Optimal. Leaf size=60 \[ -\frac {15}{8} \sqrt {-1+\frac {1}{x^2}}+\frac {5}{8} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4+\frac {15}{8} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {25, 272, 43, 52,
65, 209} \begin {gather*} \frac {15}{8} \text {ArcTan}\left (\sqrt {\frac {1}{x^2}-1}\right )+\frac {5}{8} \left (\frac {1}{x^2}-1\right )^{3/2} x^2-\frac {15}{8} \sqrt {\frac {1}{x^2}-1}+\frac {1}{4} \left (\frac {1}{x^2}-1\right )^{5/2} x^4 \end {gather*}
Antiderivative was successfully verified.
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Rule 25
Rule 43
Rule 52
Rule 65
Rule 209
Rule 272
Rubi steps
\begin {align*} \int \frac {\sqrt {-1+\frac {1}{x^2}} \left (-1+x^2\right )^2}{x} \, dx &=\int \left (-1+\frac {1}{x^2}\right )^{5/2} x^3 \, dx\\ &=-\left (\frac {1}{2} \text {Subst}\left (\int \frac {(-1+x)^{5/2}}{x^3} \, dx,x,\frac {1}{x^2}\right )\right )\\ &=\frac {1}{4} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {5}{8} \text {Subst}\left (\int \frac {(-1+x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {5}{8} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {15}{16} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {15}{8} \sqrt {-1+\frac {1}{x^2}}+\frac {5}{8} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4+\frac {15}{16} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {15}{8} \sqrt {-1+\frac {1}{x^2}}+\frac {5}{8} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4+\frac {15}{8} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+\frac {1}{x^2}}\right )\\ &=-\frac {15}{8} \sqrt {-1+\frac {1}{x^2}}+\frac {5}{8} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2+\frac {1}{4} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4+\frac {15}{8} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 65, normalized size = 1.08 \begin {gather*} \frac {1}{8} \sqrt {-1+\frac {1}{x^2}} \left (-8-9 x^2+2 x^4\right )+\frac {15 \sqrt {-1+\frac {1}{x^2}} x \tanh ^{-1}\left (\frac {\sqrt {-1+x^2}}{-1+x}\right )}{4 \sqrt {-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.36, size = 69, normalized size = 1.15
method | result | size |
trager | \(2 \left (\frac {1}{8} x^{4}-\frac {9}{16} x^{2}-\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}+\frac {15 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\left (\RootOf \left (\textit {\_Z}^{2}+1\right )+\sqrt {-\frac {x^{2}-1}{x^{2}}}\right ) x \right )}{8}\) | \(58\) |
default | \(-\frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (2 x^{2} \left (-x^{2}+1\right )^{\frac {3}{2}}+8 \left (-x^{2}+1\right )^{\frac {3}{2}}+15 x^{2} \sqrt {-x^{2}+1}+15 \arcsin \left (x \right ) x \right )}{8 \sqrt {-x^{2}+1}}\) | \(69\) |
risch | \(\frac {\left (2 x^{6}-11 x^{4}+x^{2}+8\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{8 x^{2}-8}+\frac {15 \arcsin \left (x \right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}\, x \sqrt {-x^{2}+1}}{8 \left (x^{2}-1\right )}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 67, normalized size = 1.12 \begin {gather*} -x^{2} \sqrt {\frac {1}{x^{2}} - 1} - \sqrt {\frac {1}{x^{2}} - 1} - \frac {{\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {1}{x^{2}} - 1}}{8 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {2}{x^{2}} - 1\right )}} + \frac {15}{8} \, \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.39, size = 50, normalized size = 0.83 \begin {gather*} \frac {1}{8} \, {\left (2 \, x^{4} - 9 \, x^{2} - 8\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} + \frac {15}{4} \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 65.07, size = 60, normalized size = 1.00 \begin {gather*} \frac {x^{4} \sqrt {-1 + \frac {1}{x^{2}}} \cdot \left (2 - \frac {1}{x^{2}}\right )}{8} - x^{2} \sqrt {-1 + \frac {1}{x^{2}}} - \sqrt {-1 + \frac {1}{x^{2}}} + \frac {15 \operatorname {atan}{\left (\sqrt {-1 + \frac {1}{x^{2}}} \right )}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.76, size = 67, normalized size = 1.12 \begin {gather*} \frac {1}{8} \, {\left (2 \, x^{2} \mathrm {sgn}\left (x\right ) - 9 \, \mathrm {sgn}\left (x\right )\right )} \sqrt {-x^{2} + 1} x - \frac {15}{8} \, \arcsin \left (x\right ) \mathrm {sgn}\left (x\right ) + \frac {x \mathrm {sgn}\left (x\right )}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} - \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\left (x\right )}{2 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.36, size = 44, normalized size = 0.73 \begin {gather*} \frac {15\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{8}-\sqrt {\frac {1}{x^2}-1}-\frac {7\,x^4\,\sqrt {\frac {1}{x^2}-1}}{8}-\frac {9\,x^4\,{\left (\frac {1}{x^2}-1\right )}^{3/2}}{8} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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