Optimal. Leaf size=65 \[ \frac {\sqrt {x^2}}{6 \left (1-x^2\right )}+\frac {5}{6} \coth ^{-1}\left (\sqrt {x^2}\right )-\frac {x \sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {2 x \sec ^{-1}(x)}{3 \sqrt {-1+x^2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 67, normalized size of antiderivative = 1.03, number of steps
used = 4, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {198, 197, 5336,
12, 393, 212} \begin {gather*} \frac {\sqrt {x^2}}{6 \left (1-x^2\right )}+\frac {2 x \sec ^{-1}(x)}{3 \sqrt {x^2-1}}-\frac {x \sec ^{-1}(x)}{3 \left (x^2-1\right )^{3/2}}+\frac {5 x \tanh ^{-1}(x)}{6 \sqrt {x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 198
Rule 212
Rule 393
Rule 5336
Rubi steps
\begin {align*} \int \frac {\sec ^{-1}(x)}{\left (-1+x^2\right )^{5/2}} \, dx &=-\frac {x \sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {2 x \sec ^{-1}(x)}{3 \sqrt {-1+x^2}}-\frac {x \int \frac {-3+2 x^2}{3 \left (1-x^2\right )^2} \, dx}{\sqrt {x^2}}\\ &=-\frac {x \sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {2 x \sec ^{-1}(x)}{3 \sqrt {-1+x^2}}-\frac {x \int \frac {-3+2 x^2}{\left (1-x^2\right )^2} \, dx}{3 \sqrt {x^2}}\\ &=\frac {\sqrt {x^2}}{6 \left (1-x^2\right )}-\frac {x \sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {2 x \sec ^{-1}(x)}{3 \sqrt {-1+x^2}}+\frac {(5 x) \int \frac {1}{1-x^2} \, dx}{6 \sqrt {x^2}}\\ &=\frac {\sqrt {x^2}}{6 \left (1-x^2\right )}-\frac {x \sec ^{-1}(x)}{3 \left (-1+x^2\right )^{3/2}}+\frac {2 x \sec ^{-1}(x)}{3 \sqrt {-1+x^2}}+\frac {5 x \tanh ^{-1}(x)}{6 \sqrt {x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 67, normalized size = 1.03 \begin {gather*} \frac {4 x \left (-3+2 x^2\right ) \sec ^{-1}(x)+\sqrt {1-\frac {1}{x^2}} x \left (-2 x-5 \left (-1+x^2\right ) \log (1-x)+5 \left (-1+x^2\right ) \log (1+x)\right )}{12 \left (-1+x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.42, size = 128, normalized size = 1.97
method | result | size |
default | \(\frac {\sqrt {x^{2}-1}\, x \left (4 \,\mathrm {arcsec}\left (x \right ) x^{2}-\sqrt {\frac {x^{2}-1}{x^{2}}}\, x -6 \,\mathrm {arcsec}\left (x \right )\right )}{6 x^{4}-12 x^{2}+6}-\frac {5 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \ln \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}-1\right )}{6 \sqrt {x^{2}-1}}+\frac {5 \sqrt {\frac {x^{2}-1}{x^{2}}}\, x \ln \left (\frac {1}{x}+i \sqrt {1-\frac {1}{x^{2}}}+1\right )}{6 \sqrt {x^{2}-1}}\) | \(128\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.52, size = 48, normalized size = 0.74 \begin {gather*} \frac {1}{3} \, {\left (\frac {2 \, x}{\sqrt {x^{2} - 1}} - \frac {x}{{\left (x^{2} - 1\right )}^{\frac {3}{2}}}\right )} \operatorname {arcsec}\left (x\right ) - \frac {x}{6 \, {\left (x^{2} - 1\right )}} + \frac {5}{12} \, \log \left (x + 1\right ) - \frac {5}{12} \, \log \left (x - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.01, size = 75, normalized size = 1.15 \begin {gather*} -\frac {2 \, x^{3} - 4 \, {\left (2 \, x^{3} - 3 \, x\right )} \sqrt {x^{2} - 1} \operatorname {arcsec}\left (x\right ) - 5 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x + 1\right ) + 5 \, {\left (x^{4} - 2 \, x^{2} + 1\right )} \log \left (x - 1\right ) - 2 \, x}{12 \, {\left (x^{4} - 2 \, x^{2} + 1\right )}} \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 [A]
time = 0.62, size = 58, normalized size = 0.89 \begin {gather*} \frac {{\left (2 \, x^{2} - 3\right )} x \arccos \left (\frac {1}{x}\right )}{3 \, {\left (x^{2} - 1\right )}^{\frac {3}{2}}} + \frac {5 \, \log \left ({\left | x + 1 \right |}\right )}{12 \, \mathrm {sgn}\left (x\right )} - \frac {5 \, \log \left ({\left | x - 1 \right |}\right )}{12 \, \mathrm {sgn}\left (x\right )} - \frac {x}{6 \, {\left (x^{2} - 1\right )} \mathrm {sgn}\left (x\right )} \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 = -0.02 \begin {gather*} \int \frac {\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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