Optimal. Leaf size=107 \[ -\frac{a^2 \text{sech}^{-1}(a+b x)}{2 b^2}-\frac{\sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{2 b^2}+\frac{a \tan ^{-1}\left (\frac{\sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{b^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a+b x) \]
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Rubi [A] time = 0.0617878, antiderivative size = 107, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 8, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6321, 5468, 3773, 3770, 3767, 8} \[ -\frac{a^2 \text{sech}^{-1}(a+b x)}{2 b^2}-\frac{\sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{2 b^2}+\frac{a \tan ^{-1}\left (\frac{\sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1)}{a+b x}\right )}{b^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a+b x) \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 3773
Rule 3770
Rule 3767
Rule 8
Rubi steps
\begin{align*} \int x \text{sech}^{-1}(a+b x) \, dx &=-\frac{\operatorname{Subst}\left (\int x \text{sech}(x) (-a+\text{sech}(x)) \tanh (x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^2}\\ &=\frac{1}{2} x^2 \text{sech}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int (-a+\text{sech}(x))^2 \, dx,x,\text{sech}^{-1}(a+b x)\right )}{2 b^2}\\ &=-\frac{a^2 \text{sech}^{-1}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a+b x)-\frac{\operatorname{Subst}\left (\int \text{sech}^2(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{2 b^2}+\frac{a \operatorname{Subst}\left (\int \text{sech}(x) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{b^2}\\ &=-\frac{a^2 \text{sech}^{-1}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a+b x)+\frac{a \tan ^{-1}\left (\frac{\sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2}-\frac{i \operatorname{Subst}\left (\int 1 \, dx,x,-i \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)\right )}{2 b^2}\\ &=-\frac{\sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{2 b^2}-\frac{a^2 \text{sech}^{-1}(a+b x)}{2 b^2}+\frac{1}{2} x^2 \text{sech}^{-1}(a+b x)+\frac{a \tan ^{-1}\left (\frac{\sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x)}{a+b x}\right )}{b^2}\\ \end{align*}
Mathematica [C] time = 0.161123, size = 176, normalized size = 1.64 \[ \frac{a^2 \log (a+b x)-a^2 \log \left (a \sqrt{-\frac{a+b x-1}{a+b x+1}}+b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{-\frac{a+b x-1}{a+b x+1}}+1\right )+b^2 x^2 \text{sech}^{-1}(a+b x)-\sqrt{-\frac{a+b x-1}{a+b x+1}} (a+b x+1)-2 i a \log \left (2 \sqrt{-\frac{a+b x-1}{a+b x+1}} (a+b x+1)-2 i (a+b x)\right )}{2 b^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.234, size = 111, normalized size = 1. \begin{align*}{\frac{1}{{b}^{2}} \left ({\frac{{\rm arcsech} \left (bx+a\right ) \left ( bx+a \right ) ^{2}}{2}}-{\rm arcsech} \left (bx+a\right )a \left ( bx+a \right ) -{\frac{bx+a}{2}\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}} \left ( 2\,a\arcsin \left ( bx+a \right ) +\sqrt{1- \left ( bx+a \right ) ^{2}} \right ){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \frac{2 \, b^{2} x^{2} \log \left (\sqrt{b x + a + 1} \sqrt{-b x - a + 1} b x + \sqrt{b x + a + 1} \sqrt{-b x - a + 1} a + b x + a\right ) - 2 \, b^{2} x^{2} \log \left (b x + a\right ) -{\left (a^{2} + 2 \, a + 1\right )} \log \left (b x + a + 1\right ) - 2 \,{\left (b^{2} x^{2} - a^{2}\right )} \log \left (b x + a\right ) -{\left (a^{2} - 2 \, a + 1\right )} \log \left (-b x - a + 1\right )}{4 \, b^{2}} + \int \frac{b^{2} x^{3} + a b x^{2}}{2 \,{\left (b^{2} x^{2} + 2 \, a b x + a^{2} +{\left (b^{2} x^{2} + 2 \, a b x + a^{2} - 1\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (-b x - a + 1\right )\right )} - 1\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.2629, size = 687, normalized size = 6.42 \begin{align*} \frac{2 \, b^{2} x^{2} \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^{2} \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^{2} \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 ) + 4 \, a \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 \,{\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}}}}{4 \, b^{2}} \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 \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 x \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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