\(\int \frac {a+c x^2}{2+3 x^4} \, dx\) [156]

Optimal result

Integrand size = 17, antiderivative size = 141 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=-\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1+\sqrt [4]{6} x\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}-6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (\sqrt {6}+6^{3/4} x+3 x^2\right )}{8\ 6^{3/4}} \]

[Out]

-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)

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {1182, 1176, 631, 210, 1179, 642} \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=-\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )}{4\ 6^{3/4}}+\frac {\left (\sqrt {6} a+2 c\right ) \arctan \left (\sqrt [4]{6} x+1\right )}{4\ 6^{3/4}}-\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2-6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}}+\frac {\left (\sqrt {6} a-2 c\right ) \log \left (3 x^2+6^{3/4} x+\sqrt {6}\right )}{8\ 6^{3/4}} \]

[In]

Int[(a + c*x^2)/(2 + 3*x^4),x]

[Out]

-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))

Rule 210

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])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[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])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[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]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[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]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rubi steps \begin{align*} \text {integral}& = \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 ) \text {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 ) \text {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*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.80 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=\frac {-2 \left (\sqrt {6} a+2 c\right ) \arctan \left (1-\sqrt [4]{6} x\right )+2 \left (\sqrt {6} a+2 c\right ) \arctan \left (1+\sqrt [4]{6} x\right )-\left (\sqrt {6} a-2 c\right ) \left (\log \left (2-2 \sqrt [4]{6} x+\sqrt {6} x^2\right )-\log \left (2+2 \sqrt [4]{6} x+\sqrt {6} x^2\right )\right )}{8\ 6^{3/4}} \]

[In]

Integrate[(a + c*x^2)/(2 + 3*x^4),x]

[Out]

(-2*(Sqrt[6]*a + 2*c)*ArcTan[1 - 6^(1/4)*x] + 2*(Sqrt[6]*a + 2*c)*ArcTan[1 + 6^(1/4)*x] - (Sqrt[6]*a - 2*c)*(L
og[2 - 2*6^(1/4)*x + Sqrt[6]*x^2] - Log[2 + 2*6^(1/4)*x + Sqrt[6]*x^2]))/(8*6^(3/4))

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 1.49 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.22

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (3 \textit {\_Z}^{4}+2\right )}{\sum }\frac {\left (\textit {\_R}^{2} c +a \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}\right )}{12}\)	\(31\)
default	\(\frac {a \sqrt {3}\, 6^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{48}+\frac {c \sqrt {3}\, 6^{\frac {3}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}{x^{2}+\frac {\sqrt {3}\, 6^{\frac {1}{4}} x \sqrt {2}}{3}+\frac {\sqrt {6}}{3}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {3}\, 6^{\frac {3}{4}} x}{6}-1\right )\right )}{144}\)	\(188\)
meijerg	\(\frac {54^{\frac {3}{4}} c \left (\frac {x^{3} \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}-\frac {x^{3} \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {3}{4}}}+\frac {x^{3} \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {3}{4}}}\right )}{216}+\frac {24^{\frac {3}{4}} a \left (-\frac {x \sqrt {2}\, \ln \left (1-6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8-3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \ln \left (1+6^{\frac {1}{4}} \left (x^{4}\right )^{\frac {1}{4}}+\frac {\sqrt {3}\, \sqrt {2}\, \sqrt {x^{4}}}{2}\right )}{2 \left (x^{4}\right )^{\frac {1}{4}}}+\frac {x \sqrt {2}\, \arctan \left (\frac {3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}{8+3^{\frac {1}{4}} 8^{\frac {3}{4}} \left (x^{4}\right )^{\frac {1}{4}}}\right )}{\left (x^{4}\right )^{\frac {1}{4}}}\right )}{96}\)	\(334\)

[In]

int((c*x^2+a)/(3*x^4+2),x,method=_RETURNVERBOSE)

[Out]

1/12*sum((_R^2*c+a)/_R^3*ln(x-_R),_R=RootOf(3*_Z^4+2))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 493 vs. \(2 (104) = 208\).

Time = 0.28 (sec) , antiderivative size = 493, normalized size of antiderivative = 3.50 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=-\frac {1}{24} \, \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x + {\left (9 \, a^{3} - 6 \, a c^{2} - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) + \frac {1}{24} \, \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x - {\left (9 \, a^{3} - 6 \, a c^{2} - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) - \frac {1}{24} \, \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x + {\left (9 \, a^{3} - 6 \, a c^{2} + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) + \frac {1}{24} \, \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}} \log \left (-3 \, {\left (9 \, a^{4} - 4 \, c^{4}\right )} x - {\left (9 \, a^{3} - 6 \, a c^{2} + \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}} c\right )} \sqrt {-12 \, a c - \sqrt {6} \sqrt {-9 \, a^{4} + 12 \, a^{2} c^{2} - 4 \, c^{4}}}\right ) \]

[In]

integrate((c*x^2+a)/(3*x^4+2),x, algorithm="fricas")

[Out]

-1/24*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
*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*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)*sq
rt(-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*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)))

Sympy [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.48 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=\operatorname {RootSum} {\left (55296 t^{4} + 2304 t^{2} a c + 9 a^{4} + 12 a^{2} c^{2} + 4 c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 4608 t^{3} c + 72 t a^{3} - 144 t a c^{2}}{9 a^{4} - 4 c^{4}} \right )} \right )\right )} \]

[In]

integrate((c*x**2+a)/(3*x**4+2),x)

[Out]

