Optimal. Leaf size=83 \[ \frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {sech}^{-1}(c x)\right )^2 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )-b \left (a+b \text {sech}^{-1}(c x)\right ) \text {PolyLog}\left (2,-e^{2 \text {sech}^{-1}(c x)}\right )+\frac {1}{2} b^2 \text {PolyLog}\left (3,-e^{2 \text {sech}^{-1}(c x)}\right ) \]
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Rubi [A]
time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6420, 3799,
2221, 2611, 2320, 6724} \begin {gather*} -b \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-\log \left (e^{2 \text {sech}^{-1}(c x)}+1\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {1}{2} b^2 \text {Li}_3\left (-e^{2 \text {sech}^{-1}(c x)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 2221
Rule 2320
Rule 2611
Rule 3799
Rule 6420
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \text {sech}^{-1}(c x)\right )^2}{x} \, dx &=-\text {Subst}\left (\int (a+b x)^2 \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-2 \text {Subst}\left (\int \frac {e^{2 x} (a+b x)^2}{1+e^{2 x}} \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {sech}^{-1}(c x)\right )^2 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )+(2 b) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {sech}^{-1}(c x)\right )^2 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )-b \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )+b^2 \text {Subst}\left (\int \text {Li}_2\left (-e^{2 x}\right ) \, dx,x,\text {sech}^{-1}(c x)\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {sech}^{-1}(c x)\right )^2 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )-b \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )+\frac {1}{2} b^2 \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 \text {sech}^{-1}(c x)}\right )\\ &=\frac {\left (a+b \text {sech}^{-1}(c x)\right )^3}{3 b}-\left (a+b \text {sech}^{-1}(c x)\right )^2 \log \left (1+e^{2 \text {sech}^{-1}(c x)}\right )-b \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 \text {sech}^{-1}(c x)}\right )+\frac {1}{2} b^2 \text {Li}_3\left (-e^{2 \text {sech}^{-1}(c x)}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 116, normalized size = 1.40 \begin {gather*} a^2 \log (c x)+a b \left (-\text {sech}^{-1}(c x) \left (\text {sech}^{-1}(c x)+2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )\right )+\text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )\right )+b^2 \left (-\frac {1}{3} \text {sech}^{-1}(c x)^3-\text {sech}^{-1}(c x)^2 \log \left (1+e^{-2 \text {sech}^{-1}(c x)}\right )+\text {sech}^{-1}(c x) \text {PolyLog}\left (2,-e^{-2 \text {sech}^{-1}(c x)}\right )+\frac {1}{2} \text {PolyLog}\left (3,-e^{-2 \text {sech}^{-1}(c x)}\right )\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.22, size = 250, normalized size = 3.01
method | result | size |
derivativedivides | \(a^{2} \ln \left (c x \right )+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{3}}{3}-b^{2} \mathrm {arcsech}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-b^{2} \mathrm {arcsech}\left (c x \right ) \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {b^{2} \polylog \left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}+a b \mathrm {arcsech}\left (c x \right )^{2}-2 a b \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-a b \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\) | \(250\) |
default | \(a^{2} \ln \left (c x \right )+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{3}}{3}-b^{2} \mathrm {arcsech}\left (c x \right )^{2} \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-b^{2} \mathrm {arcsech}\left (c x \right ) \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )+\frac {b^{2} \polylog \left (3, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )}{2}+a b \mathrm {arcsech}\left (c x \right )^{2}-2 a b \,\mathrm {arcsech}\left (c x \right ) \ln \left (1+\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )-a b \polylog \left (2, -\left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )^{2}\right )\) | \(250\) |
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 {\left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}}{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 {{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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