Optimal. Leaf size=175 \[ \frac {\sqrt {x^2} \left (2-3 x^2\right )}{6 \left (-1+x^2\right )}-\frac {13}{6} \coth ^{-1}\left (\sqrt {x^2}\right )-\frac {5 x^3 \sec ^{-1}(x)}{6 \left (-1+x^2\right )^{3/2}}+\frac {x^5 \sec ^{-1}(x)}{2 \left (-1+x^2\right )^{3/2}}-\frac {5 x \sec ^{-1}(x)}{2 \sqrt {-1+x^2}}-\frac {5 i \sqrt {x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x}+\frac {5 i \sqrt {x^2} \text {Li}_2\left (-i e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac {5 i \sqrt {x^2} \text {Li}_2\left (i e^{i \sec ^{-1}(x)}\right )}{2 x} \]
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Rubi [A]
time = 0.19, antiderivative size = 232, normalized size of antiderivative = 1.33, number of steps
used = 16, number of rules used = 11, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.733, Rules used = {5350, 4790,
4794, 4804, 4266, 2317, 2438, 212, 205, 296, 331} \begin {gather*} \frac {5 i \sqrt {x^2} \text {Li}_2\left (-i e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac {5 i \sqrt {x^2} \text {Li}_2\left (i e^{i \sec ^{-1}(x)}\right )}{2 x}+\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {3 \sqrt {x^2}}{4}-\frac {5}{12 \left (1-\frac {1}{x^2}\right ) \sqrt {x^2}}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{2 \sqrt {1-\frac {1}{x^2}} x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}-\frac {13 \sqrt {x^2} \coth ^{-1}(x)}{6 x}-\frac {5 i \sqrt {x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 212
Rule 296
Rule 331
Rule 2317
Rule 2438
Rule 4266
Rule 4790
Rule 4794
Rule 4804
Rule 5350
Rubi steps
\begin {align*} \int \frac {x^6 \sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx &=-\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {\cos ^{-1}(x)}{x^3 \left (1-x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{x}\\ &=\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {\sqrt {x^2} \text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{2 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\cos ^{-1}(x)}{x \left (1-x^2\right )^{5/2}} \, dx,x,\frac {1}{x}\right )}{2 x}\\ &=\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right )} \, dx,x,\frac {1}{x}\right )}{4 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2} \, dx,x,\frac {1}{x}\right )}{6 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\cos ^{-1}(x)}{x \left (1-x^2\right )^{3/2}} \, dx,x,\frac {1}{x}\right )}{2 x}\\ &=-\frac {5}{12 \left (1-\frac {1}{x^2}\right ) \sqrt {x^2}}-\frac {3 \sqrt {x^2}}{4}+\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {1}{x}\right )}{12 x}+\frac {\left (3 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {1}{x}\right )}{4 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {1}{x}\right )}{2 x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\cos ^{-1}(x)}{x \sqrt {1-x^2}} \, dx,x,\frac {1}{x}\right )}{2 x}\\ &=-\frac {5}{12 \left (1-\frac {1}{x^2}\right ) \sqrt {x^2}}-\frac {3 \sqrt {x^2}}{4}+\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {13 \sqrt {x^2} \coth ^{-1}(x)}{6 x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int x \sec (x) \, dx,x,\sec ^{-1}(x)\right )}{2 x}\\ &=-\frac {5}{12 \left (1-\frac {1}{x^2}\right ) \sqrt {x^2}}-\frac {3 \sqrt {x^2}}{4}+\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {13 \sqrt {x^2} \coth ^{-1}(x)}{6 x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {5 i \sqrt {x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x}-\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sec ^{-1}(x)\right )}{2 x}+\frac {\left (5 \sqrt {x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sec ^{-1}(x)\right )}{2 x}\\ &=-\frac {5}{12 \left (1-\frac {1}{x^2}\right ) \sqrt {x^2}}-\frac {3 \sqrt {x^2}}{4}+\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {13 \sqrt {x^2} \coth ^{-1}(x)}{6 x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {5 i \sqrt {x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x}+\frac {\left (5 i \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac {\left (5 i \sqrt {x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sec ^{-1}(x)}\right )}{2 x}\\ &=-\frac {5}{12 \left (1-\frac {1}{x^2}\right ) \sqrt {x^2}}-\frac {3 \sqrt {x^2}}{4}+\frac {\sqrt {x^2}}{4 \left (1-\frac {1}{x^2}\right )}-\frac {13 \sqrt {x^2} \coth ^{-1}(x)}{6 x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{6 \left (1-\frac {1}{x^2}\right )^{3/2} x}-\frac {5 \sqrt {x^2} \sec ^{-1}(x)}{2 \sqrt {1-\frac {1}{x^2}} x}+\frac {x \sqrt {x^2} \sec ^{-1}(x)}{2 \left (1-\frac {1}{x^2}\right )^{3/2}}-\frac {5 i \sqrt {x^2} \sec ^{-1}(x) \tan ^{-1}\left (e^{i \sec ^{-1}(x)}\right )}{x}+\frac {5 i \sqrt {x^2} \text {Li}_2\left (-i e^{i \sec ^{-1}(x)}\right )}{2 x}-\frac {5 i \sqrt {x^2} \text {Li}_2\left (i e^{i \sec ^{-1}(x)}\right )}{2 x}\\ \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. \(383\) vs. \(2(175)=350\).
