Optimal. Leaf size=321 \[ \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {\text {RootSum}\left [\text {$\#$1}^8-2 \text {$\#$1}^6-6 \text {$\#$1}^4+2 \text {$\#$1}^2+1\& ,\frac {\text {$\#$1}^6 \left (-\log \left (\sqrt {x^4+1}+x^2-1\right )\right )+\text {$\#$1}^6 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )-2 \text {$\#$1}^4 \log \left (\sqrt {x^4+1}+x^2-1\right )+2 \text {$\#$1}^4 \log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )-\log \left (\text {$\#$1} \sqrt {x^4+1}+\text {$\#$1} x^2-\text {$\#$1}+\sqrt {2} \sqrt {\sqrt {x^4+1}+x^2} x\right )+\log \left (\sqrt {x^4+1}+x^2-1\right )}{2 \text {$\#$1}^7-3 \text {$\#$1}^5-6 \text {$\#$1}^3+\text {$\#$1}}\& \right ]}{\sqrt {2}} \]
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Rubi [C] time = 3.35, antiderivative size = 710, normalized size of antiderivative = 2.21, number of steps used = 30, number of rules used = 6, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {6728, 2132, 206, 6725, 2133, 725} \begin {gather*} -\frac {i \left (3+\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(-2+i)\right )}}+\frac {i \left (3+\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {\sqrt {5}-1} x\right )}{\sqrt {2 \left (\sqrt {5}+(-1-2 i)\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(-2+i)\right )}}-\frac {\left (3-\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}+\frac {\left (3-\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {1+\sqrt {5}} x+\sqrt {2}\right )}{\sqrt {-2 \sqrt {5}+(-2-4 i)} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left (\sqrt {5}+(2-i)\right )}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {\sqrt {5}-1} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(-2-i)\right )}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {1+\sqrt {5}} x+\sqrt {2}}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left (\sqrt {5}+(2+i)\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {\sqrt {x^4+1}+x^2}}\right )}{\sqrt {2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 206
Rule 725
Rule 2132
Rule 2133
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx &=\int \left (\frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}}+\frac {\left (2+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )}\right ) \, dx\\ &=\int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4}} \, dx+\int \frac {\left (2+x^2\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx\\ &=\int \left (\frac {\left (1+\frac {3}{\sqrt {5}}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}+\frac {\left (1-\frac {3}{\sqrt {5}}\right ) \sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}}\right ) \, dx+\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1+\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (1-\sqrt {5}+2 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}+\frac {1}{5} \left (5-3 \sqrt {5}\right ) \int \left (\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {i \sqrt {x^2+\sqrt {1+x^4}}}{2 \sqrt {1+\sqrt {5}} \left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx+\frac {1}{5} \left (5+3 \sqrt {5}\right ) \int \left (\frac {\sqrt {-1+\sqrt {5}} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-\sqrt {5}\right ) \left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}}+\frac {\sqrt {-1+\sqrt {5}} \sqrt {x^2+\sqrt {1+x^4}}}{2 \left (1-\sqrt {5}\right ) \left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}--\frac {\left (-5-3 \sqrt {5}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{10 \sqrt {-1+\sqrt {5}}}--\frac {\left (-5-3 \sqrt {5}\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{10 \sqrt {-1+\sqrt {5}}}+\frac {\left (i \left (5-3 \sqrt {5}\right )\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{10 \sqrt {1+\sqrt {5}}}+\frac {\left (i \left (5-3 \sqrt {5}\right )\right ) \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+x^4}} \, dx}{10 \sqrt {1+\sqrt {5}}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}--\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {-1+\sqrt {5}}}--\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {-1+\sqrt {5}}}--\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {-1+\sqrt {5}}}--\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (\sqrt {-1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {-1+\sqrt {5}}}+-\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {1+\sqrt {5}}}+-\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1+i x^2}} \, dx}{\sqrt {1+\sqrt {5}}}+\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}-\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {1+\sqrt {5}}}+\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \int \frac {1}{\left (i \sqrt {1+\sqrt {5}}+\sqrt {2} x\right ) \sqrt {1-i x^2}} \, dx}{\sqrt {1+\sqrt {5}}}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}-\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {-1+\sqrt {5}}}-\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (-5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1+\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {-1+\sqrt {5}}}+-\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {1+\sqrt {5}}}+-\frac {\left (\left (\frac {1}{20}+\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2-i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {1-i x^2}}\right )}{\sqrt {1+\sqrt {5}}}+\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {-\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {1+\sqrt {5}}}+\frac {\left (\left (\frac {1}{20}-\frac {i}{20}\right ) \left (5-3 \sqrt {5}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2+i \left (-1-\sqrt {5}\right )-x^2} \, dx,x,\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {1+i x^2}}\right )}{\sqrt {1+\sqrt {5}}}\\ &=-\frac {i \left (3+\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}+\frac {i \left (3+\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+i \sqrt {-1+\sqrt {5}} x\right )}{\sqrt {2 \left ((-1-2 i)+\sqrt {5}\right )} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left ((-2+i)+\sqrt {5}\right )}}-\frac {\left (3-\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}-\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}+\frac {\left (3-\sqrt {5}\right ) \tan ^{-1}\left (\frac {(1+i) \left (\sqrt {2}+\sqrt {1+\sqrt {5}} x\right )}{\sqrt {(-2-4 i)-2 \sqrt {5}} \sqrt {1+i x^2}}\right )}{4 \sqrt {(-5-5 i) \left ((2-i)+\sqrt {5}\right )}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (3+\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+i \sqrt {-1+\sqrt {5}} x}{\sqrt {(2+i)-i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left ((-2-i)+\sqrt {5}\right )}}+\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}-\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left ((2+i)+\sqrt {5}\right )}}-\frac {\left (\frac {1}{4}+\frac {i}{4}\right ) \left (3-\sqrt {5}\right ) \tanh ^{-1}\left (\frac {\sqrt {2}+\sqrt {1+\sqrt {5}} x}{\sqrt {(2+i)+i \sqrt {5}} \sqrt {1-i x^2}}\right )}{\sqrt {(10+10 i) \left ((2+i)+\sqrt {5}\right )}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^2+\sqrt {1+x^4}}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [F] time = 0.86, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1+x^2\right )^2 \sqrt {x^2+\sqrt {1+x^4}}}{\sqrt {1+x^4} \left (-1+x^2+x^4\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 0.70, size = 341, normalized size = 1.06 \begin {gather*} \sqrt {2} \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 )+\frac {\text {RootSum}\left [1-2 \text {$\#$1}^2-6 \text {$\#$1}^4+2 \text {$\#$1}^6+\text {$\#$1}^8\&,\frac {\log \left (1+x^2+\sqrt {1+x^4}\right )-\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right )+2 \log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^2-2 \log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^2-\log \left (1+x^2+\sqrt {1+x^4}\right ) \text {$\#$1}^6+\log \left (\sqrt {2} x \sqrt {x^2+\sqrt {1+x^4}}-\text {$\#$1}-x^2 \text {$\#$1}-\sqrt {1+x^4} \text {$\#$1}\right ) \text {$\#$1}^6}{-\text {$\#$1}-6 \text {$\#$1}^3+3 \text {$\#$1}^5+2 \text {$\#$1}^7}\&\right ]}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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}} {\left (x^{2} + 1\right )}^{2}}{{\left (x^{4} + x^{2} - 1\right )} \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 = 0.06, size = 0, normalized size = 0.00 \[\int \frac {\left (x^{2}+1\right )^{2} \sqrt {x^{2}+\sqrt {x^{4}+1}}}{\sqrt {x^{4}+1}\, \left (x^{4}+x^{2}-1\right )}\, dx\]
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}} {\left (x^{2} + 1\right )}^{2}}{{\left (x^{4} + x^{2} - 1\right )} \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 = -0.00 \begin {gather*} \int \frac {{\left (x^2+1\right )}^2\,\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}\,\left (x^4+x^2-1\right )} \,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^{2} + 1\right )^{2} \sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\sqrt {x^{4} + 1} \left (x^{4} + x^{2} - 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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