\(\int \frac {(-3+x^4) (1+x^4)^{2/3}}{x^3 (2+x^3+2 x^4)} \, dx\) [1929]

Optimal result

Integrand size = 30, antiderivative size = 134 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3}}{4 x^2}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2 \sqrt [3]{2} \sqrt [3]{1+x^4}}\right )}{2\ 2^{2/3}}+\frac {\log \left (x+\sqrt [3]{2} \sqrt [3]{1+x^4}\right )}{2\ 2^{2/3}}-\frac {\log \left (x^2-\sqrt [3]{2} x \sqrt [3]{1+x^4}+2^{2/3} \left (1+x^4\right )^{2/3}\right )}{4\ 2^{2/3}} \]

[Out]

3/4*(x^4+1)^(2/3)/x^2+1/4*3^(1/2)*arctan(3^(1/2)*x/(-x+2*2^(1/3)*(x^4+1)^(1/3)))*2^(1/3)+1/4*ln(x+2^(1/3)*(x^4
+1)^(1/3))*2^(1/3)-1/8*ln(x^2-2^(1/3)*x*(x^4+1)^(1/3)+2^(2/3)*(x^4+1)^(2/3))*2^(1/3)

Rubi [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx \]

[In]

Int[((-3 + x^4)*(1 + x^4)^(2/3))/(x^3*(2 + x^3 + 2*x^4)),x]

[Out]

(3*(1 + x^4)^(2/3))/(4*x^2) + (3*x^2)/(1 - Sqrt[3] - (1 + x^4)^(1/3)) - (3*3^(1/4)*Sqrt[2 + Sqrt[3]]*(1 - (1 +
x^4)^(1/3))*Sqrt[(1 + (1 + x^4)^(1/3) + (1 + x^4)^(2/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2]*EllipticE[ArcSin[
(1 + Sqrt[3] - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))], -7 + 4*Sqrt[3]])/(2*x^2*Sqrt[-((1 - (1 + x^4
)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2)]) + (Sqrt[2]*3^(3/4)*(1 - (1 + x^4)^(1/3))*Sqrt[(1 + (1 + x^4)^(1/
3) + (1 + x^4)^(2/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2]*EllipticF[ArcSin[(1 + Sqrt[3] - (1 + x^4)^(1/3))/(1 -
Sqrt[3] - (1 + x^4)^(1/3))], -7 + 4*Sqrt[3]])/(x^2*Sqrt[-((1 - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3
))^2)]) + (3*Defer[Int][(1 + x^4)^(2/3)/(2 + x^3 + 2*x^4), x])/2 + 4*Defer[Int][(x*(1 + x^4)^(2/3))/(2 + x^3 +
2*x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{2 x^3}+\frac {(3+8 x) \left (1+x^4\right )^{2/3}}{2 \left (2+x^3+2 x^4\right )}\right ) \, dx \\ & = \frac {1}{2} \int \frac {(3+8 x) \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx-\frac {3}{2} \int \frac {\left (1+x^4\right )^{2/3}}{x^3} \, dx \\ & = \frac {1}{2} \int \left (\frac {3 \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4}+\frac {8 x \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4}\right ) \, dx-\frac {3}{4} \text {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{4 x^2}+\frac {3}{2} \int \frac {\left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx-\text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right ) \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{4 x^2}+\frac {3}{2} \int \frac {\left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx-\frac {\left (3 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{2 x^2} \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{4 x^2}+\frac {3}{2} \int \frac {\left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx+\frac {\left (3 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1+\sqrt {3}-x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{2 x^2}-\frac {\left (3 \left (1+\sqrt {3}\right ) \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{2 x^2} \\ & = \frac {3 \left (1+x^4\right )^{2/3}}{4 x^2}+\frac {3 x^2}{1-\sqrt {3}-\sqrt [3]{1+x^4}}-\frac {3 \sqrt [4]{3} \sqrt {2+\sqrt {3}} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} E\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right )|-7+4 \sqrt {3}\right )}{2 x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {\sqrt {2} 3^{3/4} \left (1-\sqrt [3]{1+x^4}\right ) \sqrt {\frac {1+\sqrt [3]{1+x^4}+\left (1+x^4\right )^{2/3}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}} \operatorname {EllipticF}\left (\arcsin \left (\frac {1+\sqrt {3}-\sqrt [3]{1+x^4}}{1-\sqrt {3}-\sqrt [3]{1+x^4}}\right ),-7+4 \sqrt {3}\right )}{x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {3}{2} \int \frac {\left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx+4 \int \frac {x \left (1+x^4\right )^{2/3}}{2+x^3+2 x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\frac {1}{8} \left (\frac {6 \left (1+x^4\right )^{2/3}}{x^2}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x-2 \sqrt [3]{2} \sqrt [3]{1+x^4}}\right )+2 \sqrt [3]{2} \log \left (x+\sqrt [3]{2} \sqrt [3]{1+x^4}\right )-\sqrt [3]{2} \log \left (x^2-\sqrt [3]{2} x \sqrt [3]{1+x^4}+2^{2/3} \left (1+x^4\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-3 + x^4)*(1 + x^4)^(2/3))/(x^3*(2 + x^3 + 2*x^4)),x]

