, normalized size = 1.52 \begin{align*} \begin{cases} \frac{a^{2} x^{2}}{2} + a b x^{2} \operatorname{asech}{\left (c x \right )} - \frac{a b \sqrt{- c^{2} x^{2} + 1}}{c^{2}} + \frac{b^{2} x^{2} \operatorname{asech}^{2}{\left (c x \right )}}{2} - \frac{b^{2} \sqrt{- c^{2} x^{2} + 1} \operatorname{asech}{\left (c x \right )}}{c^{2}} - \frac{b^{2} \log{\left (x \right )}}{c^{2}} & \text{for}\: c \neq 0 \\\frac{x^{2} \left (a + \infty b\right )^{2}}{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 )}^{2} x\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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