Integrand size = 22, antiderivative size = 154 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=-\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \]
1/12*d*ln(3*x^4+2)-1/48*ln(-6^(3/4)*x+3*x^2+6^(1/2))*(-2*c+a*6^(1/2))*6^(1 /4)+1/48*ln(6^(3/4)*x+3*x^2+6^(1/2))*(-2*c+a*6^(1/2))*6^(1/4)+1/24*arctan( -1+6^(1/4)*x)*(2*c+a*6^(1/2))*6^(1/4)+1/24*arctan(1+6^(1/4)*x)*(2*c+a*6^(1 /2))*6^(1/4)
Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{48} \left (-2 \sqrt [4]{6} \left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )+2 \sqrt [4]{6} \left (\sqrt {6} a+2 c\right ) \arctan \left (1+\sqrt [4]{6} x\right )-\sqrt [4]{6} \left (\sqrt {6} a-2 c\right ) \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+\sqrt [4]{6} \left (\sqrt {6} a-2 c\right ) \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+4 d \log \left (2+3 x^4\right )\right ) \]
(-2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*6^(1/4)*(Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x] - 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 - 2*6^(1/4 )*x + Sqrt[6]*x^2] + 6^(1/4)*(Sqrt[6]*a - 2*c)*Log[2 + 2*6^(1/4)*x + Sqrt[ 6]*x^2] + 4*d*Log[2 + 3*x^4])/48
Time = 0.30 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2415, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {a+c x^2+d x^3}{3 x^4+2} \, dx\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \int \left (\frac {a+c x^2}{3 x^4+2}+\frac {d x^3}{3 x^4+2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {1}{12} d \log \left (3 x^4+2\right )\) |
-1/4*((Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x])/6^(3/4) + ((Sqrt[6]*a + 2*c )*ArcTan[1 + 6^(1/4)*x])/(4*6^(3/4)) - ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] - 6^ (3/4)*x + 3*x^2])/(8*6^(3/4)) + ((Sqrt[6]*a - 2*c)*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(3/4)) + (d*Log[2 + 3*x^4])/12
3.2.64.3.1 Defintions of rubi rules used
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.48 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.23
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3} d +\textit {\_R}^{2} c +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(36\) |
| default | \(\frac {a \sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{48}+\frac {c \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{144}+\frac {d \ln \left (3 x^{4}+2\right )}{12}\) | \(199\) |
| meijerg | \(\frac {d \ln \left (\frac {3 x^{4}}{2}+1\right )}{12}+\frac {54^{\frac {3}{4}} c \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{216}+\frac {24^{\frac {3}{4}} a \left (-\frac {x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{96}\) | \(345\) |
Leaf count of result is larger than twice the leaf count of optimal. 513 vs. \(2 (115) = 230\).
Time = 0.29 (sec) , antiderivative size = 513, normalized size of antiderivative = 3.33 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (2 \, d - \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right )} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x + {\left (9 \, a^{3} - 6 \, a c^{2} - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) + \frac {1}{24} \, {\left (2 \, d + \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right )} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x - {\left (9 \, a^{3} - 6 \, a c^{2} - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) + \frac {1}{24} \, {\left (2 \, d - \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right )} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x + {\left (9 \, a^{3} - 6 \, a c^{2} + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) + \frac {1}{24} \, {\left (2 \, d + \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right )} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x - {\left (9 \, a^{3} - 6 \, a c^{2} + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) \]
1/24*(2*d - sqrt(-12*a*c + sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4)))*log (-3*(9*a^4 - 4*c^4)*x + (9*a^3 - 6*a*c^2 - sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^ 2 - 4*c^4)*c)*sqrt(-12*a*c + sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4))) + 1/24*(2*d + sqrt(-12*a*c + sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4)))*lo g(-3*(9*a^4 - 4*c^4)*x - (9*a^3 - 6*a*c^2 - sqrt(6)*sqrt(-9*a^4 + 12*a^2*c ^2 - 4*c^4)*c)*sqrt(-12*a*c + sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4))) + 1/24*(2*d - sqrt(-12*a*c - sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4)))*l og(-3*(9*a^4 - 4*c^4)*x + (9*a^3 - 6*a*c^2 + sqrt(6)*sqrt(-9*a^4 + 12*a^2* c^2 - 4*c^4)*c)*sqrt(-12*a*c - sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4))) + 1/24*(2*d + sqrt(-12*a*c - sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4)))* log(-3*(9*a^4 - 4*c^4)*x - (9*a^3 - 6*a*c^2 + sqrt(6)*sqrt(-9*a^4 + 12*a^2 *c^2 - 4*c^4)*c)*sqrt(-12*a*c - sqrt(6)*sqrt(-9*a^4 + 12*a^2*c^2 - 4*c^4)) )
