Optimal. Leaf size=378 \[ 3 \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-6 \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-6 \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 \text{PolyLog}\left (4,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 \text{PolyLog}\left (4,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}(a+b x)}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,-e^{2 \text{sech}^{-1}(a+b x)}\right )-\frac{3}{4} \text{PolyLog}\left (4,-e^{2 \text{sech}^{-1}(a+b x)}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (e^{2 \text{sech}^{-1}(a+b x)}+1\right ) \]
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Rubi [A] time = 0.51077, antiderivative size = 378, normalized size of antiderivative = 1., number of steps used = 20, number of rules used = 10, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.833, Rules used = {6321, 5595, 5570, 3718, 2190, 2531, 6609, 2282, 6589, 5562} \[ 3 \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-6 \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-6 \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 \text{PolyLog}\left (4,\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 \text{PolyLog}\left (4,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,-e^{2 \text{sech}^{-1}(a+b x)}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,-e^{2 \text{sech}^{-1}(a+b x)}\right )-\frac{3}{4} \text{PolyLog}\left (4,-e^{2 \text{sech}^{-1}(a+b x)}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (e^{2 \text{sech}^{-1}(a+b x)}+1\right ) \]
Antiderivative was successfully verified.
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Rule 6321
Rule 5595
Rule 5570
Rule 3718
Rule 2190
Rule 2531
Rule 6609
Rule 2282
Rule 6589
Rule 5562
Rubi steps
\begin{align*} \int \frac{\text{sech}^{-1}(a+b x)^3}{x} \, dx &=-\operatorname{Subst}\left (\int \frac{x^3 \text{sech}(x) \tanh (x)}{-a+\text{sech}(x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\operatorname{Subst}\left (\int \frac{x^3 \tanh (x)}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\left (a \operatorname{Subst}\left (\int \frac{x^3 \sinh (x)}{1-a \cosh (x)} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )-\operatorname{Subst}\left (\int x^3 \tanh (x) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=-\left (2 \operatorname{Subst}\left (\int \frac{e^{2 x} x^3}{1+e^{2 x}} \, dx,x,\text{sech}^{-1}(a+b x)\right )\right )-a \operatorname{Subst}\left (\int \frac{e^x x^3}{1-\sqrt{1-a^2}-a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )-a \operatorname{Subst}\left (\int \frac{e^x x^3}{1+\sqrt{1-a^2}-a e^x} \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text{sech}^{-1}(a+b x)}\right )-3 \operatorname{Subst}\left (\int x^2 \log \left (1-\frac{a e^x}{1-\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )-3 \operatorname{Subst}\left (\int x^2 \log \left (1-\frac{a e^x}{1+\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )+3 \operatorname{Subst}\left (\int x^2 \log \left (1+e^{2 x}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text{sech}^{-1}(a+b x)}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )+3 \operatorname{Subst}\left (\int x \text{Li}_2\left (-e^{2 x}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )-6 \operatorname{Subst}\left (\int x \text{Li}_2\left (\frac{a e^x}{1-\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )-6 \operatorname{Subst}\left (\int x \text{Li}_2\left (\frac{a e^x}{1+\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text{sech}^{-1}(a+b x)}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )-6 \text{sech}^{-1}(a+b x) \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-6 \text{sech}^{-1}(a+b x) \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x) \text{Li}_3\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )-\frac{3}{2} \operatorname{Subst}\left (\int \text{Li}_3\left (-e^{2 x}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )+6 \operatorname{Subst}\left (\int \text{Li}_3\left (\frac{a e^x}{1-\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )+6 \operatorname{Subst}\left (\int \text{Li}_3\left (\frac{a e^x}{1+\sqrt{1-a^2}}\right ) \, dx,x,\text{sech}^{-1}(a+b x)\right )\\ &=\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text{sech}^{-1}(a+b x)}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )-6 \text{sech}^{-1}(a+b x) \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-6 \text{sech}^{-1}(a+b x) \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x) \text{Li}_3\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )-\frac{3}{4} \operatorname{Subst}\left (\int \frac{\text{Li}_3(-x)}{x} \, dx,x,e^{2 \text{sech}^{-1}(a+b x)}\right )+6 \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{a x}{1-\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )+6 \operatorname{Subst}\left (\int \frac{\text{Li}_3\left (\frac{a x}{1+\sqrt{1-a^2}}\right )}{x} \, dx,x,e^{\text{sech}^{-1}(a+b x)}\right )\\ &=\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\text{sech}^{-1}(a+b x)^3 \log \left (1+e^{2 \text{sech}^{-1}(a+b x)}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{Li}_2\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )-6 \text{sech}^{-1}(a+b x) \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )-6 \text{sech}^{-1}(a+b x) \text{Li}_3\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x) \text{Li}_3\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )+6 \text{Li}_4\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1-\sqrt{1-a^2}}\right )+6 \text{Li}_4\left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{1+\sqrt{1-a^2}}\right )-\frac{3}{4} \text{Li}_4\left (-e^{2 \text{sech}^{-1}(a+b x)}\right )\\ \end{align*}
Mathematica [A] time = 0.318892, size = 384, normalized size = 1.02 \[ 3 \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )+3 \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-6 \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )-6 \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+6 \text{PolyLog}\left (4,-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}-1}\right )+6 \text{PolyLog}\left (4,\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x)^2 \text{PolyLog}\left (2,-e^{-2 \text{sech}^{-1}(a+b x)}\right )+\frac{3}{2} \text{sech}^{-1}(a+b x) \text{PolyLog}\left (3,-e^{-2 \text{sech}^{-1}(a+b x)}\right )+\frac{3}{4} \text{PolyLog}\left (4,-e^{-2 \text{sech}^{-1}(a+b x)}\right )+\text{sech}^{-1}(a+b x)^3 \log \left (\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}-1}+1\right )+\text{sech}^{-1}(a+b x)^3 \log \left (1-\frac{a e^{\text{sech}^{-1}(a+b x)}}{\sqrt{1-a^2}+1}\right )-\frac{1}{2} \text{sech}^{-1}(a+b x)^4-\text{sech}^{-1}(a+b x)^3 \log \left (e^{-2 \text{sech}^{-1}(a+b x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.373, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ({\rm arcsech} \left (bx+a\right ) \right ) ^{3}}{x}}\, 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*} \int \frac{\operatorname{arsech}\left (b x + a\right )^{3}}{x}\,{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}, 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}\, 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}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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