Optimal. Leaf size=38 \[ -\frac {1}{2 \sqrt {1+x^4}}-\sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2} \]
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Rubi [A]
time = 0.01, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {272, 45}
\begin {gather*} \frac {1}{6} \left (x^4+1\right )^{3/2}-\sqrt {x^4+1}-\frac {1}{2 \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 272
Rubi steps
\begin {align*} \int \frac {x^{11}}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{4} \text {Subst}\left (\int \frac {x^2}{(1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \text {Subst}\left (\int \left (\frac {1}{(1+x)^{3/2}}-\frac {2}{\sqrt {1+x}}+\sqrt {1+x}\right ) \, dx,x,x^4\right )\\ &=-\frac {1}{2 \sqrt {1+x^4}}-\sqrt {1+x^4}+\frac {1}{6} \left (1+x^4\right )^{3/2}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 23, normalized size = 0.61 \begin {gather*} \frac {-8-4 x^4+x^8}{6 \sqrt {1+x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.15, size = 20, normalized size = 0.53
method | result | size |
gosper | \(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\) | \(20\) |
default | \(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\) | \(20\) |
trager | \(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\) | \(20\) |
risch | \(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\) | \(20\) |
elliptic | \(\frac {x^{8}-4 x^{4}-8}{6 \sqrt {x^{4}+1}}\) | \(20\) |
meijerg | \(\frac {\frac {8 \sqrt {\pi }}{3}-\frac {\sqrt {\pi }\, \left (-2 x^{8}+8 x^{4}+16\right )}{6 \sqrt {x^{4}+1}}}{2 \sqrt {\pi }}\) | \(36\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 28, normalized size = 0.74 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1} - \frac {1}{2 \, \sqrt {x^{4} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 19, normalized size = 0.50 \begin {gather*} \frac {x^{8} - 4 \, x^{4} - 8}{6 \, \sqrt {x^{4} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.33, size = 39, normalized size = 1.03 \begin {gather*} \frac {x^{8}}{6 \sqrt {x^{4} + 1}} - \frac {2 x^{4}}{3 \sqrt {x^{4} + 1}} - \frac {4}{3 \sqrt {x^{4} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.08, size = 28, normalized size = 0.74 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1} - \frac {1}{2 \, \sqrt {x^{4} + 1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.32, size = 25, normalized size = 0.66 \begin {gather*} -\frac {6\,x^4-{\left (x^4+1\right )}^2+9}{6\,\sqrt {x^4+1}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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