Integrand size = 11, antiderivative size = 101 \[ \int \frac {a}{2+3 x^4} \, dx=-\frac {a \arctan \left (1-\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}+\frac {a \arctan \left (1+\sqrt [4]{6} x\right )}{4 \sqrt [4]{6}}-\frac {a \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}}+\frac {a \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8 \sqrt [4]{6}} \]
1/24*a*arctan(-1+6^(1/4)*x)*6^(3/4)+1/24*a*arctan(1+6^(1/4)*x)*6^(3/4)-1/4 8*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 = 0.04 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.77 \[ \int \frac {a}{2+3 x^4} \, dx=\frac {a \left (-2 \arctan \left (1-\sqrt [4]{6} x\right )+2 \arctan \left (1+\sqrt [4]{6} x\right )-\log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )+\log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )\right )}{8 \sqrt [4]{6}} \]
(a*(-2*ArcTan[1 - 6^(1/4)*x] + 2*ArcTan[1 + 6^(1/4)*x] - Log[2 - 2*6^(1/4) *x + Sqrt[6]*x^2] + Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(1/4))
Time = 0.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.33, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.818, Rules used = {27, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
\(\displaystyle \int \frac {a}{3 x^4+2} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \int \frac {1}{3 x^4+2}dx\) |
\(\Big \downarrow \) 755 |
\(\displaystyle a \left (\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {3} x^2+\sqrt {2}}{3 x^4+2}dx}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle a \left (\frac {\frac {\int \frac {1}{x^2-\frac {2^{3/4} x}{\sqrt [4]{3}}+\sqrt {\frac {2}{3}}}dx}{2 \sqrt {3}}+\frac {\int \frac {1}{x^2+\frac {2^{3/4} x}{\sqrt [4]{3}}+\sqrt {\frac {2}{3}}}dx}{2 \sqrt {3}}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle a \left (\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {2}}+\frac {\frac {\int \frac {1}{-\left (1-\sqrt [4]{6} x\right )^2-1}d\left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\int \frac {1}{-\left (\sqrt [4]{6} x+1\right )^2-1}d\left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle a \left (\frac {\int \frac {\sqrt {2}-\sqrt {3} x^2}{3 x^4+2}dx}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle a \left (\frac {-\frac {\int -\frac {6^{3/4}-6 x}{3 x^2-6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}-\frac {\int -\frac {6^{3/4} \left (\sqrt [4]{6} x+1\right )}{3 x^2+6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 25 |
\(\displaystyle a \left (\frac {\frac {\int \frac {6^{3/4}-6 x}{3 x^2-6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}+\frac {\int \frac {6^{3/4} \left (\sqrt [4]{6} x+1\right )}{3 x^2+6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 27 |
\(\displaystyle a \left (\frac {\frac {\int \frac {6^{3/4}-6 x}{3 x^2-6^{3/4} x+\sqrt {6}}dx}{2\ 2^{3/4} \sqrt [4]{3}}+\frac {1}{2} \sqrt {3} \int \frac {\sqrt [4]{6} x+1}{3 x^2+6^{3/4} x+\sqrt {6}}dx}{2 \sqrt {2}}+\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}\right )\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle a \left (\frac {\frac {\arctan \left (\sqrt [4]{6} x+1\right )}{2^{3/4} \sqrt [4]{3}}-\frac {\arctan \left (1-\sqrt [4]{6} x\right )}{2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}+\frac {\frac {\log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{2\ 2^{3/4} \sqrt [4]{3}}-\frac {\log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{2\ 2^{3/4} \sqrt [4]{3}}}{2 \sqrt {2}}\right )\) |
a*((-(ArcTan[1 - 6^(1/4)*x]/(2^(3/4)*3^(1/4))) + ArcTan[1 + 6^(1/4)*x]/(2^ (3/4)*3^(1/4)))/(2*Sqrt[2]) + (-1/2*Log[Sqrt[6] - 6^(3/4)*x + 3*x^2]/(2^(3 /4)*3^(1/4)) + Log[Sqrt[6] + 6^(3/4)*x + 3*x^2]/(2*2^(3/4)*3^(1/4)))/(2*Sq rt[2]))
3.2.52.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.45 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.25
method | result | size |
