\(\int \frac {(-1+x^4)^{2/3} (3+x^4) (-1-x^3+x^4)}{x^6 (-2-x^3+2 x^4)} \, dx\) [2031]

Optimal result

Integrand size = 42, antiderivative size = 145 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\frac {3 \left (-1+x^4\right )^{2/3} \left (-4-5 x^3+4 x^4\right )}{40 x^5}+\frac {\sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )}{4\ 2^{2/3}}-\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^4}\right )}{4\ 2^{2/3}}+\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^4}+2^{2/3} \left (-1+x^4\right )^{2/3}\right )}{8\ 2^{2/3}} \]

[Out]

3/40*(x^4-1)^(2/3)*(4*x^4-5*x^3-4)/x^5+1/8*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(1/3)*(x^4-1)^(1/3)))*2^(1/3)-1/8*l
n(-x+2^(1/3)*(x^4-1)^(1/3))*2^(1/3)+1/16*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]

\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx \]

[In]

Int[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-2 - x^3 + 2*x^4)),x]

[Out]

(-3*(-1 + x^4)^(2/3))/(8*x^2) + (3*x^2)/(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]*Ellipt
icE[ArcSin[(1 - Sqrt[3] + (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))], -7 - 4*Sqrt[3]])/(4*x^2*Sqrt[(1
+ (-1 + x^4)^(1/3))/(1 + Sqrt[3] + (-1 + x^4)^(1/3))^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]])/(Sqrt[2]*x^2*Sqrt[(1 + (-1 + x^4)^(1/3))/(1 + Sqrt[3]
+ (-1 + x^4)^(1/3))^2]) - (3*(-1 + x^4)^(2/3)*Hypergeometric2F1[-5/4, -2/3, -1/4, x^4])/(10*x^5*(1 - x^4)^(2/
3)) - ((-1 + x^4)^(2/3)*Hypergeometric2F1[-2/3, -1/4, 3/4, x^4])/(2*x*(1 - x^4)^(2/3)) + (3*Defer[Int][(-1 + x
^4)^(2/3)/(-2 - x^3 + 2*x^4), x])/4 - 2*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^6}+\frac {3 \left (-1+x^4\right )^{2/3}}{4 x^3}+\frac {\left (-1+x^4\right )^{2/3}}{2 x^2}+\frac {(3-8 x) \left (-1+x^4\right )^{2/3}}{4 \left (-2-x^3+2 x^4\right )}\right ) \, dx \\ & = \frac {1}{4} \int \frac {(3-8 x) \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx+\frac {1}{2} \int \frac {\left (-1+x^4\right )^{2/3}}{x^2} \, dx+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{x^3} \, dx+\frac {3}{2} \int \frac {\left (-1+x^4\right )^{2/3}}{x^6} \, dx \\ & = \frac {1}{4} \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}{8} \text {Subst}\left (\int \frac {\left (-1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right )+\frac {\left (-1+x^4\right )^{2/3} \int \frac {\left (1-x^4\right )^{2/3}}{x^2} \, dx}{2 \left (1-x^4\right )^{2/3}}+\frac {\left (3 \left (-1+x^4\right )^{2/3}\right ) \int \frac {\left (1-x^4\right )^{2/3}}{x^6} \, dx}{2 \left (1-x^4\right )^{2/3}} \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {1}{2} \text {Subst}\left (\int \frac {1}{\sqrt [3]{-1+x^2}} \, dx,x,x^2\right )+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \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 )}{4 x^2} \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \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 )}{4 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 )}{4 x^2} \\ & = -\frac {3 \left (-1+x^4\right )^{2/3}}{8 x^2}+\frac {3 x^2}{2 \left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )}-\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 )}{4 x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}+\frac {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 )}{\sqrt {2} x^2 \sqrt {\frac {1+\sqrt [3]{-1+x^4}}{\left (1+\sqrt {3}+\sqrt [3]{-1+x^4}\right )^2}}}-\frac {3 \left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},x^4\right )}{10 x^5 \left (1-x^4\right )^{2/3}}-\frac {\left (-1+x^4\right )^{2/3} \operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},x^4\right )}{2 x \left (1-x^4\right )^{2/3}}+\frac {3}{4} \int \frac {\left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx-2 \int \frac {x \left (-1+x^4\right )^{2/3}}{-2-x^3+2 x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.57 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\frac {1}{80} \left (\frac {6 \left (-1+x^4\right )^{2/3} \left (-4-5 x^3+4 x^4\right )}{x^5}+10 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{-1+x^4}}\right )-10 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{-1+x^4}\right )+5 \sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{-1+x^4}+2^{2/3} \left (-1+x^4\right )^{2/3}\right )\right ) \]

