Optimal. Leaf size=140 \[ -\frac {3 \sqrt {x^4+1}}{2 x}+\frac {1}{2 \sqrt {x^4+1} x}+\frac {3 \sqrt {x^4+1} x}{2 \left (x^2+1\right )}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {x^4+1}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.02, antiderivative size = 140, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {290, 325, 305, 220, 1196} \[ \frac {3 \sqrt {x^4+1} x}{2 \left (x^2+1\right )}-\frac {3 \sqrt {x^4+1}}{2 x}+\frac {1}{2 \sqrt {x^4+1} x}+\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {x^4+1}}-\frac {3 \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 220
Rule 290
Rule 305
Rule 325
Rule 1196
Rubi steps
\begin {align*} \int \frac {1}{x^2 \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{2 x \sqrt {1+x^4}}+\frac {3}{2} \int \frac {1}{x^2 \sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{2 x}+\frac {3}{2} \int \frac {x^2}{\sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{2 x}+\frac {3}{2} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {3}{2} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\\ &=\frac {1}{2 x \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{2 x}+\frac {3 x \sqrt {1+x^4}}{2 \left (1+x^2\right )}-\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {3 \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.00, size = 20, normalized size = 0.14 \[ -\frac {\, _2F_1\left (-\frac {1}{4},\frac {3}{2};\frac {3}{4};-x^4\right )}{x} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.85, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {x^{4} + 1}}{x^{10} + 2 \, x^{6} + x^{2}}, x\right ) \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 107, normalized size = 0.76 \[ -\frac {x^{3}}{2 \sqrt {x^{4}+1}}-\frac {\sqrt {x^{4}+1}}{x}+\frac {3 i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (-\EllipticE \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )+\EllipticF \left (\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) x , i\right )\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}} \]
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 \[ \int \frac {1}{{\left (x^{4} + 1\right )}^{\frac {3}{2}} x^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 15, normalized size = 0.11 \[ -\frac {{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {3}{2};\ \frac {3}{4};\ -x^4\right )}{x} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [C] time = 1.46, size = 31, normalized size = 0.22 \[ \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {3}{2} \\ \frac {3}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 x \Gamma \left (\frac {3}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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