Optimal. Leaf size=274 \[ 2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\text {sech}^{-1}(a+b x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )-\text {sech}^{-1}(a+b x)^2 \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right ) \]
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Rubi [A] time = 0.46, antiderivative size = 274, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 9, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.750, Rules used = {6321, 5595, 5570, 3718, 2190, 2531, 2282, 6589, 5562} \[ 2 \text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+2 \text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-2 \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 \text {PolyLog}\left (3,\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\text {sech}^{-1}(a+b x) \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(a+b x)}\right )+\text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\text {sech}^{-1}(a+b x)^2 \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right ) \]
Antiderivative was successfully verified.
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Rule 2190
Rule 2282
Rule 2531
Rule 3718
Rule 5562
Rule 5570
Rule 5595
Rule 6321
Rule 6589
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)^2}{x} \, dx &=-\operatorname {Subst}\left (\int \frac {x^2 \text {sech}(x) \tanh (x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\operatorname {Subst}\left (\int \frac {x^2 \tanh (x)}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\left (a \operatorname {Subst}\left (\int \frac {x^2 \sinh (x)}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )-\operatorname {Subst}\left (\int x^2 \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\left (2 \operatorname {Subst}\left (\int \frac {e^{2 x} x^2}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )-a \operatorname {Subst}\left (\int \frac {e^x x^2}{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^2}{1+\sqrt {1-a^2}-a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=\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 )+\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 )-\text {sech}^{-1}(a+b x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )-2 \operatorname {Subst}\left (\int x \log \left (1-\frac {a e^x}{1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )-2 \operatorname {Subst}\left (\int x \log \left (1-\frac {a e^x}{1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )+2 \operatorname {Subst}\left (\int x \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=\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 )+\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 )-\text {sech}^{-1}(a+b x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+2 \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 )+2 \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 )-\text {sech}^{-1}(a+b x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )-2 \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {a e^x}{1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )-2 \operatorname {Subst}\left (\int \text {Li}_2\left (\frac {a e^x}{1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )+\operatorname {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=\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 )+\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 )-\text {sech}^{-1}(a+b x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+2 \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 )+2 \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 )-\text {sech}^{-1}(a+b x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a+b x)}\right )-2 \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 )-2 \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 )\\ &=\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 )+\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 )-\text {sech}^{-1}(a+b x)^2 \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+2 \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 )+2 \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 )-\text {sech}^{-1}(a+b x) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )-2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )-2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )+\frac {1}{2} \text {Li}_3\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )\\ \end {align*}
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Mathematica [A] time = 0.30, size = 280, normalized size = 1.02 \[ 2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}-1}\right )+2 \text {sech}^{-1}(a+b x) \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-2 \text {Li}_3\left (-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}-1}\right )-2 \text {Li}_3\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\text {sech}^{-1}(a+b x)^2 \log \left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}-1}+1\right )+\text {sech}^{-1}(a+b x)^2 \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\text {sech}^{-1}(a+b x) \text {Li}_2\left (-e^{-2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {Li}_3\left (-e^{-2 \text {sech}^{-1}(a+b x)}\right )-\frac {2}{3} \text {sech}^{-1}(a+b x)^3-\text {sech}^{-1}(a+b x)^2 \log \left (e^{-2 \text {sech}^{-1}(a+b x)}+1\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.97, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.80, size = 0, normalized size = 0.00 \[ \int \frac {\mathrm {arcsech}\left (b x +a \right )^{2}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arsech}\left (b x + a\right )^{2}}{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}^2}{x} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {asech}^{2}{\left (a + b x \right )}}{x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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