Optimal. Leaf size=70 \[ \frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a+1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)}{a}-\frac{\text{sech}^{-1}(a+b x)}{x} \]
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Rubi [A] time = 0.105885, antiderivative size = 70, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 10, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5, Rules used = {6321, 5468, 3783, 2659, 208} \[ \frac{2 b \tanh ^{-1}\left (\frac{\sqrt{a+1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)}{a}-\frac{\text{sech}^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 3783
Rule 2659
Rule 208
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a+b x)}{x^2} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{x \text{sech}(x) \tanh (x)}{(-a+\text{sech}(x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)}{x}+b \operatorname{Subst}\left (\int \frac{1}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{b \text{sech}^{-1}(a+b x)}{a}-\frac{\text{sech}^{-1}(a+b x)}{x}+\frac{b \operatorname{Subst}\left (\int \frac{1}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)}{a}-\frac{\text{sech}^{-1}(a+b x)}{x}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{1}{1-a-(1+a) x^2} \, dx,x,\tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)}{a}-\frac{\text{sech}^{-1}(a+b x)}{x}+\frac{2 b \tanh ^{-1}\left (\frac{\sqrt{1+a} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a}}\right )}{a \sqrt{1-a^2}}\\ \end{align*}
Mathematica [B] time = 0.215134, size = 244, normalized size = 3.49 \[ \frac{b \left (\sqrt{1-a^2} \log (a+b x)-\sqrt{1-a^2} \log \left (a \sqrt{-\frac{a+b x-1}{a+b x+1}}+b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{-\frac{a+b x-1}{a+b x+1}}+1\right )+\log \left (\sqrt{1-a^2} a \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{1-a^2} b x \sqrt{-\frac{a+b x-1}{a+b x+1}}+\sqrt{1-a^2} \sqrt{-\frac{a+b x-1}{a+b x+1}}-a^2-a b x+1\right )-\log (x)\right )}{a \sqrt{1-a^2}}-\frac{\text{sech}^{-1}(a+b x)}{x} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.252, size = 542, normalized size = 7.7 \begin{align*} -{\frac{{\rm arcsech} \left (bx+a\right )}{x}}-{\frac{a{b}^{2}x}{ \left ( 1+a \right ) \left ( a-1 \right ) }\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}}{\it Artanh} \left ({\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}}-{\frac{{a}^{2}b}{ \left ( 1+a \right ) \left ( a-1 \right ) }\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}}{\it Artanh} \left ({\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}}-{\frac{{b}^{2}x}{a \left ( a-1 \right ) \left ( 1+a \right ) }\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}}\ln \left ( 2\,{\frac{\sqrt{-{a}^{2}+1}\sqrt{1- \left ( bx+a \right ) ^{2}}-a \left ( bx+a \right ) +1}{bx}} \right ) \sqrt{-{a}^{2}+1}{\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}}+{\frac{{b}^{2}x}{a \left ( a-1 \right ) \left ( 1+a \right ) }\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}}{\it Artanh} \left ({\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}}-{\frac{b}{ \left ( 1+a \right ) \left ( a-1 \right ) }\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}}\ln \left ( 2\,{\frac{\sqrt{-{a}^{2}+1}\sqrt{1- \left ( bx+a \right ) ^{2}}-a \left ( bx+a \right ) +1}{bx}} \right ) \sqrt{-{a}^{2}+1}{\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}}+{\frac{b}{ \left ( 1+a \right ) \left ( a-1 \right ) }\sqrt{-{\frac{bx+a-1}{bx+a}}}\sqrt{{\frac{bx+a+1}{bx+a}}}{\it Artanh} \left ({\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}} \right ){\frac{1}{\sqrt{1- \left ( bx+a \right ) ^{2}}}}} \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{b \log \left (x\right )}{a^{3} - a} - \frac{{\left (a^{2} b - a b\right )} x \log \left (b x + a + 1\right ) +{\left (a^{2} b + a b\right )} x \log \left (-b x - a + 1\right ) + 2 \,{\left (a^{3} - a\right )} \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 \,{\left (a^{3} +{\left (a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) - 2 \,{\left (a^{3} - a\right )} \log \left (b x + a\right )}{2 \,{\left (a^{3} - a\right )} x} - \int \frac{b^{2} x + a b}{b^{2} x^{3} + 2 \, a b x^{2} +{\left (a^{2} - 1\right )} x +{\left (b^{2} x^{3} + 2 \, a b x^{2} +{\left (a^{2} - 1\right )} x\right )} e^{\left (\frac{1}{2} \, \log \left (b x + a + 1\right ) + \frac{1}{2} \, \log \left (-b x - a + 1\right )\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.19925, size = 1462, normalized size = 20.89 \begin{align*} \left [-\frac{{\left (a^{2} - 1\right )} 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}{x}\right ) -{\left (a^{2} - 1\right )} 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}{x}\right ) + \sqrt{-a^{2} + 1} b x \log \left (\frac{{\left (2 \, a^{2} - 1\right )} b^{2} x^{2} + 2 \, a^{4} + 4 \,{\left (a^{3} - a\right )} b x - 4 \, a^{2} - 2 \,{\left (a b^{2} x^{2} + a^{3} +{\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt{-a^{2} + 1} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}} + 2}{x^{2}}\right ) + 2 \,{\left (a^{3} - a\right )} \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 )}{2 \,{\left (a^{3} - a\right )} x}, -\frac{{\left (a^{2} - 1\right )} 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}{x}\right ) -{\left (a^{2} - 1\right )} 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}{x}\right ) + 2 \, \sqrt{a^{2} - 1} b x \arctan \left (\frac{{\left (a b^{2} x^{2} + a^{3} +{\left (2 \, a^{2} - 1\right )} b x - a\right )} \sqrt{a^{2} - 1} \sqrt{-\frac{b^{2} x^{2} + 2 \, a b x + a^{2} - 1}{b^{2} x^{2} + 2 \, a b x + a^{2}}}}{{\left (a^{2} - 1\right )} b^{2} x^{2} + a^{4} + 2 \,{\left (a^{3} - a\right )} b x - 2 \, a^{2} + 1}\right ) + 2 \,{\left (a^{3} - a\right )} \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 )}{2 \,{\left (a^{3} - a\right )} x}\right ] \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 \frac{\operatorname{asech}{\left (a + b x \right )}}{x^{2}}\, 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 \frac{\operatorname{arsech}\left (b x + a\right )}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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