risch | \(\frac {a \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\) | \(25\) |
default | \(\frac {a \sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{48}\) | \(94\) |
meijerg | \(\frac {24^{\frac {3}{4}} a \left (-\frac {x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{96}\) | \(163\) |
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.19 \[ \int \frac {a}{2+3 x^4} \, dx=\frac {1}{96} \cdot 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}} \log \left (12 \, a x + 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}}\right ) + \frac {1}{96} i \cdot 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}} \log \left (12 \, a x + i \cdot 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}}\right ) - \frac {1}{96} i \cdot 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}} \log \left (12 \, a x - i \cdot 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}}\right ) - \frac {1}{96} \cdot 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}} \log \left (12 \, a x - 24^{\frac {3}{4}} \left (-a^{4}\right )^{\frac {1}{4}}\right ) \]
1/96*24^(3/4)*(-a^4)^(1/4)*log(12*a*x + 24^(3/4)*(-a^4)^(1/4)) + 1/96*I*24 ^(3/4)*(-a^4)^(1/4)*log(12*a*x + I*24^(3/4)*(-a^4)^(1/4)) - 1/96*I*24^(3/4 )*(-a^4)^(1/4)*log(12*a*x - I*24^(3/4)*(-a^4)^(1/4)) - 1/96*24^(3/4)*(-a^4 )^(1/4)*log(12*a*x - 24^(3/4)*(-a^4)^(1/4))
Time = 0.18 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.87 \[ \int \frac {a}{2+3 x^4} \, dx=a \left (- \frac {6^{\frac {3}{4}} \log {\left (x^{2} - \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{48} + \frac {6^{\frac {3}{4}} \log {\left (x^{2} + \frac {6^{\frac {3}{4}} x}{3} + \frac {\sqrt {6}}{3} \right )}}{48} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x - 1 \right )}}{24} + \frac {6^{\frac {3}{4}} \operatorname {atan}{\left (\sqrt [4]{6} x + 1 \right )}}{24}\right ) \]
a*(-6**(3/4)*log(x**2 - 6**(3/4)*x/3 + sqrt(6)/3)/48 + 6**(3/4)*log(x**2 + 6**(3/4)*x/3 + sqrt(6)/3)/48 + 6**(3/4)*atan(6**(1/4)*x - 1)/24 + 6**(3/4 )*atan(6**(1/4)*x + 1)/24)
Time = 0.28 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.22 \[ \int \frac {a}{2+3 x^4} \, dx=\frac {1}{48} \, {\left (2 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + 2 \cdot 3^{\frac {3}{4}} 2^{\frac {3}{4}} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - 3^{\frac {3}{4}} 2^{\frac {3}{4}} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right )\right )} a \]
1/48*(2*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)* 2^(3/4))) + 2*3^(3/4)*2^(3/4)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^ (1/4)*2^(3/4))) + 3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sq rt(2)) - 3^(3/4)*2^(3/4)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2)))*a
Time = 0.29 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.96 \[ \int \frac {a}{2+3 x^4} \, dx=\frac {1}{48} \, {\left (2 \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + 2 \cdot 6^{\frac {3}{4}} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + 6^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - 6^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right )\right )} a \]
1/48*(2*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x + sqrt(2)*(2/3)^(1/4)) ) + 2*6^(3/4)*arctan(3/4*sqrt(2)*(2/3)^(3/4)*(2*x - sqrt(2)*(2/3)^(1/4))) + 6^(3/4)*log(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 6^(3/4)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)))*a
Time = 0.13 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.36 \[ \int \frac {a}{2+3 x^4} \, dx=-\frac {{\left (-1\right )}^{1/4}\,{6144}^{3/4}\,a\,\left (\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,{6144}^{1/4}\,x}{8}\right )\,1{}\mathrm {i}+\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,{6144}^{1/4}\,x}{8}\right )\,1{}\mathrm {i}\right )}{3072} \]