Optimal. Leaf size=330 \[ \frac{6 b \text{sech}^{-1}(a+b x) \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{6 b \text{sech}^{-1}(a+b x) \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{6 b \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{6 b \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}+\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x} \]
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Rubi [A] time = 0.575216, antiderivative size = 330, normalized size of antiderivative = 1., number of steps used = 14, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {6321, 5468, 4191, 3320, 2264, 2190, 2531, 2282, 6589} \[ \frac{6 b \text{sech}^{-1}(a+b x) \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{6 b \text{sech}^{-1}(a+b x) \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{6 b \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{6 b \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}+\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )}{a \sqrt{1-a^2}}-\frac{b \text{sech}^{-1}(a+b x)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x} \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5468
Rule 4191
Rule 3320
Rule 2264
Rule 2190
Rule 2531
Rule 2282
Rule 6589
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a+b x)^3}{x^2} \, dx &=-\left (b \operatorname{Subst}\left (\int \frac{x^3 \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)^3}{x}+(3 b) \operatorname{Subst}\left (\int \frac{x^2}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{\text{sech}^{-1}(a+b x)^3}{x}+(3 b) \operatorname{Subst}\left (\int \left (-\frac{x^2}{a}+\frac{x^2}{a (1-a \cosh (x))}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\frac{b \text{sech}^{-1}(a+b x)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}+\frac{(3 b) \operatorname{Subst}\left (\int \frac{x^2}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{a}\\ &=-\frac{b \text{sech}^{-1}(a+b x)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{e^x x^2}{-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)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}-\frac{(6 b) \operatorname{Subst}\left (\int \frac{e^x x^2}{2-2 \sqrt{1-a^2}-2 a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )}{\sqrt{1-a^2}}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{e^x x^2}{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)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}+\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{(6 b) \operatorname{Subst}\left (\int x \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{(6 b) \operatorname{Subst}\left (\int x \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)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}+\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{6 b \text{sech}^{-1}(a+b x) \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{6 b \text{sech}^{-1}(a+b x) \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{(6 b) \operatorname{Subst}\left (\int \text{Li}_2\left (\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{(6 b) \operatorname{Subst}\left (\int \text{Li}_2\left (\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)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}+\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{6 b \text{sech}^{-1}(a+b x) \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{6 b \text{sech}^{-1}(a+b x) \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{(6 b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{1-\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )}{a \sqrt{1-a^2}}+\frac{(6 b) \operatorname{Subst}\left (\int \frac{\text{Li}_2\left (\frac{a x}{1+\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)^3}{a}-\frac{\text{sech}^{-1}(a+b x)^3}{x}+\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}-\frac{3 b \text{sech}^{-1}(a+b x)^2 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{6 b \text{sech}^{-1}(a+b x) \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{6 b \text{sech}^{-1}(a+b x) \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{6 b \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )}{a \sqrt{1-a^2}}+\frac{6 b \text{Li}_3\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 = 45.6533, size = 1849, normalized size = 5.6 \[ \text{result too large to display} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.458, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{3}}{{x}^{2}}}\, dx \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 )^{3}}{x} - \int \frac{8 \,{\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 )^{3} + 8 \,{\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 )^{3} - 3 \,{\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 )^{2} - 12 \,{\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} \sqrt{-b x - a + 1} \log \left (b x + a\right )^{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}\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^{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 )^{3}}{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}^{3}{\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 )^{3}}{x^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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