Optimal. Leaf size=44 \[ -\frac {1}{4 x^4 \sqrt {x^4+1}}-\frac {3}{4 \sqrt {x^4+1}}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]
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Rubi [A] time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.07, number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {266, 51, 63, 207} \[ -\frac {3 \sqrt {x^4+1}}{4 x^4}+\frac {1}{2 x^4 \sqrt {x^4+1}}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {x^4+1}\right ) \]
Antiderivative was successfully verified.
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Rule 51
Rule 63
Rule 207
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (1+x^4\right )^{3/2}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 (1+x)^{3/2}} \, dx,x,x^4\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}+\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{4 x^4}-\frac {3}{8} \operatorname {Subst}\left (\int \frac {1}{x \sqrt {1+x}} \, dx,x,x^4\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{4 x^4}-\frac {3}{4} \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sqrt {1+x^4}\right )\\ &=\frac {1}{2 x^4 \sqrt {1+x^4}}-\frac {3 \sqrt {1+x^4}}{4 x^4}+\frac {3}{4} \tanh ^{-1}\left (\sqrt {1+x^4}\right )\\ \end {align*}
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Mathematica [C] time = 0.01, size = 26, normalized size = 0.59 \[ -\frac {\, _2F_1\left (-\frac {1}{2},2;\frac {1}{2};x^4+1\right )}{2 \sqrt {x^4+1}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.83, size = 66, normalized size = 1.50 \[ \frac {3 \, {\left (x^{8} + x^{4}\right )} \log \left (\sqrt {x^{4} + 1} + 1\right ) - 3 \, {\left (x^{8} + x^{4}\right )} \log \left (\sqrt {x^{4} + 1} - 1\right ) - 2 \, {\left (3 \, x^{4} + 1\right )} \sqrt {x^{4} + 1}}{8 \, {\left (x^{8} + x^{4}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 53, normalized size = 1.20 \[ -\frac {3 \, x^{4} + 1}{4 \, {\left ({\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1}\right )}} + \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 33, normalized size = 0.75 \[ \frac {3 \arctanh \left (\frac {1}{\sqrt {x^{4}+1}}\right )}{4}-\frac {1}{4 \sqrt {x^{4}+1}\, x^{4}}-\frac {3}{4 \sqrt {x^{4}+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 53, normalized size = 1.20 \[ -\frac {3 \, x^{4} + 1}{4 \, {\left ({\left (x^{4} + 1\right )}^{\frac {3}{2}} - \sqrt {x^{4} + 1}\right )}} + \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} + 1\right ) - \frac {3}{8} \, \log \left (\sqrt {x^{4} + 1} - 1\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.24, size = 32, normalized size = 0.73 \[ \frac {3\,\mathrm {atanh}\left (\sqrt {x^4+1}\right )}{4}-\frac {3}{4\,\sqrt {x^4+1}}-\frac {1}{4\,x^4\,\sqrt {x^4+1}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.07, size = 42, normalized size = 0.95 \[ \frac {3 \operatorname {asinh}{\left (\frac {1}{x^{2}} \right )}}{4} - \frac {3}{4 x^{2} \sqrt {1 + \frac {1}{x^{4}}}} - \frac {1}{4 x^{6} \sqrt {1 + \frac {1}{x^{4}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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