Optimal. Leaf size=224 \[ \frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x} \]
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Rubi [A] time = 0.388906, antiderivative size = 224, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.667, Rules used = {6321, 5468, 4191, 3320, 2264, 2190, 2279, 2391} \[ \frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x} \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 4191
Rule 3320
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a+b x)^2}{x^2} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{x^2 \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)^2}{x}+(2 b) \operatorname{Subst}\left (\int \frac{x}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^2}{x}+(2 b) \operatorname{Subst}\left (\int \left (-\frac{x}{a}+\frac{x}{a (1-a \cosh (x))}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{x}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^x x}{-a+2 e^x-a e^{2 x}} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}-\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^x x}{2-2 \sqrt{1-a^2}-2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a^2}}+\frac{(4 b) \operatorname{Subst}\left (\int \frac{e^x x}{2+2 \sqrt{1-a^2}-2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a^2}}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{(2 b) \operatorname{Subst}\left (\int \log \left (1-\frac{2 a e^x}{2-2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \sqrt{1-a^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \log \left (1-\frac{2 a e^x}{2+2 \sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \sqrt{1-a^2}}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 a x}{2-2 \sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a \sqrt{1-a^2}}+\frac{(2 b) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{2 a x}{2+2 \sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a \sqrt{1-a^2}}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^2}{a}-\frac{\text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{2 b \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{2 b \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}\\ \end{align*}
Mathematica [C] time = 2.62484, size = 678, normalized size = 3.03 \[ \frac{-\frac{(a+b x) \text{sech}^{-1}(a+b x)^2}{x}+\frac{2 b \left (i \left (\text{PolyLog}\left (2,\frac{\left (-1-i \sqrt{a^2-1}\right ) \left (-i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}{a \left (i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}\right )-\text{PolyLog}\left (2,\frac{\left (\sqrt{a^2-1}+i\right ) \left (-i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}{a \left (\sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )-i (a-1)\right )}\right )\right )+2 \text{sech}^{-1}(a+b x) \tan ^{-1}\left (\frac{(a-1) \coth \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )-2 i \cos ^{-1}\left (\frac{1}{a}\right ) \tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )-\log \left (-\frac{(a-1) \left (-i \sqrt{a^2-1}+a+1\right ) \left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )-1\right )}{a \left (i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}\right ) \left (2 \tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )+\cos ^{-1}\left (\frac{1}{a}\right )\right )-\log \left (\frac{(a-1) \left (i \sqrt{a^2-1}+a+1\right ) \left (\tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+1\right )}{a \left (i \sqrt{a^2-1} \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )+a-1\right )}\right ) \left (\cos ^{-1}\left (\frac{1}{a}\right )-2 \tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )\right )+\log \left (\frac{\sqrt{a^2-1} e^{-\frac{1}{2} \text{sech}^{-1}(a+b x)}}{\sqrt{2} \sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right ) \left (2 \left (\tan ^{-1}\left (\frac{(a-1) \coth \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )+\tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )\right )+\cos ^{-1}\left (\frac{1}{a}\right )\right )+\log \left (\frac{\sqrt{a^2-1} e^{\frac{1}{2} \text{sech}^{-1}(a+b x)}}{\sqrt{2} \sqrt{a} \sqrt{-\frac{b x}{a+b x}}}\right ) \left (\cos ^{-1}\left (\frac{1}{a}\right )-2 \left (\tan ^{-1}\left (\frac{(a-1) \coth \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )+\tan ^{-1}\left (\frac{(a+1) \tanh \left (\frac{1}{2} \text{sech}^{-1}(a+b x)\right )}{\sqrt{a^2-1}}\right )\right )\right )\right )}{\sqrt{a^2-1}}}{a} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.357, size = 367, normalized size = 1.6 \begin{align*} -{\frac{b \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{2}}{a}}-{\frac{ \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{2}}{x}}+2\,{\frac{b\sqrt{-{a}^{2}+1}{\rm arcsech} \left (bx+a\right )}{a \left ({a}^{2}-1 \right ) }\ln \left ({\frac{-a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}+1}{1+\sqrt{-{a}^{2}+1}}} \right ) }-2\,{\frac{b\sqrt{-{a}^{2}+1}{\rm arcsech} \left (bx+a\right )}{a \left ({a}^{2}-1 \right ) }\ln \left ({\frac{a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}-1}{-1+\sqrt{-{a}^{2}+1}}} \right ) }+2\,{\frac{b\sqrt{-{a}^{2}+1}}{a \left ({a}^{2}-1 \right ) }{\it dilog} \left ({\frac{-a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}+1}{1+\sqrt{-{a}^{2}+1}}} \right ) }-2\,{\frac{b\sqrt{-{a}^{2}+1}}{a \left ({a}^{2}-1 \right ) }{\it dilog} \left ({\frac{a \left ( \left ( bx+a \right ) ^{-1}+\sqrt{ \left ( bx+a \right ) ^{-1}-1}\sqrt{ \left ( bx+a \right ) ^{-1}+1} \right ) +\sqrt{-{a}^{2}+1}-1}{-1+\sqrt{-{a}^{2}+1}}} \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{\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}}{x} - \int -\frac{2 \,{\left (2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1} \log \left (b x + a\right )^{2} + 2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )^{2} +{\left (b^{3} x^{3} + 2 \, a b^{2} x^{2} +{\left (a^{2} b - b\right )} x - 2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right ) -{\left ({\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \sqrt{b x + a + 1} \log \left (b x + a\right ) -{\left (2 \, b^{3} x^{3} + 4 \, a b^{2} x^{2} +{\left (2 \, a^{2} b - b\right )} x -{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + a^{3} +{\left (3 \, a^{2} b - b\right )} x - a\right )} \log \left (b x + a\right )\right )} \sqrt{b x + a + 1}\right )} \sqrt{-b x - a + 1}\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 )\right )}}{b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2} +{\left (b^{3} x^{5} + 3 \, a b^{2} x^{4} +{\left (3 \, a^{2} b - b\right )} x^{3} +{\left (a^{3} - a\right )} x^{2}\right )} \sqrt{b x + a + 1} \sqrt{-b x - a + 1}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\operatorname{arsech}\left (b x + a\right )^{2}}{x^{2}}, 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}^{2}{\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 )^{2}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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