Optimal. Leaf size=236 \[ \frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \text {ArcTan}\left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {PolyLog}\left (3,-i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {PolyLog}\left (3,i e^{i \sec ^{-1}(c x)}\right )}{c^3} \]
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Rubi [A]
time = 0.15, antiderivative size = 236, 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 = {5330, 4494,
4271, 3855, 4266, 2611, 2320, 6724} \begin {gather*} \frac {i b \text {ArcTan}\left (e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )^2}{c^3}-\frac {i b^2 \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^3}+\frac {i b^2 \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right ) \left (a+b \sec ^{-1}(c x)\right )}{c^3}+\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b x^2 \sqrt {1-\frac {1}{c^2 x^2}} \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {b^3 \text {Li}_3\left (-i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 2320
Rule 2611
Rule 3855
Rule 4266
Rule 4271
Rule 4494
Rule 5330
Rule 6724
Rubi steps
\begin {align*} \int x^2 \left (a+b \sec ^{-1}(c x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int (a+b x)^3 \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 )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \sec ^3(x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3-\frac {b \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{2 c^3}-\frac {b^3 \text {Subst}\left (\int \sec (x) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}+\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}-\frac {b^2 \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {\left (i b^3\right ) \text {Subst}\left (\int \text {Li}_2\left (-i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}-\frac {\left (i b^3\right ) \text {Subst}\left (\int \text {Li}_2\left (i e^{i x}\right ) \, dx,x,\sec ^{-1}(c x)\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2(-i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Subst}\left (\int \frac {\text {Li}_2(i x)}{x} \, dx,x,e^{i \sec ^{-1}(c x)}\right )}{c^3}\\ &=\frac {b^2 x \left (a+b \sec ^{-1}(c x)\right )}{c^2}-\frac {b \sqrt {1-\frac {1}{c^2 x^2}} x^2 \left (a+b \sec ^{-1}(c x)\right )^2}{2 c}+\frac {1}{3} x^3 \left (a+b \sec ^{-1}(c x)\right )^3+\frac {i b \left (a+b \sec ^{-1}(c x)\right )^2 \tan ^{-1}\left (e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \tanh ^{-1}\left (\sqrt {1-\frac {1}{c^2 x^2}}\right )}{c^3}-\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_2\left (-i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {Li}_2\left (i e^{i \sec ^{-1}(c x)}\right )}{c^3}+\frac {b^3 \text {Li}_3\left (-i e^{i \sec ^{-1}(c x)}\right )}{c^3}-\frac {b^3 \text {Li}_3\left (i e^{i \sec ^{-1}(c x)}\right )}{c^3}\\ \end {align*}
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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice
the leaf count of optimal. \(767\) vs. \(2(236)=472\).
time = 1.45, size = 767, normalized size = 3.25 \begin {gather*} \frac {6 a b^2 c x-3 a^2 b c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2+2 a^3 c^3 x^3+6 b^3 c x \sec ^{-1}(c x)-6 a b^2 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sec ^{-1}(c x)+6 a^2 b c^3 x^3 \sec ^{-1}(c x)-3 b^3 c^2 \sqrt {1-\frac {1}{c^2 x^2}} x^2 \sec ^{-1}(c x)^2+6 a b^2 c^3 x^3 \sec ^{-1}(c x)^2+2 b^3 c^3 x^3 \sec ^{-1}(c x)^3-6 a b^2 \sec ^{-1}(c x) \log \left (1-i e^{i \sec ^{-1}(c x)}\right )-3 b^3 \sec ^{-1}(c x)^2 \log \left (1-i e^{i \sec ^{-1}(c x)}\right )-3 b^3 \pi \sec ^{-1}(c x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \sec ^{-1}(c x)} \left (1-i e^{i \sec ^{-1}(c x)}\right )\right )+6 a b^2 \sec ^{-1}(c x) \log \left (1+i e^{i \sec ^{-1}(c x)}\right )+3 b^3 \sec ^{-1}(c x)^2 \log \left (1+i e^{i \sec ^{-1}(c x)}\right )+3 b^3 \sec ^{-1}(c x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \sec ^{-1}(c x)} \left (-i+e^{i \sec ^{-1}(c x)}\right )\right )-3 b^3 \pi \sec ^{-1}(c x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \sec ^{-1}(c x)} \left (-i+e^{i \sec ^{-1}(c x)}\right )\right )-3 b^3 \sec ^{-1}(c x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \sec ^{-1}(c x)} \left ((1+i)+(1-i) e^{i \sec ^{-1}(c x)}\right )\right )-3 a^2 b \log \left (\left (1+\sqrt {1-\frac {1}{c^2 x^2}}\right ) x\right )+3 b^3 \pi \sec ^{-1}(c x) \log \left (-\cos \left (\frac {1}{4} \left (\pi +2 \sec ^{-1}(c x)\right )\right )\right )+6 b^3 