Optimal. Leaf size=171 \[ -\frac {\tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\log \left (\sqrt {21}-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (\sqrt {21}+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \]
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Rubi [A]
time = 0.08, antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.667, Rules used = {217, 1179, 642,
1176, 631, 210} \begin {gather*} -\frac {\text {ArcTan}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\text {ArcTan}\left (\sqrt [4]{\frac {3}{7}} \sqrt {2} x+1\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\log \left (3 x^2-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (3 x^2+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+\sqrt {21}\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 217
Rule 631
Rule 642
Rule 1176
Rule 1179
Rubi steps
\begin {align*} \int \frac {1}{7+3 x^4} \, dx &=\frac {\int \frac {\sqrt {7}-\sqrt {3} x^2}{7+3 x^4} \, dx}{2 \sqrt {7}}+\frac {\int \frac {\sqrt {7}+\sqrt {3} x^2}{7+3 x^4} \, dx}{2 \sqrt {7}}\\ &=-\frac {\int \frac {\sqrt {2} \sqrt [4]{\frac {7}{3}}+2 x}{-\sqrt {\frac {7}{3}}-\sqrt {2} \sqrt [4]{\frac {7}{3}} x-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\int \frac {\sqrt {2} \sqrt [4]{\frac {7}{3}}-2 x}{-\sqrt {\frac {7}{3}}+\sqrt {2} \sqrt [4]{\frac {7}{3}} x-x^2} \, dx}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\int \frac {1}{\sqrt {\frac {7}{3}}-\sqrt {2} \sqrt [4]{\frac {7}{3}} x+x^2} \, dx}{4 \sqrt {21}}+\frac {\int \frac {1}{\sqrt {\frac {7}{3}}+\sqrt {2} \sqrt [4]{\frac {7}{3}} x+x^2} \, dx}{4 \sqrt {21}}\\ &=-\frac {\log \left (\sqrt {21}-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (\sqrt {21}+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}\\ &=-\frac {\tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\tan ^{-1}\left (1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )}{2 \sqrt {2} \sqrt [4]{3} 7^{3/4}}-\frac {\log \left (\sqrt {21}-\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}+\frac {\log \left (\sqrt {21}+\sqrt {2} 3^{3/4} \sqrt [4]{7} x+3 x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 120, normalized size = 0.70 \begin {gather*} \frac {-2 \tan ^{-1}\left (1-\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )+2 \tan ^{-1}\left (1+\sqrt [4]{\frac {3}{7}} \sqrt {2} x\right )-\log \left (7-\sqrt {2} \sqrt [4]{3} 7^{3/4} x+\sqrt {21} x^2\right )+\log \left (7+\sqrt {2} \sqrt [4]{3} 7^{3/4} x+\sqrt {21} x^2\right )}{4 \sqrt {2} \sqrt [4]{3} 7^{3/4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.02, size = 93, normalized size = 0.54
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\RootOf \left (3 \textit {\_Z}^{4}+7\right )}{\sum }\frac {\ln \left (-\textit {\_R} +x \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(24\) |
default | \(\frac {\sqrt {3}\, 21^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 21^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {21}}{3}}{x^{2}-\frac {\sqrt {3}\, 21^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {21}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 21^{\frac {3}{4}} x}{21}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 21^{\frac {3}{4}} x}{21}-1\right )\right )}{168}\) | \(93\) |