Time = 0.67 (sec) , antiderivative size = 148, normalized size of antiderivative = 0.96 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (165888 t^{4} - 55296 t^{3} d + t^{2} \cdot \left (6912 a c + 6912 d^{2}\right ) + t \left (- 1152 a c d - 384 d^{3}\right ) + 27 a^{4} + 36 a^{2} c^{2} + 48 a c d^{2} + 12 c^{4} + 8 d^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 13824 t^{3} c + 3456 t^{2} c d + 216 t a^{3} - 432 t a c^{2} - 288 t c d^{2} - 18 a^{3} d + 36 a c^{2} d + 8 c d^{3}}{27 a^{4} - 12 c^{4}} \right )} \right )\right )} \]
RootSum(165888*_t**4 - 55296*_t**3*d + _t**2*(6912*a*c + 6912*d**2) + _t*( -1152*a*c*d - 384*d**3) + 27*a**4 + 36*a**2*c**2 + 48*a*c*d**2 + 12*c**4 + 8*d**4, Lambda(_t, _t*log(x + (-13824*_t**3*c + 3456*_t**2*c*d + 216*_t*a **3 - 432*_t*a*c**2 - 288*_t*c*d**2 - 18*a**3*d + 36*a*c**2*d + 8*c*d**3)/ (27*a**4 - 12*c**4))))
Time = 0.30 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.27 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=-\frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} \sqrt {2} c - 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} \sqrt {2} c + 2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{72} \, \sqrt {3} {\left (3 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} a + 2 \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{72} \, \sqrt {3} {\left (3 \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} a + 2 \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) \]
-1/144*3^(3/4)*2^(3/4)*(sqrt(3)*sqrt(2)*c - 2*3^(1/4)*2^(1/4)*d - 3*a)*log (sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/144*3^(3/4)*2^(3/4)*(sqrt( 3)*sqrt(2)*c + 2*3^(1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4 )*x + sqrt(2)) + 1/72*sqrt(3)*(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c)* arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/72*sqrt(3) *(3*3^(1/4)*2^(3/4)*a + 2*3^(3/4)*2^(1/4)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2 *sqrt(3)*x - 3^(1/4)*2^(3/4)))
Time = 0.28 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.89 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c + 4 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c - 4 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
1/24*(6^(3/4)*a + 2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt( 2)*(2/3)^(1/4))) + 1/24*(6^(3/4)*a + 2*6^(1/4)*c)*arctan(3/4*sqrt(2)*(2/3) ^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 1/48*(6^(3/4)*a - 2*6^(1/4)*c + 4*d) *log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 2*6^(1/4 )*c - 4*d)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))
Time = 10.29 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.86 \[ \int \frac {a+c x^2+d x^3}{2+3 x^4} \, dx=\ln \left (-2\,c+\sqrt {6}\,a\,1{}\mathrm {i}+x\,\sqrt {3{}\mathrm {i}\,\sqrt {6}\,a^2-12\,a\,c-2{}\mathrm {i}\,\sqrt {6}\,c^2}\right )\,\left (\frac {d}{12}+\frac {\sqrt {\frac {3{}\mathrm {i}\,\sqrt {6}\,a^2}{4}-3\,a\,c-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{2}}}{12}\right )+\ln \left (2\,c-\sqrt {6}\,a\,1{}\mathrm {i}+x\,\sqrt {3{}\mathrm {i}\,\sqrt {6}\,a^2-12\,a\,c-2{}\mathrm {i}\,\sqrt {6}\,c^2}\right )\,\left (\frac {d}{12}-\frac {\sqrt {\frac {3{}\mathrm {i}\,\sqrt {6}\,a^2}{4}-3\,a\,c-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{2}}}{12}\right )+\ln \left (2\,c+\sqrt {6}\,a\,1{}\mathrm {i}+x\,\sqrt {-3{}\mathrm {i}\,\sqrt {6}\,a^2-12\,a\,c+2{}\mathrm {i}\,\sqrt {6}\,c^2}\right )\,\left (\frac {d}{12}-\frac {\sqrt {-\frac {3{}\mathrm {i}\,\sqrt {6}\,a^2}{4}-3\,a\,c+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{2}}}{12}\right )+\ln \left (2\,c+\sqrt {6}\,a\,1{}\mathrm {i}-x\,\sqrt {-3{}\mathrm {i}\,\sqrt {6}\,a^2-12\,a\,c+2{}\mathrm {i}\,\sqrt {6}\,c^2}\right )\,\left (\frac {d}{12}+\frac {\sqrt {-\frac {3{}\mathrm {i}\,\sqrt {6}\,a^2}{4}-3\,a\,c+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{2}}}{12}\right ) \]
log(6^(1/2)*a*1i - 2*c + x*(6^(1/2)*a^2*3i - 12*a*c - 6^(1/2)*c^2*2i)^(1/2 ))*(d/12 + ((6^(1/2)*a^2*3i)/4 - 3*a*c - (6^(1/2)*c^2*1i)/2)^(1/2)/12) + l og(2*c - 6^(1/2)*a*1i + x*(6^(1/2)*a^2*3i - 12*a*c - 6^(1/2)*c^2*2i)^(1/2) )*(d/12 - ((6^(1/2)*a^2*3i)/4 - 3*a*c - (6^(1/2)*c^2*1i)/2)^(1/2)/12) + lo g(2*c + 6^(1/2)*a*1i + x*(6^(1/2)*c^2*2i - 6^(1/2)*a^2*3i - 12*a*c)^(1/2)) *(d/12 - ((6^(1/2)*c^2*1i)/2 - (6^(1/2)*a^2*3i)/4 - 3*a*c)^(1/2)/12) + log (2*c + 6^(1/2)*a*1i - x*(6^(1/2)*c^2*2i - 6^(1/2)*a^2*3i - 12*a*c)^(1/2))* (d/12 + ((6^(1/2)*c^2*1i)/2 - (6^(1/2)*a^2*3i)/4 - 3*a*c)^(1/2)/12)