Optimal. Leaf size=124 \[ \frac {x^3}{2 \sqrt {1+x^4}}-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} E\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{4 \sqrt {1+x^4}} \]
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Rubi [A]
time = 0.02, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {296, 311, 226,
1210} \begin {gather*} -\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{4 \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} E\left (2 \text {ArcTan}(x)\left |\frac {1}{2}\right .\right )}{2 \sqrt {x^4+1}}+\frac {x^3}{2 \sqrt {x^4+1}}-\frac {\sqrt {x^4+1} x}{2 \left (x^2+1\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 296
Rule 311
Rule 1210
Rubi steps
\begin {align*} \int \frac {x^2}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {x^3}{2 \sqrt {1+x^4}}-\frac {1}{2} \int \frac {x^2}{\sqrt {1+x^4}} \, dx\\ &=\frac {x^3}{2 \sqrt {1+x^4}}-\frac {1}{2} \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{2} \int \frac {1-x^2}{\sqrt {1+x^4}} \, dx\\ &=\frac {x^3}{2 \sqrt {1+x^4}}-\frac {x \sqrt {1+x^4}}{2 \left (1+x^2\right )}+\frac {\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 {\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] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 3.44, size = 22, normalized size = 0.18 \begin {gather*} \frac {1}{3} x^3 \, _2F_1\left (\frac {3}{4},\frac {3}{2};\frac {7}{4};-x^4\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.16, size = 95, normalized size = 0.77
method | result | size |
meijerg | \(\frac {x^{3} \hypergeom \left (\left [\frac {3}{4}, \frac {3}{2}\right ], \left [\frac {7}{4}\right ], -x^{4}\right )}{3}\) | \(17\) |
default | \(\frac {x^{3}}{2 \sqrt {x^{4}+1}}-\frac {i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(95\) |
risch | \(\frac {x^{3}}{2 \sqrt {x^{4}+1}}-\frac {i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(95\) |
elliptic | \(\frac {x^{3}}{2 \sqrt {x^{4}+1}}-\frac {i \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \left (\EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )-\EllipticE \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )\right )}{2 \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}\) | \(95\) |
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*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains complex when optimal does not.
time = 0.09, size = 59, normalized size = 0.48 \begin {gather*} \frac {\sqrt {x^{4} + 1} x^{3} + \sqrt {i} {\left (i \, x^{4} + i\right )} E(\arcsin \left (\sqrt {i} x\right )\,|\,-1) + \sqrt {i} {\left (-i \, x^{4} - i\right )} F(\arcsin \left (\sqrt {i} x\right )\,|\,-1)}{2 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.31, size = 29, normalized size = 0.23 \begin {gather*} \frac {x^{3} \Gamma \left (\frac {3}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{4}, \frac {3}{2} \\ \frac {7}{4} \end {matrix}\middle | {x^{4} e^{i \pi }} \right )}}{4 \Gamma \left (\frac {7}{4}\right )} \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{{\left (x^4+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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