Optimal. Leaf size=41 \[ -\frac {3}{4} \sinh ^{-1}\left (x^2\right )-\frac {x^6}{2 \sqrt {x^4+1}}+\frac {3}{4} \sqrt {x^4+1} x^2 \]
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Rubi [A] time = 0.02, 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 = {275, 288, 321, 215} \[ -\frac {x^6}{2 \sqrt {x^4+1}}+\frac {3}{4} \sqrt {x^4+1} x^2-\frac {3}{4} \sinh ^{-1}\left (x^2\right ) \]
Antiderivative was successfully verified.
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Rule 215
Rule 275
Rule 288
Rule 321
Rubi steps
\begin {align*} \int \frac {x^9}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {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} \operatorname {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} \operatorname {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 = 0.01, size = 37, normalized size = 0.90 \[ \frac {x^6+3 x^2-3 \sqrt {x^4+1} \sinh ^{-1}\left (x^2\right )}{4 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 54, normalized size = 1.32 \[ \frac {2 \, x^{4} + 3 \, {\left (x^{4} + 1\right )} \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) + {\left (x^{6} + 3 \, x^{2}\right )} \sqrt {x^{4} + 1} + 2}{4 \, {\left (x^{4} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 34, normalized size = 0.83 \[ \frac {{\left (x^{4} + 3\right )} x^{2}}{4 \, \sqrt {x^{4} + 1}} + \frac {3}{4} \, \log \left (-x^{2} + \sqrt {x^{4} + 1}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 32, normalized size = 0.78 \[ \frac {x^{6}}{4 \sqrt {x^{4}+1}}+\frac {3 x^{2}}{4 \sqrt {x^{4}+1}}-\frac {3 \arcsinh \left (x^{2}\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.38, size = 73, normalized size = 1.78 \[ -\frac {\frac {3 \, {\left (x^{4} + 1\right )}}{x^{4}} - 2}{4 \, {\left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - \frac {{\left (x^{4} + 1\right )}^{\frac {3}{2}}}{x^{6}}\right )}} - \frac {3}{8} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} + 1\right ) + \frac {3}{8} \, \log \left (\frac {\sqrt {x^{4} + 1}}{x^{2}} - 1\right ) \]
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 \[ \int \frac {x^9}{{\left (x^4+1\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.01, size = 36, normalized size = 0.88 \[ \frac {x^{6}}{4 \sqrt {x^{4} + 1}} + \frac {3 x^{2}}{4 \sqrt {x^{4} + 1}} - \frac {3 \operatorname {asinh}{\left (x^{2} \right )}}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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