[In]

Integrate[((-1 + x^4)^(2/3)*(3 + x^4)*(-1 - x^3 + x^4))/(x^6*(-2 - x^3 + 2*x^4)),x]

[Out]

((6*(-1 + x^4)^(2/3)*(-4 - 5*x^3 + 4*x^4))/x^5 + 10*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(1/3)*(-1 + x^
4)^(1/3))] - 10*2^(1/3)*Log[-x + 2^(1/3)*(-1 + x^4)^(1/3)] + 5*2^(1/3)*Log[x^2 + 2^(1/3)*x*(-1 + x^4)^(1/3) +
2^(2/3)*(-1 + x^4)^(2/3)])/80

Maple [A] (verified)

Time = 78.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {-5 \left (\arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {1}{3}} \left (x^{4}-1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}-\frac {\ln \left (\frac {2^{\frac {2}{3}} \left (x^{4}-1\right )^{\frac {1}{3}} x +2^{\frac {1}{3}} x^{2}+2 \left (x^{4}-1\right )^{\frac {2}{3}}}{x^{2}}\right )}{2}+\ln \left (\frac {-2^{\frac {2}{3}} x +2 \left (x^{4}-1\right )^{\frac {1}{3}}}{x}\right )-\frac {\ln \left (2\right )}{2}\right ) x^{5} 2^{\frac {1}{3}}+3 \left (x^{4}-1\right )^{\frac {2}{3}} \left (4 x^{4}-5 x^{3}-4\right )}{40 x^{5}}\)	\(125\)
risch	\(\text {Expression too large to display}\)	\(809\)
trager	\(\text {Expression too large to display}\)	\(1488\)

[In]

int((x^4-1)^(2/3)*(x^4+3)*(x^4-x^3-1)/x^6/(2*x^4-x^3-2),x,method=_RETURNVERBOSE)

[Out]

1/40*(-5*(arctan(1/3*3^(1/2)*(x+2*2^(1/3)*(x^4-1)^(1/3))/x)*3^(1/2)-1/2*ln((2^(2/3)*(x^4-1)^(1/3)*x+2^(1/3)*x^
2+2*(x^4-1)^(2/3))/x^2)+ln((-2^(2/3)*x+2*(x^4-1)^(1/3))/x)-1/2*ln(2))*x^5*2^(1/3)+3*(x^4-1)^(2/3)*(4*x^4-5*x^3
-4))/x^5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 433 vs. \(2 (109) = 218\).

Time = 84.11 (sec) , antiderivative size = 433, normalized size of antiderivative = 2.99 \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=-\frac {20 \cdot 4^{\frac {1}{6}} \sqrt {3} \left (-1\right )^{\frac {1}{3}} x^{5} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {3} {\left (12 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {2}{3}} {\left (2 \, x^{9} + x^{8} - x^{7} - 4 \, x^{5} - x^{4} + 2 \, x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} - 12 \, \left (-1\right )^{\frac {1}{3}} {\left (4 \, x^{10} + 14 \, x^{9} + x^{8} - 8 \, x^{6} - 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{12} + 60 \, x^{11} + 24 \, x^{10} - x^{9} - 24 \, x^{8} - 120 \, x^{7} - 24 \, x^{6} + 24 \, x^{4} + 60 \, x^{3} - 8\right )}\right )}}{6 \, {\left (8 \, x^{12} - 12 \, x^{11} - 48 \, x^{10} - x^{9} - 24 \, x^{8} + 24 \, x^{7} + 48 \, x^{6} + 24 \, x^{4} - 12 \, x^{3} - 8\right )}}\right ) - 10 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (\frac {6 \cdot 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (x^{4} - 1\right )}^{\frac {1}{3}} x^{2} - 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (2 \, x^{4} - x^{3} - 2\right )} - 12 \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} x}{2 \, x^{4} - x^{3} - 2}\right ) + 5 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} x^{5} \log \left (-\frac {6 \cdot 4^{\frac {2}{3}} \left (-1\right )^{\frac {1}{3}} {\left (x^{5} + x^{4} - x\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}} - 4^{\frac {1}{3}} \left (-1\right )^{\frac {2}{3}} {\left (4 \, x^{8} + 14 \, x^{7} + x^{6} - 8 \, x^{4} - 14 \, x^{3} + 4\right )} - 6 \, {\left (4 \, x^{6} + x^{5} - 4 \, x^{2}\right )} {\left (x^{4} - 1\right )}^{\frac {1}{3}}}{4 \, x^{8} - 4 \, x^{7} + x^{6} - 8 \, x^{4} + 4 \, x^{3} + 4}\right ) - 36 \, {\left (4 \, x^{4} - 5 \, x^{3} - 4\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{480 \, x^{5}} \]