RootSum(55296*_t**4 + 2304*_t**2*a*c + 9*a**4 + 12*a**2*c**2 + 4*c**4, Lambda(_t, _t*log(x + (-4608*_t**3*c +
72*_t*a**3 - 144*_t*a*c**2)/(9*a**4 - 4*c**4))))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.18 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=\frac {1}{24} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a + \sqrt {2} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x + 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{24} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a + \sqrt {2} c\right )} \arctan \left (\frac {1}{6} \cdot 3^{\frac {3}{4}} 2^{\frac {1}{4}} {\left (2 \, \sqrt {3} x - 3^{\frac {1}{4}} 2^{\frac {3}{4}}\right )}\right ) + \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} + 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) - \frac {1}{48} \cdot 3^{\frac {1}{4}} 2^{\frac {3}{4}} {\left (\sqrt {3} a - \sqrt {2} c\right )} \log \left (\sqrt {3} x^{2} - 3^{\frac {1}{4}} 2^{\frac {3}{4}} x + \sqrt {2}\right ) \]

[In]

integrate((c*x^2+a)/(3*x^4+2),x, algorithm="maxima")

[Out]

1/24*3^(1/4)*2^(3/4)*(sqrt(3)*a + sqrt(2)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x + 3^(1/4)*2^(3/4))) + 1/2
4*3^(1/4)*2^(3/4)*(sqrt(3)*a + sqrt(2)*c)*arctan(1/6*3^(3/4)*2^(1/4)*(2*sqrt(3)*x - 3^(1/4)*2^(3/4))) + 1/48*3
^(1/4)*2^(3/4)*(sqrt(3)*a - sqrt(2)*c)*log(sqrt(3)*x^2 + 3^(1/4)*2^(3/4)*x + sqrt(2)) - 1/48*3^(1/4)*2^(3/4)*(
sqrt(3)*a - sqrt(2)*c)*log(sqrt(3)*x^2 - 3^(1/4)*2^(3/4)*x + sqrt(2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.93 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=\frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{24} \, {\left (6^{\frac {3}{4}} a + 2 \cdot 6^{\frac {1}{4}} c\right )} \arctan \left (\frac {3}{4} \, \sqrt {2} \left (\frac {2}{3}\right )^{\frac {3}{4}} {\left (2 \, x - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}}\right )}\right ) + \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} + \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) - \frac {1}{48} \, {\left (6^{\frac {3}{4}} a - 2 \cdot 6^{\frac {1}{4}} c\right )} \log \left (x^{2} - \sqrt {2} \left (\frac {2}{3}\right )^{\frac {1}{4}} x + \sqrt {\frac {2}{3}}\right ) \]

[In]

integrate((c*x^2+a)/(3*x^4+2),x, algorithm="giac")

[Out]

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)*log
(x^2 + sqrt(2)*(2/3)^(1/4)*x + sqrt(2/3)) - 1/48*(6^(3/4)*a - 2*6^(1/4)*c)*log(x^2 - sqrt(2)*(2/3)^(1/4)*x + s
qrt(2/3))

Mupad [B] (verification not implemented)

Time = 9.21 (sec) , antiderivative size = 315, normalized size of antiderivative = 2.23 \[ \int \frac {a+c x^2}{2+3 x^4} \, dx=-2\,\mathrm {atanh}\left (\frac {216\,a^2\,x\,\sqrt {-\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3+18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2-12\,c^3}-\frac {144\,c^2\,x\,\sqrt {-\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3+18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2-12\,c^3}\right )\,\sqrt {-\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}+\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}+2\,\mathrm {atanh}\left (\frac {216\,a^2\,x\,\sqrt {\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3-18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2+12\,c^3}-\frac {144\,c^2\,x\,\sqrt {\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}}}{9{}\mathrm {i}\,\sqrt {6}\,a^3-18\,a^2\,c-6{}\mathrm {i}\,\sqrt {6}\,a\,c^2+12\,c^3}\right )\,\sqrt {\frac {1{}\mathrm {i}\,\sqrt {6}\,a^2}{192}-\frac {a\,c}{48}-\frac {1{}\mathrm {i}\,\sqrt {6}\,c^2}{288}} \]

[In]

int((a + c*x^2)/(3*x^4 + 2),x)

[Out]

2*atanh((216*a^2*x*((6^(1/2)*a^2*1i)/192 - (a*c)/48 - (6^(1/2)*c^2*1i)/288)^(1/2))/(6^(1/2)*a^3*9i - 18*a^2*c
+ 12*c^3 - 6^(1/2)*a*c^2*6i) - (144*c^2*x*((6^(1/2)*a^2*1i)/192 - (a*c)/48 - (6^(1/2)*c^2*1i)/288)^(1/2))/(6^(
1/2)*a^3*9i - 18*a^2*c + 12*c^3 - 6^(1/2)*a*c^2*6i))*((6^(1/2)*a^2*1i)/192 - (a*c)/48 - (6^(1/2)*c^2*1i)/288)^
(1/2) - 2*atanh((216*a^2*x*((6^(1/2)*c^2*1i)/288 - (6^(1/2)*a^2*1i)/192 - (a*c)/48)^(1/2))/(6^(1/2)*a^3*9i + 1
8*a^2*c - 12*c^3 - 6^(1/2)*a*c^2*6i) - (144*c^2*x*((6^(1/2)*c^2*1i)/288 - (6^(1/2)*a^2*1i)/192 - (a*c)/48)^(1/
2))/(6^(1/2)*a^3*9i + 18*a^2*c - 12*c^3 - 6^(1/2)*a*c^2*6i))*((6^(1/2)*c^2*1i)/288 - (6^(1/2)*a^2*1i)/192 - (a
*c)/48)^(1/2)