Optimal. Leaf size=44 \[ -\frac {\left (\left (x^4+1\right )^5\right )^{9/10} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2} \left (x^4+1\right )^{9/2}} \]
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Rubi [A] time = 0.40, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 42, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {6688, 6720, 1699, 203} \begin {gather*} -\frac {\sqrt {x^4+1} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2} \sqrt [10]{\left (x^4+1\right )^5}} \end {gather*}
Antiderivative was successfully verified.
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Rule 203
Rule 1699
Rule 6688
Rule 6720
Rubi steps
\begin {align*} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{1+5 x^4+10 x^8+10 x^{12}+5 x^{16}+x^{20}}} \, dx &=\int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt [10]{\left (1+x^4\right )^5}} \, dx\\ &=\frac {\sqrt {1+x^4} \int \frac {-1+x^2}{\left (1+x^2\right ) \sqrt {1+x^4}} \, dx}{\sqrt [10]{\left (1+x^4\right )^5}}\\ &=-\frac {\sqrt {1+x^4} \operatorname {Subst}\left (\int \frac {1}{1+2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )}{\sqrt [10]{\left (1+x^4\right )^5}}\\ &=-\frac {\sqrt {1+x^4} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2} \sqrt [10]{\left (1+x^4\right )^5}}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 61, normalized size = 1.39 \begin {gather*} -\frac {\sqrt [4]{-1} \sqrt {x^4+1} \left (F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )-2 \Pi \left (-i;\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right )}{\sqrt [10]{\left (x^4+1\right )^5}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 20.56, size = 44, normalized size = 1.00 \begin {gather*} -\frac {\left (\left (1+x^4\right )^5\right )^{9/10} \tan ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2} \left (1+x^4\right )^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.49, size = 45, normalized size = 1.02 \begin {gather*} -\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} x}{x^{4} + 1}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {x^{2}-1}{\left (x^{2}+1\right ) \left (x^{20}+5 x^{16}+10 x^{12}+10 x^{8}+5 x^{4}+1\right )^{\frac {1}{10}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} - 1}{{\left (x^{20} + 5 \, x^{16} + 10 \, x^{12} + 10 \, x^{8} + 5 \, x^{4} + 1\right )}^{\frac {1}{10}} {\left (x^{2} + 1\right )}}\,{d x} \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^2-1}{\left (x^2+1\right )\,{\left (x^{20}+5\,x^{16}+10\,x^{12}+10\,x^8+5\,x^4+1\right )}^{1/10}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (x - 1\right ) \left (x + 1\right )}{\left (x^{2} + 1\right ) \sqrt [10]{\left (x^{4} + 1\right )^{5}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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