Optimal. Leaf size=141 \[ -\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}} \]
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Rubi [A] time = 0.10, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1168, 1162, 617, 204, 1165, 628} \[ -\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}} \]
Antiderivative was successfully verified.
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Rule 204
Rule 617
Rule 628
Rule 1162
Rule 1165
Rule 1168
Rubi steps
\begin {align*} \int \frac {a+c x^2}{2+3 x^4} \, dx &=\frac {1}{12} \left (\sqrt {6} a-2 c\right ) \int \frac {\sqrt {6}-3 x^2}{2+3 x^4} \, dx+\frac {1}{12} \left (\sqrt {6} a+2 c\right ) \int \frac {\sqrt {6}+3 x^2}{2+3 x^4} \, dx\\ &=-\frac {\left (\sqrt {6} a-2 c\right ) \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}+2 x}{-\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \int \frac {\frac {2^{3/4}}{\sqrt [4]{3}}-2 x}{-\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}-x^2} \, dx}{8\ 6^{3/4}}+\frac {1}{24} \left (\sqrt {6} a+2 c\right ) \int \frac {1}{\sqrt {\frac {2}{3}}-\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx+\frac {1}{24} \left (\sqrt {6} a+2 c\right ) \int \frac {1}{\sqrt {\frac {2}{3}}+\frac {2^{3/4} x}{\sqrt [4]{3}}+x^2} \, dx\\ &=-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a+2 c\right ) \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}\\ &=-\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 113, normalized size = 0.80 \[ \frac {-\left (\sqrt {6} a-2 c\right ) \left (\log \left (\sqrt {6} x^2-2 \sqrt [4]{6} x+2\right )-\log \left (\sqrt {6} x^2+2 \sqrt [4]{6} x+2\right )\right )-2 \left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt {6} a+2 c\right ) \tan ^{-1}\left (\sqrt [4]{6} x+1\right )}{8\ 6^{3/4}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.75, size = 2278, normalized size = 16.16 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 131, normalized size = 0.93 \[ \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.04, size = 226, normalized size = 1.60 \[ \frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{24}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, a \ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{48}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )}{72}+\frac {\sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, c \ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, x}{3}+\frac {\sqrt {6}}{3}}\right )}{144} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 3.04, size = 167, normalized size = 1.18 \[ \frac {1}{24} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a + \sqrt {2} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a + \sqrt {2} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.11, size = 315, normalized size = 2.23 \[ -2\,\mathrm {atanh}\left (\frac {216\,a^2\,x\,\sqrt {-\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3+18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2-12\,c^3}-\frac {144\,c^2\,x\,\sqrt {-\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3+18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2-12\,c^3}\right )\,\sqrt {-\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}+2\,\mathrm {atanh}\left (\frac {216\,a^2\,x\,\sqrt {\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3-18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2+12\,c^3}-\frac {144\,c^2\,x\,\sqrt {\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3-18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2+12\,c^3}\right )\,\sqrt {\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.57, size = 68, normalized size = 0.48 \[ \operatorname {RootSum} {\left (55296 t^{4} + 2304 t^{2} a c + 9 a^{4} + 12 a^{2} c^{2} + 4 c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 4608 t^{3} c + 72 t a^{3} - 144 t a c^{2}}{9 a^{4} - 4 c^{4}} \right )} \right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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