Optimal. Leaf size=27 \[ \frac {\sqrt {x^4+1}}{2}+\frac {1}{2 \sqrt {x^4+1}} \]
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Rubi [A] time = 0.01, antiderivative size = 27, 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 = {266, 43} \[ \frac {\sqrt {x^4+1}}{2}+\frac {1}{2 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{(1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {1}{(1+x)^{3/2}}+\frac {1}{\sqrt {1+x}}\right ) \, dx,x,x^4\right )\\ &=\frac {1}{2 \sqrt {1+x^4}}+\frac {\sqrt {1+x^4}}{2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 18, normalized size = 0.67 \[ \frac {x^4+2}{2 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.87, size = 14, normalized size = 0.52 \[ \frac {x^{4} + 2}{2 \, \sqrt {x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 19, normalized size = 0.70 \[ \frac {1}{2} \, \sqrt {x^{4} + 1} + \frac {1}{2 \, \sqrt {x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 15, normalized size = 0.56 \[ \frac {x^{4}+2}{2 \sqrt {x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.31, size = 19, normalized size = 0.70 \[ \frac {1}{2} \, \sqrt {x^{4} + 1} + \frac {1}{2 \, \sqrt {x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 14, normalized size = 0.52 \[ \frac {x^4+2}{2\,\sqrt {x^4+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.17, size = 22, normalized size = 0.81 \[ \frac {x^{4}}{2 \sqrt {x^{4} + 1}} + \frac {1}{\sqrt {x^{4} + 1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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