Optimal. Leaf size=61 \[ -\frac {4}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}+x^2+x+1}\right )-\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}+x^2-2 x+1}\right ) \]
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Rubi [C] time = 1.03, antiderivative size = 384, normalized size of antiderivative = 6.30, number of steps used = 29, number of rules used = 12, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.546, Rules used = {6725, 220, 2074, 1725, 1211, 1699, 206, 1248, 725, 6728, 1217, 1707} \begin {gather*} -\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt {x^4+1}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {x^4+1}}\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )}{3 \sqrt {2}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{3 \sqrt {2 \left (1+i \sqrt {3}\right )}}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {x^4+1}}\right )}{3 \sqrt {2 \left (1-i \sqrt {3}\right )}}-\frac {\left (1+i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {x^4+1}}-\frac {\left (1-i \sqrt {3}\right ) \left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {x^4+1}}+\frac {\left (x^2+1\right ) \sqrt {\frac {x^4+1}{\left (x^2+1\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {x^4+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 220
Rule 725
Rule 1211
Rule 1217
Rule 1248
Rule 1699
Rule 1707
Rule 1725
Rule 2074
Rule 6725
Rule 6728
Rubi steps
\begin {align*} \int \frac {1+x^3}{\left (-1+x^3\right ) \sqrt {1+x^4}} \, dx &=\int \left (\frac {1}{\sqrt {1+x^4}}+\frac {2}{\left (-1+x^3\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=2 \int \frac {1}{\left (-1+x^3\right ) \sqrt {1+x^4}} \, dx+\int \frac {1}{\sqrt {1+x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+2 \int \left (\frac {1}{3 (-1+x) \sqrt {1+x^4}}+\frac {-2-x}{3 \left (1+x+x^2\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}+\frac {2}{3} \int \frac {1}{(-1+x) \sqrt {1+x^4}} \, dx+\frac {2}{3} \int \frac {-2-x}{\left (1+x+x^2\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {2}{3} \int \frac {1}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {2}{3} \int \frac {x}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx+\frac {2}{3} \int \left (\frac {-1+i \sqrt {3}}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}+\frac {-1-i \sqrt {3}}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}}\right ) \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {1+x^4}}-\frac {1}{3} \int \frac {1}{\sqrt {1+x^4}} \, dx-\frac {1}{3} \int \frac {1+x^2}{\left (1-x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{(1-x) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \int \frac {1}{\left (1-i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (1+i \sqrt {3}+2 x\right ) \sqrt {1+x^4}} \, dx\\ &=\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {1+x^4}}-\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{1-2 x^2} \, dx,x,\frac {x}{\sqrt {1+x^4}}\right )+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{2-x^2} \, dx,x,\frac {-1-x^2}{\sqrt {1+x^4}}\right )+\frac {1}{3} \left (4 \left (1-i \sqrt {3}\right )\right ) \int \frac {x}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {x}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )^2\right ) \int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {1+x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{3 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {1+x^4}}+\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1-i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (1+i \sqrt {3}\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx+\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\left (\left (1+i \sqrt {3}\right )^2-4 x\right ) \sqrt {1+x^2}} \, dx,x,x^2\right )-\frac {1}{3} \left (4 \left (1+i \sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (\left (1+i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx+\frac {\left (2 \left (i-\sqrt {3}\right )\right ) \int \frac {1}{\sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}+\frac {\left (8 \left (i-\sqrt {3}\right )\right ) \int \frac {1+x^2}{\left (\left (1-i \sqrt {3}\right )^2-4 x^2\right ) \sqrt {1+x^4}} \, dx}{3 \left (i+\sqrt {3}\right )}\\ &=-\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {1+x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{3 \sqrt {2}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {1+x^4}}-\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {1+x^4}}+\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {1}{3} \left (2 \left (1-i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (1-i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (1-i \sqrt {3}\right )^2 x^2}{\sqrt {1+x^4}}\right )-\frac {1}{3} \left (2 \left (1+i \sqrt {3}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{16+\left (1+i \sqrt {3}\right )^4-x^2} \, dx,x,\frac {-4-\left (1+i \sqrt {3}\right )^2 x^2}{\sqrt {1+x^4}}\right )\\ &=-\frac {2}{3} \tan ^{-1}\left (\frac {x}{\sqrt {1+x^4}}\right )-\frac {\tanh ^{-1}\left (\frac {\sqrt {2} x}{\sqrt {1+x^4}}\right )}{3 \sqrt {2}}-\frac {\tanh ^{-1}\left (\frac {1+x^2}{\sqrt {2} \sqrt {1+x^4}}\right )}{3 \sqrt {2}}+\frac {\left (1-i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {2-\left (1+i \sqrt {3}\right ) x^2}{\sqrt {2 \left (1+i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1+i \sqrt {3}\right )}}+\frac {\left (1+i \sqrt {3}\right ) \tanh ^{-1}\left (\frac {4+\left (1+i \sqrt {3}\right )^2 x^2}{2 \sqrt {2 \left (1-i \sqrt {3}\right )} \sqrt {1+x^4}}\right )}{3 \sqrt {2 \left (1-i \sqrt {3}\right )}}+\frac {\left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \sqrt {1+x^4}}-\frac {\left (1+i \sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \sqrt {1+x^4}}+\frac {\left (i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} F\left (2 \tan ^{-1}(x)|\frac {1}{2}\right )}{3 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}-\frac {i \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{2 \sqrt {3} \sqrt {1+x^4}}+\frac {\left (3 i-\sqrt {3}\right ) \left (1+x^2\right ) \sqrt {\frac {1+x^4}{\left (1+x^2\right )^2}} \Pi \left (\frac {1}{4};2 \tan ^{-1}(x)|\frac {1}{2}\right )}{6 \left (i+\sqrt {3}\right ) \sqrt {1+x^4}}\\ \end {align*}
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Mathematica [C] time = 0.65, size = 525, normalized size = 8.61 \begin {gather*} \frac {1}{18} \left (4 (-1)^{2/3} \sqrt {1-\sqrt [3]{-1}} \tanh ^{-1}\left (\frac {\sqrt [3]{-1}-x^2}{\sqrt {1+(-1)^{2/3}} \sqrt {x^4+1}}\right )+2 \sqrt [3]{-1} \sqrt {1-\sqrt [3]{-1}} \tanh ^{-1}\left (\frac {\sqrt [3]{-1}-x^2}{\sqrt {1+(-1)^{2/3}} \sqrt {x^4+1}}\right )-2 \sqrt {1-\sqrt [3]{-1}} \tanh ^{-1}\left (\frac {\sqrt [3]{-1}-x^2}{\sqrt {1+(-1)^{2/3}} \sqrt {x^4+1}}\right )+(-1)^{2/3} \sqrt {2} \tanh ^{-1}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )-\sqrt [3]{-1} \sqrt {2} \tanh ^{-1}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )-2 \sqrt {2} \tanh ^{-1}\left (\frac {x^2+1}{\sqrt {2} \sqrt {x^4+1}}\right )+2 (-1)^{2/3} \sqrt {1+(-1)^{2/3}} \tanh ^{-1}\left (\frac {x^2+(-1)^{2/3}}{\sqrt {1-\sqrt [3]{-1}} \sqrt {x^4+1}}\right )+4 \sqrt [3]{-1} \sqrt {1+(-1)^{2/3}} \tanh ^{-1}\left (\frac {x^2+(-1)^{2/3}}{\sqrt {1-\sqrt [3]{-1}} \sqrt {x^4+1}}\right )+2 \sqrt {1+(-1)^{2/3}} \tanh ^{-1}\left (\frac {x^2+(-1)^{2/3}}{\sqrt {1-\sqrt [3]{-1}} \sqrt {x^4+1}}\right )-18 \sqrt [4]{-1} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+12 \sqrt [4]{-1} \Pi \left (i;\left .\sin ^{-1}\left ((-1)^{3/4} x\right )\right |-1\right )-12 (-1)^{11/12} \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+12 (-1)^{7/12} \Pi \left (-\sqrt [6]{-1};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )+12 \sqrt [4]{-1} \Pi \left (-(-1)^{5/6};\left .i \sinh ^{-1}\left (\sqrt [4]{-1} x\right )\right |-1\right )\right ) \end {gather*}
Warning: Unable to verify antiderivative.
