Optimal. Leaf size=76 \[ -\frac {35}{48} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {35}{16} \sqrt {\frac {1}{x^2}-1}-\frac {35}{16} \tan ^{-1}\left (\sqrt {\frac {1}{x^2}-1}\right )-\frac {1}{6} \left (\frac {1}{x^2}-1\right )^{7/2} x^6-\frac {7}{24} \left (\frac {1}{x^2}-1\right )^{5/2} x^4 \]
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Rubi [A] time = 0.03, 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, 266, 47, 50, 63, 203} \[ -\frac {1}{6} \left (\frac {1}{x^2}-1\right )^{7/2} x^6-\frac {7}{24} \left (\frac {1}{x^2}-1\right )^{5/2} x^4-\frac {35}{48} \left (\frac {1}{x^2}-1\right )^{3/2} x^2+\frac {35}{16} \sqrt {\frac {1}{x^2}-1}-\frac {35}{16} \tan ^{-1}\left (\sqrt {\frac {1}{x^2}-1}\right ) \]
Antiderivative was successfully verified.
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Rule 25
Rule 47
Rule 50
Rule 63
Rule 203
Rule 266
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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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} \operatorname {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 [C] time = 0.01, size = 34, normalized size = 0.45 \[ \frac {\sqrt {\frac {1}{x^2}-1} \, _2F_1\left (-\frac {7}{2},-\frac {1}{2};\frac {1}{2};x^2\right )}{\sqrt {1-x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.48, size = 55, normalized size = 0.72 \[ \frac {1}{48} \, {\left (8 \, x^{6} - 38 \, x^{4} + 87 \, x^{2} + 48\right )} \sqrt {-\frac {x^{2} - 1}{x^{2}}} - \frac {35}{8} \, \arctan \left (\frac {x \sqrt {-\frac {x^{2} - 1}{x^{2}}} - 1}{x}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.39, size = 77, normalized size = 1.01 \[ \frac {1}{48} \, {\left (2 \, {\left (4 \, x^{2} \mathrm {sgn}\relax (x) - 19 \, \mathrm {sgn}\relax (x)\right )} x^{2} + 87 \, \mathrm {sgn}\relax (x)\right )} \sqrt {-x^{2} + 1} x + \frac {35}{16} \, \arcsin \relax (x) \mathrm {sgn}\relax (x) - \frac {x \mathrm {sgn}\relax (x)}{2 \, {\left (\sqrt {-x^{2} + 1} - 1\right )}} + \frac {{\left (\sqrt {-x^{2} + 1} - 1\right )} \mathrm {sgn}\relax (x)}{2 \, x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 83, normalized size = 1.09 \[ \frac {\sqrt {-\frac {x^{2}-1}{x^{2}}}\, \left (-8 \left (-x^{2}+1\right )^{\frac {3}{2}} x^{4}+30 \left (-x^{2}+1\right )^{\frac {3}{2}} x^{2}+105 \sqrt {-x^{2}+1}\, x^{2}+105 x \arcsin \relax (x )+48 \left (-x^{2}+1\right )^{\frac {3}{2}}\right )}{48 \sqrt {-x^{2}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 2.07, size = 120, normalized size = 1.58 \[ \frac {3}{2} \, x^{2} \sqrt {\frac {1}{x^{2}} - 1} + \sqrt {\frac {1}{x^{2}} - 1} - \frac {3 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {5}{2}} + 8 \, {\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - 3 \, \sqrt {\frac {1}{x^{2}} - 1}}{48 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{3} + 3 \, {\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {3}{x^{2}} - 2\right )}} + \frac {3 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{\frac {3}{2}} - \sqrt {\frac {1}{x^{2}} - 1}\right )}}{8 \, {\left ({\left (\frac {1}{x^{2}} - 1\right )}^{2} + \frac {2}{x^{2}} - 1\right )}} - \frac {35}{16} \, \arctan \left (\sqrt {\frac {1}{x^{2}} - 1}\right ) \]
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 \[ \sqrt {\frac {1}{x^2}-1}-\frac {35\,\mathrm {atan}\left (\sqrt {\frac {1}{x^2}-1}\right )}{16}+\frac {19\,x^6\,\sqrt {\frac {1}{x^2}-1}}{16}+\frac {17\,x^6\,{\left (\frac {1}{x^2}-1\right )}^{3/2}}{6}+\frac {29\,x^6\,{\left (\frac {1}{x^2}-1\right )}^{5/2}}{16} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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