Optimal. Leaf size=170 \[ \text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+\text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {PolyLog}\left (2,\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {1}{2} \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(a+b x)}\right ) \]
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Rubi [A]
time = 0.21, antiderivative size = 170, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 8, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.800, Rules used = {6456, 5714,
5689, 3799, 2221, 2317, 2438, 5681} \begin {gather*} \text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{\sqrt {1-a^2}+1}\right )-\frac {1}{2} \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )-\text {sech}^{-1}(a+b x) \log \left (e^{2 \text {sech}^{-1}(a+b x)}+1\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2317
Rule 2438
Rule 3799
Rule 5681
Rule 5689
Rule 5714
Rule 6456
Rubi steps
\begin {align*} \int \frac {\text {sech}^{-1}(a+b x)}{x} \, dx &=-\text {Subst}\left (\int \frac {x \text {sech}(x) \tanh (x)}{-a+\text {sech}(x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\text {Subst}\left (\int \frac {x \tanh (x)}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\left (a \text {Subst}\left (\int \frac {x \sinh (x)}{1-a \cosh (x)} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )-\text {Subst}\left (\int x \tanh (x) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=-\left (2 \text {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(a+b x)\right )\right )-a \text {Subst}\left (\int \frac {e^x x}{1-\sqrt {1-a^2}-a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )-a \text {Subst}\left (\int \frac {e^x x}{1+\sqrt {1-a^2}-a e^x} \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )-\text {Subst}\left (\int \log \left (1-\frac {a e^x}{1-\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )-\text {Subst}\left (\int \log \left (1-\frac {a e^x}{1+\sqrt {1-a^2}}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )+\text {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(a+b x)\right )\\ &=\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+\frac {1}{2} \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(a+b x)}\right )-\text {Subst}\left (\int \frac {\log \left (1-\frac {a x}{1-\sqrt {1-a^2}}\right )}{x} \, dx,x,e^{\text {sech}^{-1}(a+b x)}\right )-\text {Subst}\left (\int \frac {\log \left (1-\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) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{2 \text {sech}^{-1}(a+b x)}\right )+\text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1-\sqrt {1-a^2}}\right )+\text {Li}_2\left (\frac {a e^{\text {sech}^{-1}(a+b x)}}{1+\sqrt {1-a^2}}\right )-\frac {1}{2} \text {Li}_2\left (-e^{2 \text {sech}^{-1}(a+b x)}\right )\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 0.15, size = 332, normalized size = 1.95 \begin {gather*} -4 i \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \tanh ^{-1}\left (\frac {(1+a) \tanh \left (\frac {1}{2} \text {sech}^{-1}(a+b x)\right )}{\sqrt {1-a^2}}\right )-\text {sech}^{-1}(a+b x) \log \left (1+e^{-2 \text {sech}^{-1}(a+b x)}\right )+\text {sech}^{-1}(a+b x) \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )+2 i \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1+\frac {\left (-1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )+\text {sech}^{-1}(a+b x) \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )-2 i \text {ArcSin}\left (\frac {\sqrt {\frac {-1+a}{a}}}{\sqrt {2}}\right ) \log \left (1-\frac {\left (1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )+\frac {1}{2} \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(a+b x)}\right )-\text {PolyLog}\left (2,-\frac {\left (-1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right )-\text {PolyLog}\left (2,\frac {\left (1+\sqrt {1-a^2}\right ) e^{-\text {sech}^{-1}(a+b x)}}{a}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [C] Result contains complex when optimal does not.
time = 0.89, size = 882, normalized size = 5.19
method | result | size |
derivativedivides | \(-\mathrm {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\mathrm {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\dilog \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\dilog \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )+\frac {\mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2}+\frac {\mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2}-\frac {\sqrt {-a^{2}+1}\, \mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2 \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2 a^{2}-2}+\dilog \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )+\dilog \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )+\frac {\left (a^{2}+\sqrt {-a^{2}+1}-1\right ) \mathrm {arcsech}\left (b x +a \right ) \left (\ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right ) a^{2}+\ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right ) a^{2}-2 \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )-2 \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right ) \sqrt {-a^{2}+1}\right )}{2 a^{2} \left (a^{2}-1\right )}\) | \(882\) |
default | \(-\mathrm {arcsech}\left (b x +a \right ) \ln \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\mathrm {arcsech}\left (b x +a \right ) \ln \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\dilog \left (1+i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )-\dilog \left (1-i \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )\right )+\frac {\mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2}+\frac {\mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2}-\frac {\sqrt {-a^{2}+1}\, \mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )}{2 \left (a^{2}-1\right )}+\frac {\sqrt {-a^{2}+1}\, \mathrm {arcsech}\left (b x +a \right ) \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )}{2 a^{2}-2}+\dilog \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )+\dilog \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right )+\frac {\left (a^{2}+\sqrt {-a^{2}+1}-1\right ) \mathrm {arcsech}\left (b x +a \right ) \left (\ln \left (\frac {-a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}+1}{1+\sqrt {-a^{2}+1}}\right ) a^{2}+\ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right ) a^{2}-2 \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right )-2 \ln \left (\frac {a \left (\frac {1}{b x +a}+\sqrt {\frac {1}{b x +a}-1}\, \sqrt {\frac {1}{b x +a}+1}\right )+\sqrt {-a^{2}+1}-1}{-1+\sqrt {-a^{2}+1}}\right ) \sqrt {-a^{2}+1}\right )}{2 a^{2} \left (a^{2}-1\right )}\) | \(882\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\operatorname {asech}{\left (a + b x \right )}}{x}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\mathrm {acosh}\left (\frac {1}{a+b\,x}\right )}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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