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IntegrateAlgebraic [A] time = 1.07, size = 61, normalized size = 1.00 \begin {gather*} -\frac {4}{3} \tan ^{-1}\left (\frac {x}{1+x+x^2+\sqrt {1+x^4}}\right )-\frac {1}{3} \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {2} x}{1-2 x+x^2+\sqrt {1+x^4}}\right ) \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.53, size = 90, normalized size = 1.48 \begin {gather*} \frac {1}{12} \, \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 ) + \frac {2}{3} \, \arctan \left (\frac {\sqrt {x^{4} + 1}}{x^{2} + 2 \, 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^{3} + 1}{\sqrt {x^{4} + 1} {\left (x^{3} - 1\right )}}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.44, size = 244, normalized size = 4.00
method | result | size |
elliptic | \(-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {\left (x^{2}-1\right )^{2}+2 x^{2}}}\right )}{6}+\frac {\sqrt {2}\, \sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \arctan \left (\frac {\sqrt {\frac {2 \left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+2}\, \left (x^{2}+1\right )}{\left (\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1\right ) \left (-x^{2}+1\right )}\right )}{3 \sqrt {\frac {\frac {\left (x^{2}+1\right )^{2}}{\left (-x^{2}+1\right )^{2}}+1}{\left (\frac {x^{2}+1}{-x^{2}+1}+1\right )^{2}}}\, \left (\frac {x^{2}+1}{-x^{2}+1}+1\right )}+\frac {\left (\frac {\ln \left (-1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}-\frac {\ln \left (1+\frac {\sqrt {2}\, \sqrt {x^{4}+1}}{2 x}\right )}{6}+\frac {2 \sqrt {2}\, \arctan \left (\frac {\sqrt {x^{4}+1}}{x}\right )}{3}\right ) \sqrt {2}}{2}\) | \(244\) |
default | \(\frac {\sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticF \left (x \left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ), i\right )}{\left (\frac {\sqrt {2}}{2}+\frac {i \sqrt {2}}{2}\right ) \sqrt {x^{4}+1}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\left (2 x^{2}+2\right ) \sqrt {2}}{4 \sqrt {x^{4}+1}}\right )}{6}+\frac {2 \left (-1\right )^{\frac {3}{4}} \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i, -\sqrt {-i}\, \left (-1\right )^{\frac {3}{4}}\right )}{3 \sqrt {x^{4}+1}}+\frac {2 \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}+\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{3}+\frac {2 \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ) \left (\frac {\arctanh \left (\frac {\sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}\, \left (x^{2}-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right )}{\sqrt {x^{4}+1}}\right )}{2 \sqrt {\frac {1}{2}-\frac {i \sqrt {3}}{2}}}+\frac {\left (-1\right )^{\frac {3}{4}} \left (-\frac {1}{2}+\frac {i \sqrt {3}}{2}\right ) \sqrt {-i x^{2}+1}\, \sqrt {i x^{2}+1}\, \EllipticPi \left (\left (-1\right )^{\frac {1}{4}} x , -i \left (-\frac {1}{2}-\frac {i \sqrt {3}}{2}\right ), i\right )}{\sqrt {x^{4}+1}}\right )}{3}\) | \(358\) |
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^{3} + 1}{\sqrt {x^{4} + 1} {\left (x^{3} - 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 = -0.02 \begin {gather*} \int \frac {x^3+1}{\left (x^3-1\right )\,\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 {\left (x + 1\right ) \left (x^{2} - x + 1\right )}{\left (x - 1\right ) \sqrt {x^{4} + 1} \left (x^{2} + x + 1\right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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