Optimal. Leaf size=74 \[ 3 \tanh ^{-1}\left (\sqrt {\frac {1-2 x^2}{2 x^2+1}}\right )-\frac {2 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^4-x^2-1\right )}{3 \left (2 x^4+x^2-1\right )} \]
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Rubi [A] time = 3.70, antiderivative size = 129, normalized size of antiderivative = 1.74, number of steps used = 17, number of rules used = 12, integrand size = 62, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {6688, 6719, 1586, 1606, 37, 96, 92, 206, 103, 21, 93, 203} \begin {gather*} -\frac {4}{3 \left (x^2+1\right ) \sqrt {\frac {1-2 x^2}{2 x^2+1}}}+\frac {2}{3 \sqrt {\frac {1-2 x^2}{2 x^2+1}}}+\frac {3 \sqrt {1-2 x^2} \tanh ^{-1}\left (\sqrt {1-2 x^2} \sqrt {2 x^2+1}\right )}{2 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \sqrt {2 x^2+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 21
Rule 37
Rule 92
Rule 93
Rule 96
Rule 103
Rule 203
Rule 206
Rule 1586
Rule 1606
Rule 6688
Rule 6719
Rubi steps
\begin {align*} \int \frac {3-9 x^4+2 x^6}{x \left (1+x^2\right )^2 \left (-1+2 x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1+2 x^2\right )} \, dx &=\int \frac {-3+9 x^4-2 x^6}{x \left (1+x^2\right )^2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \left (1-4 x^4\right )} \, dx\\ &=\frac {\sqrt {1-2 x^2} \int \frac {\sqrt {1+2 x^2} \left (-3+9 x^4-2 x^6\right )}{x \sqrt {1-2 x^2} \left (1+x^2\right )^2 \left (1-4 x^4\right )} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} \int \frac {-3+9 x^4-2 x^6}{x \left (1-2 x^2\right )^{3/2} \left (1+x^2\right )^2 \sqrt {1+2 x^2}} \, dx}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} \operatorname {Subst}\left (\int \frac {-3+9 x^2-2 x^3}{(1-2 x)^{3/2} x (1+x)^2 \sqrt {1+2 x}} \, dx,x,x^2\right )}{2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {\sqrt {1-2 x^2} \operatorname {Subst}\left (\int \left (-\frac {2}{(1-2 x)^{3/2} \sqrt {1+2 x}}-\frac {3}{(1-2 x)^{3/2} x \sqrt {1+2 x}}-\frac {8}{(1-2 x)^{3/2} (1+x)^2 \sqrt {1+2 x}}+\frac {16}{(1-2 x)^{3/2} (1+x) \sqrt {1+2 x}}\right ) \, dx,x,x^2\right )}{2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=-\frac {\sqrt {1-2 x^2} \operatorname {Subst}\left (\int \frac {1}{(1-2 x)^{3/2} \sqrt {1+2 x}} \, dx,x,x^2\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (3 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-2 x)^{3/2} x \sqrt {1+2 x}} \, dx,x,x^2\right )}{2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (4 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-2 x)^{3/2} (1+x)^2 \sqrt {1+2 x}} \, dx,x,x^2\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-2 x)^{3/2} (1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {2}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}}}-\frac {4}{3 \left (1+x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}}}-\frac {\left (4 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {2-4 x}{(1-2 x)^{3/2} (1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (3 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x} x \sqrt {1+2 x}} \, dx,x,x^2\right )}{2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (8 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x} (1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {2}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}}}-\frac {4}{3 \left (1+x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}}}-\frac {\left (8 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-2 x} (1+x) \sqrt {1+2 x}} \, dx,x,x^2\right )}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (3 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{2-2 x^2} \, dx,x,\sqrt {1-2 x^2} \sqrt {1+2 x^2}\right )}{\sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {\left (16 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\frac {\sqrt {1+2 x^2}}{\sqrt {1-2 x^2}}\right )}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {2}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}}}-\frac {4}{3 \left (1+x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}}}+\frac {16 \sqrt {1-2 x^2} \tan ^{-1}\left (\frac {\sqrt {3} \sqrt {1+2 x^2}}{\sqrt {1-2 x^2}}\right )}{3 \sqrt {3} \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}+\frac {3 \sqrt {1-2 x^2} \tanh ^{-1}\left (\sqrt {1-2 x^2} \sqrt {1+2 x^2}\right )}{2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}-\frac {\left (16 \sqrt {1-2 x^2}\right ) \operatorname {Subst}\left (\int \frac {1}{1+3 x^2} \, dx,x,\frac {\sqrt {1+2 x^2}}{\sqrt {1-2 x^2}}\right )}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ &=\frac {2}{3 \sqrt {\frac {1-2 x^2}{1+2 x^2}}}-\frac {4}{3 \left (1+x^2\right ) \sqrt {\frac {1-2 x^2}{1+2 x^2}}}+\frac {3 \sqrt {1-2 x^2} \tanh ^{-1}\left (\sqrt {1-2 x^2} \sqrt {1+2 x^2}\right )}{2 \sqrt {\frac {1-2 x^2}{1+2 x^2}} \sqrt {1+2 x^2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 80, normalized size = 1.08 \begin {gather*} \frac {8 x^4-4 x^2+9 \sqrt {4-\frac {1}{x^4}} \left (x^2+1\right ) x^2 \sin ^{-1}\left (\frac {1}{2 x^2}\right )-4}{6 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^4+3 x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.12, size = 74, normalized size = 1.00 \begin {gather*} 3 \tanh ^{-1}\left (\sqrt {\frac {1-2 x^2}{2 x^2+1}}\right )-\frac {2 \sqrt {\frac {1-2 x^2}{2 x^2+1}} \left (2 x^4-x^2-1\right )}{3 \left (2 x^4+x^2-1\right )} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 107, normalized size = 1.45 \begin {gather*} -\frac {8 \, x^{4} + 4 \, x^{2} + 9 \, {\left (2 \, x^{4} + x^{2} - 1\right )} \log \left (\frac {{\left (2 \, x^{2} + 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} - 1}{x^{2}}\right ) + 4 \, {\left (2 \, x^{4} - x^{2} - 1\right )} \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}} - 4}{6 \, {\left (2 \, x^{4} + x^{2} - 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.55, size = 44, normalized size = 0.59 \begin {gather*} -\frac {32}{3 \, {\left (\frac {{\left (\sqrt {-4 \, x^{4} + 1} - 1\right )}^{3}}{x^{6}} + 8\right )}} - \frac {3}{2} \, \log \left (-\frac {\sqrt {-4 \, x^{4} + 1} - 1}{2 \, x^{2}}\right ) + \frac {2}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.10, size = 169, normalized size = 2.28 \begin {gather*} -\frac {64 \sqrt {-4 x^{4}+1}\, x^{6}+16 \left (-4 x^{4}+1\right )^{\frac {3}{2}} x^{2}-32 \sqrt {-4 x^{4}+1}\, x^{4}+162 \arctanh \left (\frac {1}{\sqrt {-4 x^{4}+1}}\right ) x^{4}-8 \left (-4 x^{4}+1\right )^{\frac {3}{2}}-52 x^{2} \sqrt {-4 x^{4}+1}+81 \arctanh \left (\frac {1}{\sqrt {-4 x^{4}+1}}\right ) x^{2}+44 \sqrt {-4 x^{4}+1}-81 \arctanh \left (\frac {1}{\sqrt {-4 x^{4}+1}}\right )}{54 \left (x^{2}+1\right ) \sqrt {-\left (2 x^{2}+1\right ) \left (2 x^{2}-1\right )}\, \sqrt {-\frac {2 x^{2}-1}{2 x^{2}+1}}} \end {gather*}
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*} \int \frac {2 \, x^{6} - 9 \, x^{4} + 3}{{\left (2 \, x^{2} + 1\right )} {\left (2 \, x^{2} - 1\right )} {\left (x^{2} + 1\right )}^{2} x \sqrt {-\frac {2 \, x^{2} - 1}{2 \, x^{2} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.42, size = 270, normalized size = 3.65 \begin {gather*} 3\,\mathrm {atanh}\left (\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\right )-\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}}{3}\right )+\frac {2\,\sqrt {3}\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {1-2\,x^2}\,\sqrt {\frac {1}{2\,x^2+1}}}{3}\right )}{2\,x^2+2}-\frac {\left (x^2+\frac {1}{2}\right )\,\left (\frac {x^2}{3}-\frac {1}{3}\right )\,\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}}{2\,x^4+x^2-1}+\frac {3\,x^2}{\sqrt {1-2\,x^2}\,\left (2\,x^2+2\right )\,\sqrt {\frac {1}{2\,x^2+1}}}+\frac {2\,\sqrt {3}\,x^2\,\mathrm {atan}\left (\frac {\sqrt {3}\,\sqrt {1-2\,x^2}\,\sqrt {\frac {1}{2\,x^2+1}}}{3}\right )}{2\,x^2+2}-\frac {2{}\mathrm {i}}{\sqrt {-\frac {2\,x^2-1}{2\,x^2+1}}\,3{}\mathrm {i}+{\left (-\frac {2\,x^2-1}{2\,x^2+1}\right )}^{3/2}\,1{}\mathrm {i}} \end {gather*}
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 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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