Optimal. Leaf size=147 \[ \frac {2 i b \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{3 c^3}-\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {i b^2 \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 x}{3 c^2} \]
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Rubi [A] time = 0.12, antiderivative size = 147, 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 = {5222, 4409, 4185, 4181, 2279, 2391} \[ -\frac {i b^2 \text {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \text {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {2 i b \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{3 c^3}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {b^2 x}{3 c^2} \]
Antiderivative was successfully verified.
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Rule 2279
Rule 2391
Rule 4181
Rule 4185
Rule 4409
Rule 5222
Rubi steps
\begin {align*} \int x^2 \left (a+b \sec ^{-1}(c x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int (a+b x)^2 \sec ^3(x) \tan (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}\\ &=\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {(2 b) \operatorname {Subst}\left (\int (a+b x) \sec ^3(x) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2-\frac {b \operatorname {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {b^2 \operatorname {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}-\frac {b^2 \operatorname {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sec ^{-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^{i \sec ^{-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^{i \sec ^{-1}(c x)}\right )}{3 c^3}\\ &=\frac {b^2 x}{3 c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )}{3 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^2+\frac {2 i b \left (a+b \sec ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{3 c^3}-\frac {i b^2 \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}+\frac {i b^2 \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )}{3 c^3}\\ \end {align*}
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Mathematica [A] time = 1.27, size = 225, normalized size = 1.53 \[ \frac {1}{3} \left (a^2 x^3+\frac {a b \left (2 x^4 \sec ^{-1}(c x)-\frac {c^3 x^3+\sqrt {c^2 x^2-1} \tanh ^{-1}\left (\frac {c x}{\sqrt {c^2 x^2-1}}\right )-c x}{c^4 \sqrt {1-\frac {1}{c^2 x^2}}}\right )}{x}+\frac {b^2 \left (c^3 x^3 \sec ^{-1}(c x)^2-c^2 x^2 \sqrt {1-\frac {1}{c^2 x^2}} \sec ^{-1}(c x)-i \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )+i \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )+c x-\sec ^{-1}(c x) \log \left (1-i e^{i \sec ^{-1}(c x)}\right )+\sec ^{-1}(c x) \log \left (1+i e^{i \sec ^{-1}(c x)}\right )\right )}{c^3}\right ) \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (b^{2} x^{2} \operatorname {arcsec}\left (c x\right )^{2} + 2 \, a b x^{2} \operatorname {arcsec}\left (c x\right ) + a^{2} x^{2}, x\right ) \]
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 \[ \int {\left (b \operatorname {arcsec}\left (c x\right ) + a\right )}^{2} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.91, size = 343, normalized size = 2.33 \[ \frac {x^{3} a^{2}}{3}+\frac {b^{2} x^{3} \mathrm {arcsec}\left (c x \right )^{2}}{3}-\frac {b^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, \mathrm {arcsec}\left (c x \right ) x^{2}}{3 c}+\frac {b^{2} x}{3 c^{2}}+\frac {b^{2} \mathrm {arcsec}\left (c x \right ) \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3 c^{3}}-\frac {b^{2} \mathrm {arcsec}\left (c x \right ) \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3 c^{3}}-\frac {i b^{2} \dilog \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3 c^{3}}+\frac {i b^{2} \dilog \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{3 c^{3}}+\frac {2 x^{3} a b \,\mathrm {arcsec}\left (c x \right )}{3}-\frac {a b \,x^{2}}{3 c \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}+\frac {a b}{3 c^{3} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {a b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{3 c^{4} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, x} \]
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}{6} \, {\left (4 \, x^{3} \operatorname {arcsec}\left (c x\right ) - \frac {\frac {2 \, \sqrt {-\frac {1}{c^{2} x^{2}} + 1}}{c^{2} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{2}} + \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} + 1\right )}{c^{2}} - \frac {\log \left (\sqrt {-\frac {1}{c^{2} x^{2}} + 1} - 1\right )}{c^{2}}}{c}\right )} a b + \frac {1}{12} \, {\left (4 \, x^{3} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )^{2} - x^{3} \log \left (c^{2} x^{2}\right )^{2} - 2 \, c^{2} {\left (\frac {2 \, {\left (c^{2} x^{3} + 3 \, x\right )}}{c^{4}} - \frac {3 \, \log \left (c x + 1\right )}{c^{5}} + \frac {3 \, \log \left (c x - 1\right )}{c^{5}}\right )} \log \relax (c)^{2} + 36 \, c^{2} \int \frac {x^{4} \log \left (c^{2} x^{2}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \log \relax (c) - 72 \, c^{2} \int \frac {x^{4} \log \relax (x)}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \log \relax (c) + 36 \, c^{2} \int \frac {x^{4} \log \left (c^{2} x^{2}\right ) \log \relax (x)}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 36 \, c^{2} \int \frac {x^{4} \log \relax (x)^{2}}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} + 12 \, c^{2} \int \frac {x^{4} \log \left (c^{2} x^{2}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} + 6 \, {\left (\frac {2 \, x}{c^{2}} - \frac {\log \left (c x + 1\right )}{c^{3}} + \frac {\log \left (c x - 1\right )}{c^{3}}\right )} \log \relax (c)^{2} - 36 \, \int \frac {x^{2} \log \left (c^{2} x^{2}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \log \relax (c) + 72 \, \int \frac {x^{2} \log \relax (x)}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} \log \relax (c) - 24 \, \int \frac {\sqrt {c x + 1} \sqrt {c x - 1} x^{2} \arctan \left (\sqrt {c x + 1} \sqrt {c x - 1}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 36 \, \int \frac {x^{2} \log \left (c^{2} x^{2}\right ) \log \relax (x)}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} + 36 \, \int \frac {x^{2} \log \relax (x)^{2}}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x} - 12 \, \int \frac {x^{2} \log \left (c^{2} x^{2}\right )}{3 \, {\left (c^{2} x^{2} - 1\right )}}\,{d x}\right )} b^{2} \]
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 \[ \int x^2\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]
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 \[ \int x^{2} \left (a + b \operatorname {asec}{\left (c x \right )}\right )^{2}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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