Optimal. Leaf size=312 \[ \frac {2 x^4+x^2+\left (-x^2-1\right ) \left (\sqrt {x^4+1}+x^2\right ) x+\sqrt {x^4+1} \left (2 x^2-\left (\sqrt {x^4+1}+x^2\right ) x+1\right )+1}{2 \left (x^2-1\right ) \left (\sqrt {x^4+1}+x^2\right )^{3/2}}+\frac {\tan ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {\sqrt {2}-1}}\right )}{2 \sqrt {\sqrt {2}-1}}-\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \left (\sqrt {2}-1\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {1+\sqrt {2}}}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{2} \sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {2 \left (1+\sqrt {2}\right )} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \]
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Rubi [C] time = 0.19, antiderivative size = 125, normalized size of antiderivative = 0.40, number of steps used = 7, number of rules used = 4, integrand size = 32, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2133, 731, 725, 206} \begin {gather*} -\frac {\sqrt {1-i x^2}}{2 (x+1)}-\frac {\sqrt {1+i x^2}}{2 (x+1)}-\frac {1}{4} (1-i)^{3/2} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{4} (1+i)^{3/2} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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Rule 206
Rule 725
Rule 731
Rule 2133
Rubi steps
\begin {align*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx &=\left (\frac {1}{2}-\frac {i}{2}\right ) \int \frac {1}{(1+x)^2 \sqrt {1-i x^2}} \, dx+\left (\frac {1}{2}+\frac {i}{2}\right ) \int \frac {1}{(1+x)^2 \sqrt {1+i x^2}} \, dx\\ &=-\frac {\sqrt {1-i x^2}}{2 (1+x)}-\frac {\sqrt {1+i x^2}}{2 (1+x)}-\frac {1}{2} i \int \frac {1}{(1+x) \sqrt {1-i x^2}} \, dx+\frac {1}{2} i \int \frac {1}{(1+x) \sqrt {1+i x^2}} \, dx\\ &=-\frac {\sqrt {1-i x^2}}{2 (1+x)}-\frac {\sqrt {1+i x^2}}{2 (1+x)}+\frac {1}{2} i \operatorname {Subst}\left (\int \frac {1}{(1-i)-x^2} \, dx,x,\frac {1+i x}{\sqrt {1-i x^2}}\right )-\frac {1}{2} i \operatorname {Subst}\left (\int \frac {1}{(1+i)-x^2} \, dx,x,\frac {1-i x}{\sqrt {1+i x^2}}\right )\\ &=-\frac {\sqrt {1-i x^2}}{2 (1+x)}-\frac {\sqrt {1+i x^2}}{2 (1+x)}-\frac {1}{4} (1-i)^{3/2} \tanh ^{-1}\left (\frac {1+i x}{\sqrt {1-i} \sqrt {1-i x^2}}\right )-\frac {1}{4} (1+i)^{3/2} \tanh ^{-1}\left (\frac {1-i x}{\sqrt {1+i} \sqrt {1+i x^2}}\right )\\ \end {align*}
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Mathematica [F] time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^2+\sqrt {1+x^4}}}{(1+x)^2 \sqrt {1+x^4}} \, dx \end {gather*}
Verification is not applicable to the result.
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IntegrateAlgebraic [A] time = 3.22, size = 312, normalized size = 1.00 \begin {gather*} \frac {2 x^4+x^2+\left (-x^2-1\right ) \left (\sqrt {x^4+1}+x^2\right ) x+\sqrt {x^4+1} \left (2 x^2-\left (\sqrt {x^4+1}+x^2\right ) x+1\right )+1}{2 \left (x^2-1\right ) \left (\sqrt {x^4+1}+x^2\right )^{3/2}}+\frac {\tan ^{-1}\left (\sqrt {1+\sqrt {2}} \sqrt {\sqrt {x^4+1}+x^2}\right )}{2 \sqrt {\sqrt {2}-1}}-\frac {1}{2} \sqrt {1+\sqrt {2}} \tan ^{-1}\left (\frac {\sqrt {2 \sqrt {2}-2} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right )-\frac {\tanh ^{-1}\left (\sqrt {\sqrt {2}-1} \sqrt {\sqrt {x^4+1}+x^2}\right )}{2 \sqrt {1+\sqrt {2}}}+\frac {1}{2} \sqrt {\sqrt {2}-1} \tanh ^{-1}\left (\frac {\sqrt {2+2 \sqrt {2}} x \sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}+x^2+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 2.07, size = 394, normalized size = 1.26 \begin {gather*} \frac {4 \, {\left (x + 1\right )} \sqrt {\sqrt {2} + 1} \arctan \left (\frac {2 \, {\left (x^{3} + x^{2} - \sqrt {2} {\left (x^{3} + 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} x - x - 1\right )} - x + 1\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} \sqrt {\sqrt {2} + 1} + {\left (2 \, x^{2} - \sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1} {\left (\sqrt {2} - 1\right )} + 2\right )} \sqrt {2 \, \sqrt {2} + 2} \sqrt {\sqrt {2} + 1}}{2 \, {\left (x^{2} - 2 \, x + 1\right )}}\right ) + {\left (x + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} + {\left (\sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} - 1}}{x^{2} + 2 \, x + 1}\right ) - {\left (x + 1\right )} \sqrt {\sqrt {2} - 1} \log \left (-\frac {{\left (2 \, x^{3} - \sqrt {2} {\left (x^{3} - x^{2} - x - 1\right )} + \sqrt {x^{4} + 1} {\left (\sqrt {2} {\left (x - 1\right )} - 2 \, x\right )} - 2\right )} \sqrt {x^{2} + \sqrt {x^{4} + 1}} - {\left (\sqrt {2} {\left (x^{2} + 1\right )} + 2 \, \sqrt {x^{4} + 1}\right )} \sqrt {\sqrt {2} - 1}}{x^{2} + 2 \, x + 1}\right ) + 4 \, \sqrt {x^{2} + \sqrt {x^{4} + 1}} {\left (x^{2} - \sqrt {x^{4} + 1} - 1\right )}}{8 \, {\left (x + 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 + 1\right )}^{2}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.48, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {x^{2}+\sqrt {x^{4}+1}}}{\left (1+x \right )^{2} \sqrt {x^{4}+1}}\, dx \end {gather*}
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 + 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 = -0.00 \begin {gather*} \int \frac {\sqrt {\sqrt {x^4+1}+x^2}}{\sqrt {x^4+1}\,{\left (x+1\right )}^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 {\sqrt {x^{2} + \sqrt {x^{4} + 1}}}{\left (x + 1\right )^{2} \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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