Optimal. Leaf size=422 \[ -\frac {2 \sqrt {2} \sqrt {2 x^4+2 x^2+1} x}{3 \left (\sqrt {2} x^2+1\right )}+\frac {2 \sqrt {2 x^4+2 x^2+1}}{3 x}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{9 \sqrt {15}}-\frac {\left (1+19 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{63 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+\frac {2 \sqrt [4]{2} \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {2 x^4+2 x^2+1}}-\frac {\left (3+\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{189 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}-\frac {\sqrt {2 x^4+2 x^2+1}}{9 x^3} \]
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Rubi [A] time = 0.52, antiderivative size = 422, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.241, Rules used = {1329, 1683, 1714, 1195, 1708, 1103, 1706} \[ -\frac {2 \sqrt {2} \sqrt {2 x^4+2 x^2+1} x}{3 \left (\sqrt {2} x^2+1\right )}+\frac {2 \sqrt {2 x^4+2 x^2+1}}{3 x}-\frac {\sqrt {2 x^4+2 x^2+1}}{9 x^3}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {2 x^4+2 x^2+1}}\right )}{9 \sqrt {15}}-\frac {\left (1+19 \sqrt {2}\right ) \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{63 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}}+\frac {2 \sqrt [4]{2} \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {2 x^4+2 x^2+1}}-\frac {\left (3+\sqrt {2}\right )^2 \left (\sqrt {2} x^2+1\right ) \sqrt {\frac {2 x^4+2 x^2+1}{\left (\sqrt {2} x^2+1\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{189 \sqrt [4]{2} \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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Rule 1103
Rule 1195
Rule 1329
Rule 1683
Rule 1706
Rule 1708
Rule 1714
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx &=-\frac {\sqrt {1+2 x^2+2 x^4}}{9 x^3}+\frac {1}{9} \int \frac {-18-14 x^2-4 x^4}{x^2 \left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{9 x^3}+\frac {2 \sqrt {1+2 x^2+2 x^4}}{3 x}-\frac {1}{27} \int \frac {6+120 x^2+72 x^4}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{9 x^3}+\frac {2 \sqrt {1+2 x^2+2 x^4}}{3 x}-\frac {1}{108} \int \frac {24+216 \sqrt {2}+\left (480-72 \left (6-2 \sqrt {2}\right )\right ) x^2}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx+\frac {1}{3} \left (2 \sqrt {2}\right ) \int \frac {1-\sqrt {2} x^2}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{9 x^3}+\frac {2 \sqrt {1+2 x^2+2 x^4}}{3 x}-\frac {2 \sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{3 \left (1+\sqrt {2} x^2\right )}+\frac {2 \sqrt [4]{2} \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {1+2 x^2+2 x^4}}-\frac {1}{63} \left (4 \left (2+3 \sqrt {2}\right )\right ) \int \frac {1+\sqrt {2} x^2}{\left (3+2 x^2\right ) \sqrt {1+2 x^2+2 x^4}} \, dx-\frac {1}{63} \left (2 \left (1+19 \sqrt {2}\right )\right ) \int \frac {1}{\sqrt {1+2 x^2+2 x^4}} \, dx\\ &=-\frac {\sqrt {1+2 x^2+2 x^4}}{9 x^3}+\frac {2 \sqrt {1+2 x^2+2 x^4}}{3 x}-\frac {2 \sqrt {2} x \sqrt {1+2 x^2+2 x^4}}{3 \left (1+\sqrt {2} x^2\right )}+\frac {2 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{3}} x}{\sqrt {1+2 x^2+2 x^4}}\right )}{9 \sqrt {15}}+\frac {2 \sqrt [4]{2} \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{3 \sqrt {1+2 x^2+2 x^4}}-\frac {\left (1+19 \sqrt {2}\right ) \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{63 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}-\frac {\left (3+\sqrt {2}\right )^2 \left (1+\sqrt {2} x^2\right ) \sqrt {\frac {1+2 x^2+2 x^4}{\left (1+\sqrt {2} x^2\right )^2}} \Pi \left (\frac {1}{24} \left (12-11 \sqrt {2}\right );2 \tan ^{-1}\left (\sqrt [4]{2} x\right )|\frac {1}{4} \left (2-\sqrt {2}\right )\right )}{189 \sqrt [4]{2} \sqrt {1+2 x^2+2 x^4}}\\ \end {align*}
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Mathematica [C] time = 0.18, size = 219, normalized size = 0.52 \[ \frac {36 x^6+30 x^4+12 x^2-(3+15 i) \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} x^3 F\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+18 i \sqrt {1-i} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} x^3 E\left (\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )+2 (1-i)^{3/2} \sqrt {1+(1-i) x^2} \sqrt {1+(1+i) x^2} x^3 \Pi \left (\frac {1}{3}+\frac {i}{3};\left .i \sinh ^{-1}\left (\sqrt {1-i} x\right )\right |i\right )-3}{27 x^3 \sqrt {2 x^4+2 x^2+1}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.36, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {2 \, x^{4} + 2 \, x^{2} + 1}}{4 \, x^{10} + 10 \, x^{8} + 8 \, x^{6} + 3 \, x^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 1} {\left (2 \, x^{2} + 3\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 260, normalized size = 0.62 \[ -\frac {2 \sqrt {\left (1-i\right ) x^{2}+1}\, \sqrt {\left (1+i\right ) x^{2}+1}\, \EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )}{9 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {4 \sqrt {-i x^{2}+x^{2}+1}\, \sqrt {i x^{2}+x^{2}+1}\, \EllipticPi \left (\sqrt {-1+i}\, x , \frac {1}{3}+\frac {i}{3}, \frac {\sqrt {-1-i}}{\sqrt {-1+i}}\right )}{27 \sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}}+\frac {2 \sqrt {2 x^{4}+2 x^{2}+1}}{3 x}-\frac {\sqrt {2 x^{4}+2 x^{2}+1}}{9 x^{3}}+\frac {\left (\frac {2}{3}-\frac {2 i}{3}\right ) \sqrt {\left (1-i\right ) x^{2}+1}\, \sqrt {\left (1+i\right ) x^{2}+1}\, \left (-\EllipticE \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )+\EllipticF \left (\sqrt {-1+i}\, x , \frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right )\right )}{\sqrt {-1+i}\, \sqrt {2 x^{4}+2 x^{2}+1}} \]
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 \[ \int \frac {1}{\sqrt {2 \, x^{4} + 2 \, x^{2} + 1} {\left (2 \, x^{2} + 3\right )} x^{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^4\,\left (2\,x^2+3\right )\,\sqrt {2\,x^4+2\,x^2+1}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{4} \left (2 x^{2} + 3\right ) \sqrt {2 x^{4} + 2 x^{2} + 1}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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