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sec ^{-1}(c x)\right )\right )-3 b^3 \sec ^{-1}(c x)^2 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(c x)\right )-\sin \left (\frac {1}{2} \sec ^{-1}(c x)\right )\right )-6 b^3 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(c x)\right )\right )+3 b^3 \sec ^{-1}(c x)^2 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(c x)\right )+\sin \left (\frac {1}{2} \sec ^{-1}(c x)\right )\right )+3 b^3 \pi \sec ^{-1}(c x) \log \left (\sin \left (\frac {1}{4} \left (\pi +2 \sec ^{-1}(c x)\right )\right )\right )-6 i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {PolyLog}\left (2,-i e^{i \sec ^{-1}(c x)}\right )+6 i b^2 \left (a+b \sec ^{-1}(c x)\right ) \text {PolyLog}\left (2,i e^{i \sec ^{-1}(c x)}\right )+6 b^3 \text {PolyLog}\left (3,-i e^{i \sec ^{-1}(c x)}\right )-6 b^3 \text {PolyLog}\left (3,i e^{i \sec ^{-1}(c x)}\right )}{6 c^3} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 641 vs. \(2 (286 ) = 572\).
time = 0.61, size = 642, normalized size = 2.72
method | result | size |
derivativedivides | \(\frac {\frac {c^{3} x^{3} a^{3}}{3}+\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{3} c^{3} x^{3}}{3}-\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}{2}+b^{3} \mathrm {arcsec}\left (c x \right ) c x +\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+i b^{3} \mathrm {arcsec}\left (c x \right ) \polylog \left (2, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+b^{3} \polylog \left (3, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+2 i b^{3} \arctan \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \polylog \left (3, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-i b^{3} \mathrm {arcsec}\left (c x \right ) \polylog \left (2, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2} c^{3} x^{3}-a \,b^{2} \mathrm {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}-a \,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 )+a \,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 )+i a \,b^{2} \dilog \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-i a \,b^{2} \dilog \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+a \,b^{2} c x +a^{2} b \,c^{3} x^{3} \mathrm {arcsec}\left (c x \right )-\frac {a^{2} b \left (c^{2} x^{2}-1\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {a^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{3}}\) | \(642\) |
default | \(\frac {\frac {c^{3} x^{3} a^{3}}{3}+\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{3} c^{3} x^{3}}{3}-\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}}{2}+b^{3} \mathrm {arcsec}\left (c x \right ) c x +\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \ln \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+i b^{3} \mathrm {arcsec}\left (c x \right ) \polylog \left (2, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+b^{3} \polylog \left (3, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-\frac {b^{3} \mathrm {arcsec}\left (c x \right )^{2} \ln \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )}{2}+2 i b^{3} \arctan \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )-b^{3} \polylog \left (3, i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-i b^{3} \mathrm {arcsec}\left (c x \right ) \polylog \left (2, -i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+a \,b^{2} \mathrm {arcsec}\left (c x \right )^{2} c^{3} x^{3}-a \,b^{2} \mathrm {arcsec}\left (c x \right ) \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c^{2} x^{2}-a \,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 )+a \,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 )+i a \,b^{2} \dilog \left (1-i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )-i a \,b^{2} \dilog \left (1+i \left (\frac {1}{c x}+i \sqrt {1-\frac {1}{c^{2} x^{2}}}\right )\right )+a \,b^{2} c x +a^{2} b \,c^{3} x^{3} \mathrm {arcsec}\left (c x \right )-\frac {a^{2} b \left (c^{2} x^{2}-1\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}}-\frac {a^{2} b \sqrt {c^{2} x^{2}-1}\, \ln \left (c x +\sqrt {c^{2} x^{2}-1}\right )}{2 \sqrt {\frac {c^{2} x^{2}-1}{c^{2} x^{2}}}\, c x}}{c^{3}}\) | \(642\) |
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 {asec}{\left (c x \right )}\right )^{3}\, 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 = -0.00 \begin {gather*} \int x^2\,{\left (a+b\,\mathrm {acos}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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