Optimal. Leaf size=34 \[ -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}+x^2-2 x+1}\right ) \]
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Rubi [A] time = 0.09, antiderivative size = 53, normalized size of antiderivative = 1.56, number of steps used = 7, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1742, 12, 1248, 725, 206, 1699} \begin {gather*} -\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{\sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )}{\sqrt {2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 206
Rule 725
Rule 1248
Rule 1699
Rule 1742
Rubi steps
\begin {align*} \int \frac {1+x}{(-1+x) \sqrt {1+x^4}} \, dx &=\int -\frac {2 x}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx+\int \frac {-1-x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\left (2 \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx\right )-\operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}-\operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x^2}} \, dx,x,x^2\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}+\operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1+x^4}}\right )\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{\sqrt {2}}+\frac {\tanh ^{-1}\left (\frac {-1-x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{\sqrt {2}}\\ \end {align*}
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Mathematica [C] time = 0.26, size = 178, normalized size = 5.24 \begin {gather*} -\frac {(-1)^{3/4} \sqrt {2} \sqrt {-\frac {i \left (\sqrt {2}+(1-i) x\right )}{\sqrt {2}-(1-i) x}} \left (x^2+i\right ) \left (\left (1+\sqrt {2}\right ) F\left (\left .\sin ^{-1}\left (\sqrt {-\frac {i \left ((1-i) x+\sqrt {2}\right )}{\sqrt {2}-(1-i) x}}\right )\right |-1\right )-2 \sqrt {2} \Pi \left (1-\sqrt {2};\left .\sin ^{-1}\left (\sqrt {-\frac {i \left ((1-i) x+\sqrt {2}\right )}{\sqrt {2}-(1-i) x}}\right )\right |-1\right )\right )}{\sqrt {\frac {x^2+i}{\left (\sqrt [4]{-1}-x\right )^2}} \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.66, size = 34, normalized size = 1.00 \begin {gather*} -\sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}+x^2-2 x+1}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 68, normalized size = 2.00 \begin {gather*} \frac {1}{4} \, \sqrt {2} \log \left (-\frac {3 \, x^{4} - 4 \, x^{3} - 2 \, \sqrt {2} \sqrt {x^{4} + 1} {\left (x^{2} - x + 1\right )} + 6 \, x^{2} - 4 \, x + 3}{x^{4} - 4 \, x^{3} + 6 \, x^{2} - 4 \, 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 {x + 1}{\sqrt {x^{4} + 1} {\left (x - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.01, size = 136, normalized size = 4.00 \begin {gather*} \frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {x^{4}+1}}\right )}{2}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{\sqrt {x^{4}+1}} \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 {x + 1}{\sqrt {x^{4} + 1} {\left (x - 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 = -0.03 \begin {gather*} \int \frac {x+1}{\sqrt {x^4+1}\,\left (x-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 {x + 1}{\left (x - 1\right ) \sqrt {x^{4} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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