Optimal. Leaf size=76 \[ \frac {35}{16} \sqrt {-1+\frac {1}{x^2}}-\frac {35}{48} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6-\frac {35}{16} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 8, 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 {35}{16} \text {ArcTan}\left (\sqrt {\frac {1}{x^2}-1}\right )-\frac {35}{48} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {35}{16} \sqrt {\frac {1}{x^2}-1}-\frac {1}{6} \left (\frac {1}{x^2}-1\right )^{7/2} x^6-\frac {7}{24} \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 )^3}{x} \, dx &=-\int \left (-1+\frac {1}{x^2}\right )^{7/2} x^5 \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {(-1+x)^{7/2}}{x^4} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6+\frac {7}{12} \text {Subst}\left (\int \frac {(-1+x)^{5/2}}{x^3} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6+\frac {35}{48} \text {Subst}\left (\int \frac {(-1+x)^{3/2}}{x^2} \, dx,x,\frac {1}{x^2}\right )\\ &=-\frac {35}{48} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6+\frac {35}{32} \text {Subst}\left (\int \frac {\sqrt {-1+x}}{x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {35}{16} \sqrt {-1+\frac {1}{x^2}}-\frac {35}{48} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6-\frac {35}{32} \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x} \, dx,x,\frac {1}{x^2}\right )\\ &=\frac {35}{16} \sqrt {-1+\frac {1}{x^2}}-\frac {35}{48} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6-\frac {35}{16} \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+\frac {1}{x^2}}\right )\\ &=\frac {35}{16} \sqrt {-1+\frac {1}{x^2}}-\frac {35}{48} \left (-1+\frac {1}{x^2}\right )^{3/2} x^2-\frac {7}{24} \left (-1+\frac {1}{x^2}\right )^{5/2} x^4-\frac {1}{6} \left (-1+\frac {1}{x^2}\right )^{7/2} x^6-\frac {35}{16} \tan ^{-1}\left (\sqrt {-1+\frac {1}{x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 70, normalized size = 0.92 \begin {gather*} \frac {1}{48} \sqrt {-1+\frac {1}{x^2}} \left (48+87 x^2-38 x^4+8 x^6\right )-\frac {35 \sqrt {-1+\frac {1}{x^2}} x \tanh ^{-1}\left (\frac {\sqrt {-1+x^2}}{-1+x}\right )}{8 \sqrt {-1+x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 83, normalized size = 1.09
method | result | size |
trager | \(2 \left (\frac {1}{12} x^{6}-\frac {19}{48} x^{4}+\frac {29}{32} x^{2}+\frac {1}{2}\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}+\frac {35 \RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (-\left (\RootOf \left (\textit {\_Z}^{2}+1\right )-\sqrt {-\frac {x^{2}-1}{x^{2}}}\right ) x \right )}{16}\) | \(66\) |
risch | \(\frac {\left (8 x^{8}-46 x^{6}+125 x^{4}-39 x^{2}-48\right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}}{48 x^{2}-48}-\frac {35 \arcsin \left (x \right ) \sqrt {-\frac {x^{2}-1}{x^{2}}}\, x \sqrt {-x^{2}+1}}{16 \left (x^{2}-1\right )}\) | \(78\) |
default | \(-\frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (8 x^{4} \left (-x^{2}+1\right )^{\frac {3}{2}}-30 x^{2} \left (-x^{2}+1\right )^{\frac {3}{2}}-48 \left (-x^{2}+1\right )^{\frac {3}{2}}-105 x^{2} \sqrt {-x^{2}+1}-105 \arcsin \left (x \right ) x \right )}{48 \sqrt {-x^{2}+1}}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 120 vs.
\(2 (56) = 112\).
time = 0.50, size = 120, normalized size = 1.58 \begin {gather*} \frac {3}{2} \, x^{2} \sqrt {\frac {1}{x^{2}} - 1} + \sqrt {\frac {1}{x^{2}} - 1} - \frac {3 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\frac {1}{x^{2}} - 1}}{48 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{3} + 3 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {3}{x^{2}} - 2\right )}} + \frac {3 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {1}{x^{2}} - 1}\right )}}{8 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {2}{x^{2}} - 1\right )}} - \frac {35}{16} \, \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 = 0.36, size = 55, normalized size = 0.72 \begin {gather*} \frac {1}{48} \, {\left (8 \, x^{6} - 38 \, x^{4} + 87 \, x^{2} + 48\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} - \frac {35}{8} \, \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 = 168.85, size = 80, normalized size = 1.05 \begin {gather*} - \frac {x^{6} \left (-1 + \frac {1}{x^{2}}\right )^{\frac {3}{2}}}{6} - \frac {5 x^{4} \sqrt {-1 + \frac {1}{x^{2}}} \cdot \left (2 - \frac {1}{x^{2}}\right )}{16} + \frac {3 x^{2} \sqrt {-1 + \frac {1}{x^{2}}}}{2} + \sqrt {-1 + \frac {1}{x^{2}}} - \frac {35 \operatorname {atan}{\left (\sqrt {-1 + \frac {1}{x^{2}}} \right )}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.51, size = 77, normalized size = 1.01 \begin {gather*} \frac {1}{48} \, {\left (2 \, {\left (4 \, x^{2} \mathrm {sgn}\left (x\right ) - 19 \, \mathrm {sgn}\left (x\right )\right )} x^{2} + 87 \, \mathrm {sgn}\left (x\right )\right )} \sqrt {-x^{2} + 1} x + \frac {35}{16} \, \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 = 3.50, size = 54, normalized size = 0.71 \begin {gather*} \sqrt {\frac {1}{x^2}-1}-\frac {35\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{16}+\frac {19\,x^6\,\sqrt {\frac {1}{x^2}-1}}{16}+\frac {17\,x^6\,{\left (\frac {1}{x^2}-1\right )}^{3/2}}{6}+\frac {29\,x^6\,{\left (\frac {1}{x^2}-1\right )}^{5/2}}{16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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