Optimal. Leaf size=40 \[ a x+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}+b x \text{sech}^{-1}(c x) \]
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Rubi [A] time = 0.0151617, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {6277, 216} \[ a x+\frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \sin ^{-1}(c x)}{c}+b x \text{sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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Rule 6277
Rule 216
Rubi steps
\begin{align*} \int \left (a+b \text{sech}^{-1}(c x)\right ) \, dx &=a x+b \int \text{sech}^{-1}(c x) \, dx\\ &=a x+b x \text{sech}^{-1}(c x)+\left (b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x}\right ) \int \frac{1}{\sqrt{1-c^2 x^2}} \, dx\\ &=a x+b x \text{sech}^{-1}(c x)+\frac{b \sqrt{\frac{1}{1+c x}} \sqrt{1+c x} \sin ^{-1}(c x)}{c}\\ \end{align*}
Mathematica [A] time = 0.0874106, size = 60, normalized size = 1.5 \[ a x-\frac{b \sqrt{\frac{1-c x}{c x+1}} \sqrt{1-c^2 x^2} \sin ^{-1}(c x)}{c (c x-1)}+b x \text{sech}^{-1}(c x) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.165, size = 42, normalized size = 1.1 \begin{align*} ax+bx{\rm arcsech} \left (cx\right )-{\frac{b}{c}\arctan \left ( \sqrt{-1+{\frac{1}{cx}}}\sqrt{1+{\frac{1}{cx}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.970528, size = 42, normalized size = 1.05 \begin{align*} a x + \frac{{\left (c x \operatorname{arsech}\left (c x\right ) - \arctan \left (\sqrt{\frac{1}{c^{2} x^{2}} - 1}\right )\right )} b}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.69404, size = 262, normalized size = 6.55 \begin{align*} \frac{a c x - b c \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 2 \, b \arctan \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) +{\left (b c x - b c\right )} \log \left (\frac{c x \sqrt{-\frac{c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right )}{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*} \int \left (a + b \operatorname{asech}{\left (c x \right )}\right )\, dx \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 b \operatorname{arsech}\left (c x\right ) + a\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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