Integrand size = 17, antiderivative size = 114 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=-\frac {a \arctan \left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \arctan \left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {a \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {1}{12} d \log \left (2+3 x^4\right ) \]
1/24*a*arctan(-1+6^(1/4)*x)*6^(3/4)+1/24*a*arctan(1+6^(1/4)*x)*6^(3/4)+1/1 2*d*ln(3*x^4+2)-1/48*a*ln(-6^(3/4)*x+3*x^2+6^(1/2))*6^(3/4)+1/48*a*ln(6^(3 /4)*x+3*x^2+6^(1/2))*6^(3/4)
Time = 0.03 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=\frac {1}{48} \left (-2 6^{3/4} a \arctan \left (1-\sqrt [4]{6} x\right )+2\ 6^{3/4} a \arctan \left (1+\sqrt [4]{6} x\right )-6^{3/4} a \log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+6^{3/4} a \log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+4 d \log \left (2+3 x^4\right )\right ) \]
(-2*6^(3/4)*a*ArcTan[1 - 6^(1/4)*x] + 2*6^(3/4)*a*ArcTan[1 + 6^(1/4)*x] - 6^(3/4)*a*Log[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] + 6^(3/4)*a*Log[2 + 2*6^(1/4) *x + Sqrt[6]*x^2] + 4*d*Log[2 + 3*x^4])/48
Time = 0.27 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, 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.
\(\displaystyle \int \frac {a+d x^3}{3 x^4+2} \, dx\) |
\(\Big \downarrow \) 2415 |
\(\displaystyle \int \left (\frac {a}{3 x^4+2}+\frac {d x^3}{3 x^4+2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {a \arctan \left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \arctan \left (\sqrt [4]{6} x+1\right )}{4 \sqrt [4]{6}}-\frac {a \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8 \sqrt [4]{6}}+\frac {1}{12} d \log \left (3 x^4+2\right )\) |
-1/4*(a*ArcTan[1 - 6^(1/4)*x])/6^(1/4) + (a*ArcTan[1 + 6^(1/4)*x])/(4*6^(1 /4)) - (a*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2])/(8*6^(1/4)) + (a*Log[Sqrt[6] + 6^(3/4)*x + 3*x^2])/(8*6^(1/4)) + (d*Log[2 + 3*x^4])/12
3.2.60.3.1 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 = 1.46 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.27
method | result | size |
risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{3} d +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(31\) |
default | \(\frac {a \sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{48}+\frac {d \ln \left (3 x^{4}+2\right )}{12}\) | \(106\) |
meijerg | \(\frac {d \ln \left (\frac {3 x^{4}}{2}+1\right )}{12}+\frac {24^{\frac {3}{4}} a \left (-\frac {x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{96}\) | \(175\) |
Leaf count of result is larger than twice the leaf count of optimal. 191 vs. \(2 (83) = 166\).
