Optimal. Leaf size=537 \[ -\frac{2 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \sqrt{1-a^2}}-\frac{b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{b^2 \log \left (\frac{x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}+\frac{b^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}-\frac{2 b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \sqrt{1-a^2}}-\frac{b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2} \]
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Rubi [A] time = 0.749842, antiderivative size = 537, normalized size of antiderivative = 1., number of steps used = 23, number of rules used = 11, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.917, Rules used = {6321, 5468, 4191, 3324, 3320, 2264, 2190, 2279, 2391, 2668, 31} \[ -\frac{2 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \sqrt{1-a^2}}-\frac{b^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{b^2 \log \left (\frac{x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}+\frac{b^2 \sqrt{\frac{-a-b x+1}{a+b x+1}} (a+b x+1) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}-\frac{2 b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \sqrt{1-a^2}}-\frac{b^2 \text{sech}^{-1}(a+b x) \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2} \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 4191
Rule 3324
Rule 3320
Rule 2264
Rule 2190
Rule 2279
Rule 2391
Rule 2668
Rule 31
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a+b x)^2}{x^3} \, dx &=-\left (b^2 \operatorname{Subst}\left (\int \frac{x^2 \text{sech}(x) \tanh (x)}{(-a+\text{sech}(x))^3} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+b^2 \operatorname{Subst}\left (\int \frac{x}{(-a+\text{sech}(x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+b^2 \operatorname{Subst}\left (\int \left (\frac{x}{a^2}+\frac{x}{a^2 (-1+a \cosh (x))^2}+\frac{2 x}{a^2 (-1+a \cosh (x))}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{x}{(-1+a \cosh (x))^2} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2}+\frac{\left (2 b^2\right ) \operatorname{Subst}\left (\int \frac{x}{-1+a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2}\\ &=\frac{b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+\frac{\left (4 b^2\right ) \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^2}-\frac{b^2 \operatorname{Subst}\left (\int \frac{x}{-1+a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a^2 \left (1-a^2\right )}+\frac{b^2 \operatorname{Subst}\left (\int \frac{\sinh (x)}{-1+a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a \left (1-a^2\right )}\\ &=\frac{b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \operatorname{Subst}\left (\int \frac{1}{-1+x} \, dx,x,\frac{a}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac{\left (2 b^2\right ) \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^2 \left (1-a^2\right )}+\frac{\left (4 b^2\right ) \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 )}{a \sqrt{1-a^2}}-\frac{\left (4 b^2\right ) \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 )}{a \sqrt{1-a^2}}\\ &=\frac{b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}-\frac{2 b^2 \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^2 \sqrt{1-a^2}}+\frac{2 b^2 \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^2 \sqrt{1-a^2}}+\frac{b^2 \log \left (\frac{x}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac{\left (2 b^2\right ) \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 )}{a \left (1-a^2\right )^{3/2}}+\frac{\left (2 b^2\right ) \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 )}{a \left (1-a^2\right )^{3/2}}-\frac{\left (2 b^2\right ) \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^2 \sqrt{1-a^2}}+\frac{\left (2 b^2\right ) \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^2 \sqrt{1-a^2}}\\ &=\frac{b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}-\frac{2 b^2 \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^2 \sqrt{1-a^2}}-\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \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^2 \sqrt{1-a^2}}+\frac{b^2 \log \left (\frac{x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}-\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}-\frac{\left (2 b^2\right ) \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^2 \sqrt{1-a^2}}+\frac{\left (2 b^2\right ) \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^2 \sqrt{1-a^2}}\\ &=\frac{b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}-\frac{2 b^2 \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^2 \sqrt{1-a^2}}-\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \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^2 \sqrt{1-a^2}}+\frac{b^2 \log \left (\frac{x}{a+b x}\right )}{a^2 \left (1-a^2\right )}-\frac{2 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{2 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}+\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}-\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}\\ &=\frac{b^2 \sqrt{\frac{1-a-b x}{1+a+b x}} (1+a+b x) \text{sech}^{-1}(a+b x)}{a \left (1-a^2\right ) (a+b x) \left (1-\frac{a}{a+b x}\right )}+\frac{b^2 \text{sech}^{-1}(a+b x)^2}{2 a^2}-\frac{\text{sech}^{-1}(a+b x)^2}{2 x^2}+\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}-\frac{2 b^2 \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^2 \sqrt{1-a^2}}-\frac{b^2 \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^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \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^2 \sqrt{1-a^2}}+\frac{b^2 \log \left (\frac{x}{a+b x}\right )}{a^2 \left (1-a^2\right )}+\frac{b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}-\frac{2 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}-\frac{b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \left (1-a^2\right )^{3/2}}+\frac{2 b^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a^2 \sqrt{1-a^2}}\\ \end{align*}
Mathematica [C] time = 7.76266, size = 1439, normalized size = 2.68 \[ \text{result too large to display} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.563, size = 1026, normalized size = 1.9 \begin{align*} \text{result too large to display} \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}}{2 \, x^{2}} - \int -\frac{4 \,{\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} + 4 \,{\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 - 4 \,{\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 (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} \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 - 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 )\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 )}{b^{3} x^{6} + 3 \, a b^{2} x^{5} +{\left (3 \, a^{2} b - b\right )} x^{4} +{\left (a^{3} - a\right )} x^{3} +{\left (b^{3} x^{6} + 3 \, a b^{2} x^{5} +{\left (3 \, a^{2} b - b\right )} x^{4} +{\left (a^{3} - a\right )} x^{3}\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^{3}}, 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^{3}}\, 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^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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