[Out]

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

Maple [A] (verified)

Time = 74.04 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(\frac {2 \,2^{\frac {1}{3}} x^{2} \ln \left (\frac {2^{\frac {2}{3}} x +2 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right )+6 \left (x^{4}+1\right )^{\frac {2}{3}}-2^{\frac {1}{3}} x^{2} \left (-2 \arctan \left (\frac {\sqrt {3}\, \left (x -2 \,2^{\frac {1}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+\ln \left (\frac {-2^{\frac {2}{3}} x \left (x^{4}+1\right )^{\frac {1}{3}}+2^{\frac {1}{3}} x^{2}+2 \left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right )+\ln \left (2\right )\right )}{8 x^{2}}\)	\(118\)
risch	\(\text {Expression too large to display}\)	\(645\)
trager	\(\text {Expression too large to display}\)	\(1469\)

[In]

int((x^4-3)*(x^4+1)^(2/3)/x^3/(2*x^4+x^3+2),x,method=_RETURNVERBOSE)

[Out]

1/8*(2*2^(1/3)*x^2*ln((2^(2/3)*x+2*(x^4+1)^(1/3))/x)+6*(x^4+1)^(2/3)-2^(1/3)*x^2*(-2*arctan(1/3*3^(1/2)*(x-2*2
^(1/3)*(x^4+1)^(1/3))/x)*3^(1/2)+ln((-2^(2/3)*x*(x^4+1)^(1/3)+2^(1/3)*x^2+2*(x^4+1)^(2/3))/x^2)+ln(2)))/x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 391 vs. \(2 (97) = 194\).

Time = 64.75 (sec) , antiderivative size = 391, normalized size of antiderivative = 2.92 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\frac {4 \cdot 4^{\frac {1}{6}} \sqrt {3} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{9} - x^{8} - x^{7} + 4 \, x^{5} - x^{4} + 2 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (8 \, x^{12} - 60 \, x^{11} + 24 \, x^{10} + x^{9} + 24 \, x^{8} - 120 \, x^{7} + 24 \, x^{6} + 24 \, x^{4} - 60 \, x^{3} + 8\right )} - 12 \, {\left (4 \, x^{10} - 14 \, x^{9} + x^{8} + 8 \, x^{6} - 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}\right )}}{6 \, {\left (8 \, x^{12} + 12 \, x^{11} - 48 \, x^{10} + x^{9} + 24 \, x^{8} + 24 \, x^{7} - 48 \, x^{6} + 24 \, x^{4} + 12 \, x^{3} + 8\right )}}\right ) + 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{4} + x^{3} + 2\right )} + 12 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{4} + x^{3} + 2}\right ) - 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{5} - x^{4} + x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} {\left (4 \, x^{8} - 14 \, x^{7} + x^{6} + 8 \, x^{4} - 14 \, x^{3} + 4\right )} - 6 \, {\left (4 \, x^{6} - x^{5} + 4 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{4 \, x^{8} + 4 \, x^{7} + x^{6} + 8 \, x^{4} + 4 \, x^{3} + 4}\right ) + 36 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{48 \, x^{2}} \]

