Optimal. Leaf size=242 \[ \frac {i b^2 \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^3}-\frac {i b^2 \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^3}-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}-\frac {b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^3}-\frac {b x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {i b^3 \text {Li}_3\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {i b^3 \text {Li}_3\left (i e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )}{c^3} \]
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Rubi [A] time = 0.20, antiderivative size = 242, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 8, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {6285, 5451, 4186, 3770, 4180, 2531, 2282, 6589} \[ \frac {i b^2 \text {PolyLog}\left (2,-i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^3}-\frac {i b^2 \text {PolyLog}\left (2,i e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )}{c^3}-\frac {i b^3 \text {PolyLog}\left (3,-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {i b^3 \text {PolyLog}\left (3,i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{c^2}-\frac {b x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \left (a+b \text {sech}^{-1}(c x)\right )^2}{2 c^2}-\frac {b \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right ) \left (a+b \text {sech}^{-1}(c x)\right )^2}{c^3}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {\frac {1-c x}{c x+1}} (c x+1)}{c x}\right )}{c^3} \]
Antiderivative was successfully verified.
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Rule 2282
Rule 2531
Rule 3770
Rule 4180
Rule 4186
Rule 5451
Rule 6285
Rule 6589
Rubi steps
\begin {align*} \int x^2 \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx &=-\frac {\operatorname {Subst}\left (\int (a+b x)^3 \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 )^3-\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}^3(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{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 )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \operatorname {Subst}\left (\int (a+b x)^2 \text {sech}(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{2 c^3}+\frac {b^3 \operatorname {Subst}\left (\int \text {sech}(x) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{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 )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1-i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}-\frac {\left (i b^2\right ) \operatorname {Subst}\left (\int (a+b x) \log \left (1+i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{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 )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (-i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \text {Li}_2\left (i e^x\right ) \, dx,x,\text {sech}^{-1}(c x)\right )}{c^3}\\ &=-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{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 )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {\left (i b^3\right ) \operatorname {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{\text {sech}^{-1}(c x)}\right )}{c^3}\\ &=-\frac {b^2 x \left (a+b \text {sech}^{-1}(c x)\right )}{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 )^2}{2 c^2}+\frac {1}{3} x^3 \left (a+b \text {sech}^{-1}(c x)\right )^3-\frac {b \left (a+b \text {sech}^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {b^3 \tan ^{-1}\left (\frac {\sqrt {\frac {1-c x}{1+c x}} (1+c x)}{c x}\right )}{c^3}+\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {i b^2 \left (a+b \text {sech}^{-1}(c x)\right ) \text {Li}_2\left (i e^{\text {sech}^{-1}(c x)}\right )}{c^3}-\frac {i b^3 \text {Li}_3\left (-i e^{\text {sech}^{-1}(c x)}\right )}{c^3}+\frac {i b^3 \text {Li}_3\left (i e^{\text {sech}^{-1}(c x)}\right )}{c^3}\\ \end {align*}
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Mathematica [A] time = 1.11, size = 440, normalized size = 1.82 \[ \frac {2 a^3 c^3 x^3+6 a^2 b c^3 x^3 \text {sech}^{-1}(c x)-3 a^2 b c x \sqrt {\frac {1-c x}{c x+1}} (c x+1)+3 i a^2 b \log \left (2 \sqrt {\frac {1-c x}{c x+1}} (c x+1)-2 i c x\right )-6 a b^2 \left (-c^3 x^3 \text {sech}^{-1}(c x)^2-i \text {Li}_2\left (-i e^{-\text {sech}^{-1}(c x)}\right )+i \text {Li}_2\left (i e^{-\text {sech}^{-1}(c x)}\right )+c x+c x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \text {sech}^{-1}(c x)-i \text {sech}^{-1}(c x) \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )+i \text {sech}^{-1}(c x) \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )\right )-\left (b^3 \left (-2 c^3 x^3 \text {sech}^{-1}(c x)^3-3 i \left (2 \text {sech}^{-1}(c x) \text {Li}_2\left (-i e^{-\text {sech}^{-1}(c x)}\right )-2 \text {sech}^{-1}(c x) \text {Li}_2\left (i e^{-\text {sech}^{-1}(c x)}\right )+2 \text {Li}_3\left (-i e^{-\text {sech}^{-1}(c x)}\right )-2 \text {Li}_3\left (i e^{-\text {sech}^{-1}(c x)}\right )+\text {sech}^{-1}(c x)^2 \log \left (1-i e^{-\text {sech}^{-1}(c x)}\right )-\text {sech}^{-1}(c x)^2 \log \left (1+i e^{-\text {sech}^{-1}(c x)}\right )-4 i \tan ^{-1}\left (\tanh \left (\frac {1}{2} \text {sech}^{-1}(c x)\right )\right )\right )+3 c x \sqrt {\frac {1-c x}{c x+1}} (c x+1) \text {sech}^{-1}(c x)^2+6 c x \text {sech}^{-1}(c x)\right )\right )}{6 c^3} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{3} x^{2} \operatorname {arsech}\left (c x\right )^{3} + 3 \, a b^{2} x^{2} \operatorname {arsech}\left (c x\right )^{2} + 3 \, a^{2} b x^{2} \operatorname {arsech}\left (c x\right ) + a^{3} x^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.02, size = 0, normalized size = 0.00 \[ \int x^{2} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{3}\, dx \]
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 \[ \frac {1}{3} \, a^{3} x^{3} + \frac {1}{2} \, {\left (2 \, x^{3} \operatorname {arsech}\left (c x\right ) - \frac {\frac {\sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{2}}}{c}\right )} a^{2} b + \int b^{3} x^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{3} + 3 \, a b^{2} x^{2} \log \left (\sqrt {\frac {1}{c x} + 1} \sqrt {\frac {1}{c x} - 1} + \frac {1}{c x}\right )^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int x^2\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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