Time = 0.27 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.68 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (\sqrt {3} \sqrt {\sqrt {6} \sqrt {-a^{4}}} + 2 \, d\right )} \log \left (3 \, a x + \sqrt {3} \sqrt {\sqrt {6} \sqrt {-a^{4}}}\right ) - \frac {1}{24} \, {\left (\sqrt {3} \sqrt {\sqrt {6} \sqrt {-a^{4}}} - 2 \, d\right )} \log \left (3 \, a x - \sqrt {3} \sqrt {\sqrt {6} \sqrt {-a^{4}}}\right ) + \frac {1}{24} \, {\left (\sqrt {3} \sqrt {-\sqrt {6} \sqrt {-a^{4}}} + 2 \, d\right )} \log \left (3 \, a x + \sqrt {3} \sqrt {-\sqrt {6} \sqrt {-a^{4}}}\right ) - \frac {1}{24} \, {\left (\sqrt {3} \sqrt {-\sqrt {6} \sqrt {-a^{4}}} - 2 \, d\right )} \log \left (3 \, a x - \sqrt {3} \sqrt {-\sqrt {6} \sqrt {-a^{4}}}\right ) \]
1/24*(sqrt(3)*sqrt(sqrt(6)*sqrt(-a^4)) + 2*d)*log(3*a*x + sqrt(3)*sqrt(sqr t(6)*sqrt(-a^4))) - 1/24*(sqrt(3)*sqrt(sqrt(6)*sqrt(-a^4)) - 2*d)*log(3*a* x - sqrt(3)*sqrt(sqrt(6)*sqrt(-a^4))) + 1/24*(sqrt(3)*sqrt(-sqrt(6)*sqrt(- a^4)) + 2*d)*log(3*a*x + sqrt(3)*sqrt(-sqrt(6)*sqrt(-a^4))) - 1/24*(sqrt(3 )*sqrt(-sqrt(6)*sqrt(-a^4)) - 2*d)*log(3*a*x - sqrt(3)*sqrt(-sqrt(6)*sqrt( -a^4)))
Time = 0.16 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.45 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (165888 t^{4} - 55296 t^{3} d + 6912 t^{2} d^{2} - 384 t d^{3} + 27 a^{4} + 8 d^{4}, \left ( t \mapsto t \log {\left (x + \frac {24 t - 2 d}{3 a} \right )} \right )\right )} \]
RootSum(165888*_t**4 - 55296*_t**3*d + 6912*_t**2*d**2 - 384*_t*d**3 + 27* a**4 + 8*d**4, Lambda(_t, _t*log(x + (24*_t - 2*d)/(3*a))))
Time = 0.28 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.31 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} a \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} a \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d + 3 \, a\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) + \frac {1}{144} \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} {\left (2 \cdot 3^{\frac {1}{4}} 2^{\frac {1}{4}} d - 3 \, a\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]
1/24*3^(3/4)*2^(3/4)*a*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2 ^(3/4))) + 1/24*3^(3/4)*2^(3/4)*a*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 1/144*3^(3/4)*2^(3/4)*(2*3^(1/4)*2^(1/4)*d + 3*a)*lo g(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) + 1/144*3^(3/4)*2^(3/4)*(2*3^ (1/4)*2^(1/4)*d - 3*a)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))
Time = 0.28 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.96 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=\frac {1}{24} \cdot 6^{\frac {3}{4}} a \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \cdot 6^{\frac {3}{4}} a \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a + 4 \, d\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 4 \, d\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]
1/24*6^(3/4)*a*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4))) + 1/24*6^(3/4)*a*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4 ))) + 1/48*(6^(3/4)*a + 4*d)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 4*d)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3))
Time = 0.29 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.03 \[ \int \frac {a+d x^3}{2+3 x^4} \, dx=\ln \left (x-\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {\frac {3}{4}{}\mathrm {i}}\,a}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{1/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {\frac {3}{4}{}\mathrm {i}}\,a}{12}\right )+\ln \left (x-\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}+\frac {6^{1/4}\,\sqrt {-\frac {3}{4}{}\mathrm {i}}\,a}{12}\right )+\ln \left (x+\frac {{\left (-1\right )}^{3/4}\,2^{1/4}\,3^{3/4}}{3}\right )\,\left (\frac {d}{12}-\frac {6^{1/4}\,\sqrt {-\frac {3}{4}{}\mathrm {i}}\,a}{12}\right ) \]
log(x - ((-1)^(1/4)*2^(1/4)*3^(3/4))/3)*(d/12 - (6^(1/4)*(3i/4)^(1/2)*a)/1 2) + log(x + ((-1)^(1/4)*2^(1/4)*3^(3/4))/3)*(d/12 + (6^(1/4)*(3i/4)^(1/2) *a)/12) + log(x - ((-1)^(3/4)*2^(1/4)*3^(3/4))/3)*(d/12 + (6^(1/4)*(-3i/4) ^(1/2)*a)/12) + log(x + ((-1)^(3/4)*2^(1/4)*3^(3/4))/3)*(d/12 - (6^(1/4)*( -3i/4)^(1/2)*a)/12)