[In]

integrate((x^4-3)*(x^4+1)^(2/3)/x^3/(2*x^4+x^3+2),x, algorithm="fricas")

[Out]

1/48*(4*4^(1/6)*sqrt(3)*x^2*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(2*x^9 - x^8 - x^7 + 4*x^5 - x^4 + 2*x)*(x^
4 + 1)^(2/3) - 4^(1/3)*(8*x^12 - 60*x^11 + 24*x^10 + x^9 + 24*x^8 - 120*x^7 + 24*x^6 + 24*x^4 - 60*x^3 + 8) -
12*(4*x^10 - 14*x^9 + x^8 + 8*x^6 - 14*x^5 + 4*x^2)*(x^4 + 1)^(1/3))/(8*x^12 + 12*x^11 - 48*x^10 + x^9 + 24*x^
8 + 24*x^7 - 48*x^6 + 24*x^4 + 12*x^3 + 8)) + 2*4^(2/3)*x^2*log(-(6*4^(1/3)*(x^4 + 1)^(1/3)*x^2 + 4^(2/3)*(2*x
^4 + x^3 + 2) + 12*(x^4 + 1)^(2/3)*x)/(2*x^4 + x^3 + 2)) - 4^(2/3)*x^2*log(-(6*4^(2/3)*(x^5 - x^4 + x)*(x^4 +
1)^(2/3) - 4^(1/3)*(4*x^8 - 14*x^7 + x^6 + 8*x^4 - 14*x^3 + 4) - 6*(4*x^6 - x^5 + 4*x^2)*(x^4 + 1)^(1/3))/(4*x
^8 + 4*x^7 + x^6 + 8*x^4 + 4*x^3 + 4)) + 36*(x^4 + 1)^(2/3))/x^2

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**4-3)*(x**4+1)**(2/3)/x**3/(2*x**4+x**3+2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (2 \, x^{4} + x^{3} + 2\right )} x^{3}} \,d x } \]

[In]

integrate((x^4-3)*(x^4+1)^(2/3)/x^3/(2*x^4+x^3+2),x, algorithm="maxima")

[Out]

integrate((x^4 + 1)^(2/3)*(x^4 - 3)/((2*x^4 + x^3 + 2)*x^3), x)

Giac [F]

\[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} + 1\right )}^{\frac {2}{3}} {\left (x^{4} - 3\right )}}{{\left (2 \, x^{4} + x^{3} + 2\right )} x^{3}} \,d x } \]

[In]

integrate((x^4-3)*(x^4+1)^(2/3)/x^3/(2*x^4+x^3+2),x, algorithm="giac")

[Out]

integrate((x^4 + 1)^(2/3)*(x^4 - 3)/((2*x^4 + x^3 + 2)*x^3), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3}}{x^3 \left (2+x^3+2 x^4\right )} \, dx=\int \frac {{\left (x^4+1\right )}^{2/3}\,\left (x^4-3\right )}{x^3\,\left (2\,x^4+x^3+2\right )} \,d x \]

[In]

int(((x^4 + 1)^(2/3)*(x^4 - 3))/(x^3*(x^3 + 2*x^4 + 2)),x)

[Out]

int(((x^4 + 1)^(2/3)*(x^4 - 3))/(x^3*(x^3 + 2*x^4 + 2)), x)