Optimal. Leaf size=140 \[ -\frac {b^2 x}{3 c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right ) \text {ArcTan}\left (e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3} \]
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Rubi [A]
time = 0.09, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {6420, 5559,
4270, 4265, 2317, 2438} \begin {gather*} -\frac {2 b \text {ArcTan}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3}-\frac {b x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 x}{3 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2317
Rule 2438
Rule 4265
Rule 4270
Rule 5559
Rule 6420
Rubi steps
\begin {align*} \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=-\frac {\text {Subst}\left (\int (a+b x)^2 \text {sech}^3(x) \tanh (x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}\\ &=\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {(2 b) \text {Subst}\left (\int (a+b x) \text {sech}^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{3 c^3}\\ &=-\frac {b^2 x}{3 c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b \text {Subst}\left (\int (a+b x) \text {sech}(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{3 c^3}\\ &=-\frac {b^2 x}{3 c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right ) \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{3 c^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{3 c^3}\\ &=-\frac {b^2 x}{3 c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right ) \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}+\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {\left (i b^2\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}\\ &=-\frac {b^2 x}{3 c^2}-\frac {b x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {2 b \left (a+b \text {sech}^{-1}(c x)\right ) \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.87, size = 241, normalized size = 1.72 \begin {gather*} \frac {1}{3} \left (a^2 x^3+a b \left (2 x^3 \text {sech}^{-1}(c x)+\frac {\sqrt {\frac {1-c x}{1+c x}} \left (c x-c^3 x^3+2 \sqrt {1-c^2 x^2} \text {ArcTan}\left (\frac {\sqrt {1-c x}}{\sqrt {1+c x}}\right )\right )}{c^3 (-1+c x)}\right )+\frac {b^2 \left (-c x-c x \sqrt {\frac {1-c x}{1+c x}} (1+c x) \text {sech}^{-1}(c x)+c^3 x^3 \text {sech}^{-1}(c x)^2+i \text {sech}^{-1}(c x) \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-i \text {sech}^{-1}(c x) \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )+i \text {PolyLog}\left (2,-i e^{-\text {sech}^{-1}(c x)}\right )-i \text {PolyLog}\left (2,i e^{-\text {sech}^{-1}(c x)}\right )\right )}{c^3}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.55, size = 349, normalized size = 2.49
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{2} x^{2}}{3}-\frac {b^{2} c x}{3}+\frac {i b^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i b^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+\frac {i b^{2} \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i b^{2} \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+2 a b \left (\frac {c^{3} x^{3} \mathrm {arcsech}\left (c x \right )}{3}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\) | \(349\) |
default | \(\frac {\frac {a^{2} c^{3} x^{3}}{3}+\frac {b^{2} \mathrm {arcsech}\left (c x \right )^{2} c^{3} x^{3}}{3}-\frac {b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, c^{2} x^{2}}{3}-\frac {b^{2} c x}{3}+\frac {i b^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i b^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+\frac {i b^{2} \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}-\frac {i b^{2} \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3}+2 a b \left (\frac {c^{3} x^{3} \mathrm {arcsech}\left (c x \right )}{3}-\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (c x \sqrt {-c^{2} x^{2}+1}-\arcsin \left (c x \right )\right )}{6 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{3}}\) | \(349\) |
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 x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}\, 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 x^2\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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