Optimal. Leaf size=189 \[ -\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )}{\sqrt {2}}+\frac {\left (6 x^8+12 x^4+4\right ) x^2+\sqrt {x^4+1} \left (6 x^6+9 x^2\right ) x^2}{2 x \left (2 x^8+3 x^4+1\right ) \sqrt {\sqrt {x^4+1}+x^2}+2 x \sqrt {x^4+1} \sqrt {\sqrt {x^4+1}+x^2} \left (2 x^6+2 x^2\right )} \]
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Rubi [F] time = 2.75, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx &=\int \left (\frac {1}{\sqrt {x^2+\sqrt {1+x^4}}}+\frac {4}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {4}{\left (1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=4 \int \frac {1}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-4 \int \frac {1}{\left (1+x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=4 \int \left (-\frac {1}{4 \left (i-x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{4 \left (i+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {1}{2 \left (-1-x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx-4 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \int \frac {1}{\left (-1-x^4\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{\left (i-x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \frac {1}{\left (i+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )-2 \int \left (-\frac {i}{2 \left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 \left (i+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx-\int \left (\frac {i}{4 \left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {i}{4 \left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}+\frac {i}{2 \left (-i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx-\int \left (-\frac {i}{4 \left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{4 \left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}-\frac {i}{2 \left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} i \int \frac {1}{\left (-i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} i \int \frac {1}{\left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{\left (i-x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+i \int \frac {1}{\left (i+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx-\frac {1}{2} i \int \left (\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+i \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+i \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}}\right ) \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ &=-4 \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}-x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} i \int \frac {1}{\left (\sqrt [4]{-1}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} i \int \frac {1}{\left (-(-1)^{3/4}+x\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{4} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\frac {1}{2} \sqrt [4]{-1} \int \frac {1}{\left (\sqrt [4]{-1}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{4} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx-\frac {1}{2} (-1)^{3/4} \int \frac {1}{\left (-(-1)^{3/4}+x\right ) \sqrt {x^2+\sqrt {1+x^4}}} \, dx+\int \frac {1}{\sqrt {x^2+\sqrt {1+x^4}}} \, dx\\ \end {align*}
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Mathematica [F] time = 0.76, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (-1+x^4\right )^2}{\left (1+x^4\right )^2 \sqrt {x^2+\sqrt {1+x^4}}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.70, size = 202, normalized size = 1.07 \begin {gather*} \frac {x^2 \sqrt {1+x^4} \left (9 x^2+6 x^6\right )+x^2 \left (4+12 x^4+6 x^8\right )}{2 x \sqrt {1+x^4} \left (2 x^2+2 x^6\right ) \sqrt {x^2+\sqrt {1+x^4}}+2 x \left (1+3 x^4+2 x^8\right ) \sqrt {x^2+\sqrt {1+x^4}}}-\frac {3}{2} \tanh ^{-1}\left (\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {\tanh ^{-1}\left (\frac {-\frac {1}{\sqrt {2}}+\frac {x^2}{\sqrt {2}}+\frac {\sqrt {1+x^4}}{\sqrt {2}}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 1.37, size = 180, normalized size = 0.95 \begin {gather*} \frac {\sqrt {2} {\left (x^{4} + 1\right )} \log \left (4 \, x^{4} + 4 \, \sqrt {x^{4} + 1} x^{2} + 2 \, {\left (\sqrt {2} x^{3} + \sqrt {2} \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1\right ) + 3 \, {\left (x^{4} + 1\right )} \log \left (-\frac {9 \, x^{4} + 8 \, \sqrt {x^{4} + 1} x^{2} - 4 \, {\left (2 \, x^{3} + \sqrt {x^{4} + 1} x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + 1}{x^{4} + 1}\right ) - 4 \, {\left (x^{7} + 3 \, x^{3} - {\left (x^{5} + 4 \, x\right )} \sqrt {x^{4} + 1}\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{8 \, {\left (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 {{\left (x^{4} - 1\right )}^{2}}{{\left (x^{4} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{4}-1\right )^{2}}{\left (x^{4}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}\, dx\]
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 {{\left (x^{4} - 1\right )}^{2}}{{\left (x^{4} + 1\right )}^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}\,{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 = -0.01 \begin {gather*} \int \frac {{\left (x^4-1\right )}^2}{{\left (x^4+1\right )}^2\,\sqrt {\sqrt {x^4+1}+x^2}} \,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 )^{2} \left (x + 1\right )^{2} \left (x^{2} + 1\right )^{2}}{\sqrt {x^{2} + \sqrt {x^{4} + 1}} \left (x^{4} + 1\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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