Optimal. Leaf size=140 \[ -\frac {2 b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3}-\frac {b x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2+\frac {i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {b^2 x}{3 c^2} \]
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Rubi [A] time = 0.12, 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 = {6285, 5451, 4185, 4180, 2279, 2391} \[ \frac {i b^2 \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right )}{3 c^3}-\frac {b x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^2}-\frac {2 b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{3 c^3}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^2-\frac {b^2 x}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4180
Rule 4185
Rule 5451
Rule 6285
Rubi steps
\begin {align*} \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=-\frac {\operatorname {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) \operatorname {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 \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \operatorname {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 = 1.27, size = 224, normalized size = 1.60 \[ \frac {1}{3} \left (a^2 x^3+a b \left (2 x^3 \text {sech}^{-1}(c x)-\frac {\sqrt {\frac {1-c x}{c x+1}} \left (c^3 x^3+\sqrt {1-c^2 x^2} \sin ^{-1}(c x)-c x\right )}{c^3 (c x-1)}\right )+\frac {b^2 \left (c^3 x^3 \text {sech}^{-1}(c x)^2+i \text {Li}_2\left (-i e^{-\text {sech}^{-1}(c x)}\right )-i \text {Li}_2\left (i e^{-\text {sech}^{-1}(c x)}\right )-c x-c x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \text {sech}^{-1}(c x)+i \text {sech}^{-1}(c x) \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-i \text {sech}^{-1}(c x) \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )}{c^3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.50, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \operatorname {arsech}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {arsech}\left (c x\right ) + a^{2} x^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.64, size = 372, normalized size = 2.66 \[ \frac {x^{3} a^{2}}{3}+\frac {x^{3} b^{2} \mathrm {arcsech}\left (c x \right )^{2}}{3}-\frac {b^{2} \mathrm {arcsech}\left (c x \right ) \sqrt {-\frac {c x -1}{c x}}\, \sqrt {\frac {c x +1}{c x}}\, x^{2}}{3 c}-\frac {b^{2} x}{3 c^{2}}+\frac {i b^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3 c^{3}}-\frac {i b^{2} \mathrm {arcsech}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3 c^{3}}+\frac {i b^{2} \dilog \left (1+i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3 c^{3}}-\frac {i b^{2} \dilog \left (1-i \left (\frac {1}{c x}+\sqrt {-1+\frac {1}{c x}}\, \sqrt {1+\frac {1}{c x}}\right )\right )}{3 c^{3}}+\frac {2 a b \,x^{3} \mathrm {arcsech}\left (c x \right )}{3}-\frac {a b \sqrt {-\frac {c x -1}{c x}}\, x^{2} \sqrt {\frac {c x +1}{c x}}}{3 c}+\frac {a b \sqrt {-\frac {c x -1}{c x}}\, x \sqrt {\frac {c x +1}{c x}}\, \arcsin \left (c x \right )}{3 c^{2} \sqrt {-c^{2} x^{2}+1}} \]
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 \[ \frac {1}{3} \, a^{2} x^{3} + \frac {1}{3} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} a b + b^{2} \int x^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^2\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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