Optimal. Leaf size=189 \[ \frac {-\left (\left (x^2-1\right ) x^2\right )-\sqrt {x^4+1} x^2}{4 x \left (x^2+1\right ) \sqrt {\sqrt {x^4+1}+x^2}}+\frac {1}{4} \sqrt {5 \sqrt {2}-1} \tan ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )+\frac {1}{4} \sqrt {1+5 \sqrt {2}} \tanh ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [C] time = 1.46, antiderivative size = 405, normalized size of antiderivative = 2.14, number of steps used = 28, number of rules used = 6, integrand size = 34, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {6742, 2133, 731, 725, 206, 6725} \begin {gather*} \frac {i \sqrt {1-i x^2}}{8 (-x+i)}-\frac {i \sqrt {1-i x^2}}{8 (x+i)}-\frac {i \sqrt {1+i x^2}}{8 (-x+i)}+\frac {i \sqrt {1+i x^2}}{8 (x+i)}-\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{4 (1+i)^{5/2}}+\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x+1}{\sqrt {1+i} \sqrt {1-i x^2}}\right )}{4 (1+i)^{5/2}}-\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {\tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{4 (1-i)^{5/2}}+\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {\tanh ^{-1}\left (\frac {x+1}{\sqrt {1-i} \sqrt {1+i x^2}}\right )}{4 (1-i)^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 206
Rule 725
Rule 731
Rule 2133
Rule 6725
Rule 6742
Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx &=\int \left (-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 (i-x)^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{4 (i+x)^2 \sqrt {1+x^4}}-\frac {\sqrt {x^2+\sqrt {1+x^4}}}{2 \left (-1-x^2\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i-x)^2 \sqrt {1+x^4}} \, dx\right )-\frac {1}{4} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i+x)^2 \sqrt {1+x^4}} \, dx-\frac {1}{2} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (-1-x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\left (\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{(i-x)^2 \sqrt {1-i x^2}} \, dx\right )-\left (\frac {1}{8}-\frac {i}{8}\right ) \int \frac {1}{(i+x)^2 \sqrt {1-i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x)^2 \sqrt {1+i x^2}} \, dx-\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x)^2 \sqrt {1+i x^2}} \, dx-\frac {1}{2} \int \left (-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i-x) \sqrt {1+x^4}}-\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 (i+x) \sqrt {1+x^4}}\right ) \, dx\\ &=\frac {i \sqrt {1-i x^2}}{8 (i-x)}-\frac {i \sqrt {1-i x^2}}{8 (i+x)}-\frac {i \sqrt {1+i x^2}}{8 (i-x)}+\frac {i \sqrt {1+i x^2}}{8 (i+x)}+\frac {1}{8} i \int \frac {1}{(i-x) \sqrt {1-i x^2}} \, dx+\frac {1}{8} i \int \frac {1}{(i+x) \sqrt {1-i x^2}} \, dx+\frac {1}{8} i \int \frac {1}{(i-x) \sqrt {1+i x^2}} \, dx+\frac {1}{8} i \int \frac {1}{(i+x) \sqrt {1+i x^2}} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i-x) \sqrt {1+x^4}} \, dx+\frac {1}{4} i \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(i+x) \sqrt {1+x^4}} \, dx\\ &=\frac {i \sqrt {1-i x^2}}{8 (i-x)}-\frac {i \sqrt {1-i x^2}}{8 (i+x)}-\frac {i \sqrt {1+i x^2}}{8 (i-x)}+\frac {i \sqrt {1+i x^2}}{8 (i+x)}+\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x) \sqrt {1+i x^2}} \, dx+\left (-\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x) \sqrt {1+i x^2}} \, dx-\frac {1}{8} i \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+x}{\sqrt {1+i x^2}}\right )-\frac {1}{8} i \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+x}{\sqrt {1+i x^2}}\right )-\frac {1}{8} i \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-x}{\sqrt {1-i x^2}}\right )-\frac {1}{8} i \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-x}{\sqrt {1-i x^2}}\right )+\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i-x) \sqrt {1-i x^2}} \, dx+\left (\frac {1}{8}+\frac {i}{8}\right ) \int \frac {1}{(i+x) \sqrt {1-i x^2}} \, dx\\ &=\frac {i \sqrt {1-i x^2}}{8 (i-x)}-\frac {i \sqrt {1-i x^2}}{8 (i+x)}-\frac {i \sqrt {1+i x^2}}{8 (i-x)}+\frac {i \sqrt {1+i x^2}}{8 (i+x)}-\frac {1}{16} (1+i)^{3/2} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{16} (1+i)^{3/2} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{16} (1-i)^{3/2} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{16} (1-i)^{3/2} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {-1-x}{\sqrt {1-i x^2}}\right )+\left (-\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-x}{\sqrt {1-i x^2}}\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {-1+x}{\sqrt {1+i x^2}}\right )+\left (\frac {1}{8}-\frac {i}{8}\right ) \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+x}{\sqrt {1+i x^2}}\right )\\ &=\frac {i \sqrt {1-i x^2}}{8 (i-x)}-\frac {i \sqrt {1-i x^2}}{8 (i+x)}-\frac {i \sqrt {1+i x^2}}{8 (i-x)}+\frac {i \sqrt {1+i x^2}}{8 (i+x)}-\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{16} (1+i)^{3/2} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{8} \sqrt {1+i} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )+\frac {1}{16} (1+i)^{3/2} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1+i} \sqrt {1-i x^2}}\right )-\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )-\frac {1}{16} (1-i)^{3/2} \tanh ^{-1}\left (\frac {1-x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{8} \sqrt {1-i} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )+\frac {1}{16} (1-i)^{3/2} \tanh ^{-1}\left (\frac {1+x}{\sqrt {1-i} \sqrt {1+i x^2}}\right )\\ \end {align*}
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Mathematica [F] time = 0.25, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+x^2\right )^2 \sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 1.38, size = 247, normalized size = 1.31 \begin {gather*} \frac {-x^2 \left (-1+x^2\right )-x^2 \sqrt {1+x^4}}{4 x \left (1+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}+\frac {1}{4} \sqrt {-1+5 \sqrt {2}} \tan ^{-1}\left (\frac {-\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right )+\frac {1}{4} \sqrt {1+5 \sqrt {2}} \tanh ^{-1}\left (\frac {-\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}}+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} x^2+\sqrt {-\frac {1}{2}+\frac {1}{\sqrt {2}}} \sqrt {1+x^4}}{x \sqrt {x^2+\sqrt {1+x^4}}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 4.95, size = 384, normalized size = 2.03 \begin {gather*} \frac {4 \, {\left (x^{2} + 1\right )} \sqrt {5 \, \sqrt {2} - 1} \arctan \left (\frac {{\left (21 \, x^{2} + 7 \, \sqrt {2} {\left (x^{2} + 2\right )} + \sqrt {x^{4} + 1} {\left ({\left (\sqrt {2} + 3\right )} \sqrt {-98 \, \sqrt {2} + 147} - 7 \, \sqrt {2} - 21\right )} - {\left (3 \, x^{2} + \sqrt {2} {\left (x^{2} + 4\right )} + 5\right )} \sqrt {-98 \, \sqrt {2} + 147} - 7\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} - 1}}{98 \, x}\right ) + {\left (x^{2} + 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (\frac {7 \, \sqrt {2} x^{2} + 14 \, x^{2} + {\left (x^{3} + \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} - \sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} x + x\right )} + 5 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} + 1} + 7 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - {\left (x^{2} + 1\right )} \sqrt {5 \, \sqrt {2} + 1} \log \left (\frac {7 \, \sqrt {2} x^{2} + 14 \, x^{2} - {\left (x^{3} + \sqrt {2} {\left (2 \, x^{3} + 3 \, x\right )} - \sqrt {x^{4} + 1} {\left (2 \, \sqrt {2} x + x\right )} + 5 \, x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {5 \, \sqrt {2} + 1} + 7 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} + 1\right )}}{x^{2} + 1}\right ) - 4 \, {\left (x^{3} - \sqrt {x^{4} + 1} x + x\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{16 \, {\left (x^{2} + 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 {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.04, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (x^{2}+1\right )^{2} \sqrt {x^{4}+1}}\, dx\]
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 {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} {\left (x^{2} + 1\right )}^{2}}\,{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 {\sqrt {\sqrt {x^4+1}+x^2}}{{\left (x^2+1\right )}^2\,\sqrt {x^4+1}} \,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 {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\left (x^{2} + 1\right )^{2} \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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