[In]

integrate((x^4-1)^(2/3)*(x^4+3)*(x^4-x^3-1)/x^6/(2*x^4-x^3-2),x, algorithm="fricas")

[Out]

-1/480*(20*4^(1/6)*sqrt(3)*(-1)^(1/3)*x^5*arctan(1/6*4^(1/6)*sqrt(3)*(12*4^(2/3)*(-1)^(2/3)*(2*x^9 + x^8 - x^7
- 4*x^5 - x^4 + 2*x)*(x^4 - 1)^(2/3) - 12*(-1)^(1/3)*(4*x^10 + 14*x^9 + x^8 - 8*x^6 - 14*x^5 + 4*x^2)*(x^4 -
1)^(1/3) + 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))/(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)) - 10*4^(2/3)*(-1)^(1/3)*x^5*lo
g((6*4^(1/3)*(-1)^(2/3)*(x^4 - 1)^(1/3)*x^2 - 4^(2/3)*(-1)^(1/3)*(2*x^4 - x^3 - 2) - 12*(x^4 - 1)^(2/3)*x)/(2*
x^4 - x^3 - 2)) + 5*4^(2/3)*(-1)^(1/3)*x^5*log(-(6*4^(2/3)*(-1)^(1/3)*(x^5 + x^4 - x)*(x^4 - 1)^(2/3) - 4^(1/3
)*(-1)^(2/3)*(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*(4*x^4 - 5*x^3 - 4)*(x^4 - 1)^(2/3))/x^5

Sympy [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\text {Timed out} \]

[In]

integrate((x**4-1)**(2/3)*(x**4+3)*(x**4-x**3-1)/x**6/(2*x**4-x**3-2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{4} - x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-1)^(2/3)*(x^4+3)*(x^4-x^3-1)/x^6/(2*x^4-x^3-2),x, algorithm="maxima")

[Out]

integrate((x^4 - x^3 - 1)*(x^4 + 3)*(x^4 - 1)^(2/3)/((2*x^4 - x^3 - 2)*x^6), x)

Giac [F]

\[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\int { \frac {{\left (x^{4} - x^{3} - 1\right )} {\left (x^{4} + 3\right )} {\left (x^{4} - 1\right )}^{\frac {2}{3}}}{{\left (2 \, x^{4} - x^{3} - 2\right )} x^{6}} \,d x } \]

[In]

integrate((x^4-1)^(2/3)*(x^4+3)*(x^4-x^3-1)/x^6/(2*x^4-x^3-2),x, algorithm="giac")

[Out]

integrate((x^4 - x^3 - 1)*(x^4 + 3)*(x^4 - 1)^(2/3)/((2*x^4 - x^3 - 2)*x^6), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (-1+x^4\right )^{2/3} \left (3+x^4\right ) \left (-1-x^3+x^4\right )}{x^6 \left (-2-x^3+2 x^4\right )} \, dx=\int \frac {{\left (x^4-1\right )}^{2/3}\,\left (x^4+3\right )\,\left (-x^4+x^3+1\right )}{x^6\,\left (-2\,x^4+x^3+2\right )} \,d x \]

[In]

int(((x^4 - 1)^(2/3)*(x^4 + 3)*(x^3 - x^4 + 1))/(x^6*(x^3 - 2*x^4 + 2)),x)

[Out]

int(((x^4 - 1)^(2/3)*(x^4 + 3)*(x^3 - x^4 + 1))/(x^6*(x^3 - 2*x^4 + 2)), x)