meijerg | \(\frac {1029^{\frac {3}{4}} \left (-\frac {x \sqrt {2}\, \ln \left (1-\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{7}+\frac {\sqrt {3}\, \sqrt {7}\, \sqrt {x^{4}}}{7}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{14-\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{7}+\frac {\sqrt {3}\, \sqrt {7}\, \sqrt {x^{4}}}{7}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{14+\sqrt {2}\, 3^{\frac {1}{4}} 7^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{4116}\) | \(187\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 2.67, size = 151, normalized size = 0.88 \begin {gather*} \frac {1}{84} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{42} \cdot 7^{\frac {3}{4}} 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} x + 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{84} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{42} \cdot 7^{\frac {3}{4}} 3^{\frac {3}{4}} \sqrt {2} {\left (2 \, \sqrt {3} x - 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{168} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {3} x^{2} + 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) - \frac {1}{168} \cdot 7^{\frac {1}{4}} 3^{\frac {3}{4}} \sqrt {2} \log \left (\sqrt {3} x^{2} - 7^{\frac {1}{4}} 3^{\frac {1}{4}} \sqrt {2} x + \sqrt {7}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.75, size = 161, normalized size = 0.94 \begin {gather*} -\frac {1}{2058} \cdot 1029^{\frac {3}{4}} \sqrt {2} \arctan \left (\frac {1}{147} \cdot 1029^{\frac {1}{4}} \sqrt {3} \sqrt {2} \sqrt {1029^{\frac {3}{4}} \sqrt {2} x + 147 \, x^{2} + 49 \, \sqrt {21}} - \frac {1}{7} \cdot 1029^{\frac {1}{4}} \sqrt {2} x - 1\right ) - \frac {1}{2058} \cdot 1029^{\frac {3}{4}} \sqrt {2} \arctan \left (-\frac {1}{7} \cdot 1029^{\frac {1}{4}} \sqrt {2} x + \frac {1}{2058} \cdot 1029^{\frac {1}{4}} \sqrt {2} \sqrt {-588 \cdot 1029^{\frac {3}{4}} \sqrt {2} x + 86436 \, x^{2} + 28812 \, \sqrt {21}} + 1\right ) + \frac {1}{8232} \cdot 1029^{\frac {3}{4}} \sqrt {2} \log \left (588 \cdot 1029^{\frac {3}{4}} \sqrt {2} x + 86436 \, x^{2} + 28812 \, \sqrt {21}\right ) - \frac {1}{8232} \cdot 1029^{\frac {3}{4}} \sqrt {2} \log \left (-588 \cdot 1029^{\frac {3}{4}} \sqrt {2} x + 86436 \, x^{2} + 28812 \, \sqrt {21}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 151, normalized size = 0.88 \begin {gather*} - \frac {\sqrt [4]{189} \sqrt {2} \log {\left (x^{2} - \frac {\sqrt [4]{189} \sqrt {2} x}{3} + \frac {\sqrt {21}}{3} \right )}}{168} + \frac {\sqrt [4]{189} \sqrt {2} \log {\left (x^{2} + \frac {\sqrt [4]{189} \sqrt {2} x}{3} + \frac {\sqrt {21}}{3} \right )}}{168} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot \sqrt [4]{3} \cdot 7^{\frac {3}{4}} x}{7} - 1 \right )}}{84} + \frac {\sqrt {2} \cdot 3^{\frac {3}{4}} \cdot \sqrt [4]{7} \operatorname {atan}{\left (\frac {\sqrt {2} \cdot \sqrt [4]{3} \cdot 7^{\frac {3}{4}} x}{7} + 1 \right )}}{84} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.49, size = 95, normalized size = 0.56 \begin {gather*} \frac {1}{84} \cdot 756^{\frac {1}{4}} \arctan \left (\frac {3}{14} \, \left (\frac {7}{3}\right )^{\frac {3}{4}} \sqrt {2} {\left (2 \, x + \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{84} \cdot 756^{\frac {1}{4}} \arctan \left (\frac {3}{14} \, \left (\frac {7}{3}\right )^{\frac {3}{4}} \sqrt {2} {\left (2 \, x - \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2}\right )}\right ) + \frac {1}{168} \cdot 756^{\frac {1}{4}} \log \left (x^{2} + \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2} x + \sqrt {\frac {7}{3}}\right ) - \frac {1}{168} \cdot 756^{\frac {1}{4}} \log \left (x^{2} - \left (\frac {7}{3}\right )^{\frac {1}{4}} \sqrt {2} x + \sqrt {\frac {7}{3}}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.11, size = 45, normalized size = 0.26 \begin {gather*} \sqrt {2}\,{189}^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,{189}^{3/4}\,x\,\left (\frac {1}{126}-\frac {1}{126}{}\mathrm {i}\right )\right )\,\left (\frac {1}{84}+\frac {1}{84}{}\mathrm {i}\right )+\sqrt {2}\,{189}^{1/4}\,\mathrm {atan}\left (\sqrt {2}\,{189}^{3/4}\,x\,\left (\frac {1}{126}+\frac {1}{126}{}\mathrm {i}\right )\right )\,\left (\frac {1}{84}-\frac {1}{84}{}\mathrm {i}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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