Optimal. Leaf size=41 \[ -\frac {x^6}{2 \sqrt {1+x^4}}+\frac {3}{4} x^2 \sqrt {1+x^4}-\frac {3}{4} \sinh ^{-1}\left (x^2\right ) \]
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Rubi [A]
time = 0.01, antiderivative size = 41, 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 = {281, 294, 327,
221} \begin {gather*} -\frac {3}{4} \sinh ^{-1}\left (x^2\right )-\frac {x^6}{2 \sqrt {x^4+1}}+\frac {3}{4} \sqrt {x^4+1} x^2 \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 281
Rule 294
Rule 327
Rubi steps
\begin {align*} \int \frac {x^9}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x^4}{\left (1+x^2\right )^{3/2}} \, dx,x,x^2\right )\\ &=-\frac {x^6}{2 \sqrt {1+x^4}}+\frac {3}{2} \text {Subst}\left (\int \frac {x^2}{\sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^6}{2 \sqrt {1+x^4}}+\frac {3}{4} x^2 \sqrt {1+x^4}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=-\frac {x^6}{2 \sqrt {1+x^4}}+\frac {3}{4} x^2 \sqrt {1+x^4}-\frac {3}{4} \sinh ^{-1}\left (x^2\right )\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 41, normalized size = 1.00 \begin {gather*} \frac {3 x^2+x^6}{4 \sqrt {1+x^4}}-\frac {3}{4} \tanh ^{-1}\left (\frac {x^2}{\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 32, normalized size = 0.78
method | result | size |
risch | \(\frac {x^{2} \left (x^{4}+3\right )}{4 \sqrt {x^{4}+1}}-\frac {3 \arcsinh \left (x^{2}\right )}{4}\) | \(25\) |
default | \(\frac {x^{6}}{4 \sqrt {x^{4}+1}}+\frac {3 x^{2}}{4 \sqrt {x^{4}+1}}-\frac {3 \arcsinh \left (x^{2}\right )}{4}\) | \(32\) |
elliptic | \(\frac {x^{6}}{4 \sqrt {x^{4}+1}}+\frac {3 x^{2}}{4 \sqrt {x^{4}+1}}-\frac {3 \arcsinh \left (x^{2}\right )}{4}\) | \(32\) |
trager | \(\frac {x^{2} \left (x^{4}+3\right )}{4 \sqrt {x^{4}+1}}+\frac {3 \ln \left (x^{2}-\sqrt {x^{4}+1}\right )}{4}\) | \(35\) |
meijerg | \(\frac {\frac {\sqrt {\pi }\, x^{2} \left (5 x^{4}+15\right )}{10 \sqrt {x^{4}+1}}-\frac {3 \sqrt {\pi }\, \arcsinh \left (x^{2}\right )}{2}}{2 \sqrt {\pi }}\) | \(38\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 73 vs.
\(2 (31) = 62\).
time = 0.29, size = 73, normalized size = 1.78 \begin {gather*} -\frac {\frac {3 \, {\left (x^{4} + 1\right )}}{x^{4}} - 2}{4 \, {\left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{x^{6}}\right )}} - \frac {3}{8} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) + \frac {3}{8} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 54, normalized size = 1.32 \begin {gather*} \frac {2 \, x^{4} + 3 \, {\left (x^{4} + 1\right )} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) + {\left (x^{6} + 3 \, x^{2}\right )} \sqrt {x^{4} + 1} + 2}{4 \, {\left (x^{4} + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.39, size = 36, normalized size = 0.88 \begin {gather*} \frac {x^{6}}{4 \sqrt {x^{4} + 1}} + \frac {3 x^{2}}{4 \sqrt {x^{4} + 1}} - \frac {3 \operatorname {asinh}{\left (x^{2} \right )}}{4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.53, size = 34, normalized size = 0.83 \begin {gather*} \frac {{\left (x^{4} + 3\right )} x^{2}}{4 \, \sqrt {x^{4} + 1}} + \frac {3}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \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 = -0.02 \begin {gather*} \int \frac {x^9}{{\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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