Optimal. Leaf size=44 \[ \frac{(a+b x) \text{sech}^{-1}(a+b x)}{b}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{-a-b x+1}{a+b x+1}}\right )}{b} \]
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Rubi [A] time = 0.0556131, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 6, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6313, 1961, 12, 203} \[ \frac{(a+b x) \text{sech}^{-1}(a+b x)}{b}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{-a-b x+1}{a+b x+1}}\right )}{b} \]
Antiderivative was successfully verified.
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Rule 6313
Rule 1961
Rule 12
Rule 203
Rubi steps
\begin{align*} \int \text{sech}^{-1}(a+b x) \, dx &=\frac{(a+b x) \text{sech}^{-1}(a+b x)}{b}+\int \frac{\sqrt{\frac{1-a-b x}{1+a+b x}}}{1-a-b x} \, dx\\ &=\frac{(a+b x) \text{sech}^{-1}(a+b x)}{b}-(4 b) \operatorname{Subst}\left (\int \frac{1}{2 b^2 \left (1+x^2\right )} \, dx,x,\sqrt{\frac{1-a-b x}{1+a+b x}}\right )\\ &=\frac{(a+b x) \text{sech}^{-1}(a+b x)}{b}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\sqrt{\frac{1-a-b x}{1+a+b x}}\right )}{b}\\ &=\frac{(a+b x) \text{sech}^{-1}(a+b x)}{b}-\frac{2 \tan ^{-1}\left (\sqrt{\frac{1-a-b x}{1+a+b x}}\right )}{b}\\ \end{align*}
Mathematica [B] time = 0.195596, size = 95, normalized size = 2.16 \[ x \text{sech}^{-1}(a+b x)+\frac{2 \sqrt{-\frac{a+b x-1}{a+b x+1}} \left (a \tan ^{-1}\left (\sqrt{\frac{a+b x-1}{a+b x+1}}\right )-\sinh ^{-1}\left (\frac{\sqrt{a+b x-1}}{\sqrt{2}}\right )\right )}{b \sqrt{\frac{a+b x-1}{a+b x+1}}} \]
Warning: Unable to verify antiderivative.
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Maple [A] time = 0.213, size = 50, normalized size = 1.1 \begin{align*} x{\rm arcsech} \left (bx+a\right )+{\frac{{\rm arcsech} \left (bx+a\right )a}{b}}-{\frac{1}{b}\arctan \left ( \sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.9777, size = 42, normalized size = 0.95 \begin{align*} \frac{{\left (b x + a\right )} \operatorname{arsech}\left (b x + a\right ) - \arctan \left (\sqrt{\frac{1}{{\left (b x + a\right )}^{2}} - 1}\right )}{b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.1473, size = 567, normalized size = 12.89 \begin{align*} \frac{2 \, b x \log \left (\frac{{\left (b x + a\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{b x + a}\right ) + a \log \left (\frac{{\left (b x + a\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 1}{x}\right ) - a \log \left (\frac{{\left (b x + a\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} - 1}{x}\right ) - 2 \, \arctan \left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}\right )}{2 \, b} \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 \operatorname{asech}{\left (a + b x \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 \operatorname{arsech}\left (b x + a\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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