time = 1.13, size = 383, normalized size = 2.19 \begin {gather*} -\frac {x^5 \left (22 \sec ^{-1}(x)+40 \sec ^{-1}(x) \cos \left (2 \sec ^{-1}(x)\right )-30 \sec ^{-1}(x) \cos \left (4 \sec ^{-1}(x)\right )-30 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right )+30 \sqrt {1-\frac {1}{x^2}} \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right )+26 \sqrt {1-\frac {1}{x^2}} \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(x)\right )\right )-26 \sqrt {1-\frac {1}{x^2}} \log \left (\sin \left (\frac {1}{2} \sec ^{-1}(x)\right )\right )+16 \sin \left (2 \sec ^{-1}(x)\right )-60 i \sqrt {1-\frac {1}{x^2}} \text {Li}_2\left (-i e^{i \sec ^{-1}(x)}\right ) \sin ^2\left (2 \sec ^{-1}(x)\right )+60 i \sqrt {1-\frac {1}{x^2}} \text {Li}_2\left (i e^{i \sec ^{-1}(x)}\right ) \sin ^2\left (2 \sec ^{-1}(x)\right )-15 \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right ) \sin \left (3 \sec ^{-1}(x)\right )+15 \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right ) \sin \left (3 \sec ^{-1}(x)\right )+13 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (3 \sec ^{-1}(x)\right )-13 \log \left (\sin \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (3 \sec ^{-1}(x)\right )-4 \sin \left (4 \sec ^{-1}(x)\right )+15 \sec ^{-1}(x) \log \left (1-i e^{i \sec ^{-1}(x)}\right ) \sin \left (5 \sec ^{-1}(x)\right )-15 \sec ^{-1}(x) \log \left (1+i e^{i \sec ^{-1}(x)}\right ) \sin \left (5 \sec ^{-1}(x)\right )-13 \log \left (\cos \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (5 \sec ^{-1}(x)\right )+13 \log \left (\sin \left (\frac {1}{2} \sec ^{-1}(x)\right )\right ) \sin \left (5 \sec ^{-1}(x)\right )\right )}{96 \left (-1+x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Mathics [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.84, size = 240, normalized size = 1.37
method | result | size |
default | \(\frac {\sqrt {x^{2}-1}\, x \left (3 x^{4} \mathrm {arcsec}\left (x \right )-3 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x^{3}-20 \,\mathrm {arcsec}\left (x \right ) x^{2}+2 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x +15 \,\mathrm {arcsec}\left (x \right )\right )}{6 x^{4}-12 x^{2}+6}-\frac {i \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \left (15 i \mathrm {arcsec}\left (x \right ) \ln \left (1-i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )-15 i \mathrm {arcsec}\left (x \right ) \ln \left (1+i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )+13 i \ln \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}-1\right )-13 i \ln \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}+1\right )+15 \dilog \left (1-i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )-15 \dilog \left (1+i \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}\right )\right )\right )}{6 \sqrt {x^{2}-1}}\) | \(240\) |
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 = 0.32, 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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] N/A
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 \frac {x^6\,\mathrm {acos}\left (\frac {1}{x}\right )}{{\left (x^2-1\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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