\(\int \frac {(-3+x^4) (1+x^4)^{2/3} (2+x^3+2 x^4)}{x^6 (4-x^3+4 x^4)} \, dx\) [2042]

Optimal result

Integrand size = 42, antiderivative size = 146 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\frac {3 \left (1+x^4\right )^{2/3} \left (8+15 x^3+8 x^4\right )}{80 x^5}-\frac {3 \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1+x^4}}\right )}{16 \sqrt [3]{2}}+\frac {3 \log \left (-x+2^{2/3} \sqrt [3]{1+x^4}\right )}{16 \sqrt [3]{2}}-\frac {3 \log \left (x^2+2^{2/3} x \sqrt [3]{1+x^4}+2 \sqrt [3]{2} \left (1+x^4\right )^{2/3}\right )}{32 \sqrt [3]{2}} \]

[Out]

3/80*(x^4+1)^(2/3)*(8*x^4+15*x^3+8)/x^5-3/32*3^(1/2)*arctan(3^(1/2)*x/(x+2*2^(2/3)*(x^4+1)^(1/3)))*2^(2/3)+3/3
2*ln(-x+2^(2/3)*(x^4+1)^(1/3))*2^(2/3)-3/64*ln(x^2+2^(2/3)*x*(x^4+1)^(1/3)+2*2^(1/3)*(x^4+1)^(2/3))*2^(2/3)

Rubi [F]

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

[In]

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

[Out]

(9*(1 + x^4)^(2/3))/(16*x^2) + (9*x^2)/(4*(1 - Sqrt[3] - (1 + x^4)^(1/3))) - (9*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[Ar
cSin[(1 + Sqrt[3] - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))], -7 + 4*Sqrt[3]])/(8*x^2*Sqrt[-((1 - (1
+ x^4)^(1/3))/(1 - Sqrt[3] - (1 + x^4)^(1/3))^2)]) + (3*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]])/(2*Sqrt[2]*x^2*Sqrt[-((1 - (1 + x^4)^(1/3))/(1 - Sqrt[3] - (1 +
x^4)^(1/3))^2)]) + (3*Hypergeometric2F1[-5/4, -2/3, -1/4, -x^4])/(10*x^5) - Hypergeometric2F1[-2/3, -1/4, 3/4,
-x^4]/(2*x) - (9*Defer[Int][(1 + x^4)^(2/3)/(4 - x^3 + 4*x^4), x])/8 + 6*Defer[Int][(x*(1 + x^4)^(2/3))/(4 -
x^3 + 4*x^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{2 x^6}-\frac {9 \left (1+x^4\right )^{2/3}}{8 x^3}+\frac {\left (1+x^4\right )^{2/3}}{2 x^2}+\frac {3 (-3+16 x) \left (1+x^4\right )^{2/3}}{8 \left (4-x^3+4 x^4\right )}\right ) \, dx \\ & = \frac {3}{8} \int \frac {(-3+16 x) \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+\frac {1}{2} \int \frac {\left (1+x^4\right )^{2/3}}{x^2} \, dx-\frac {9}{8} \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 {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}+\frac {3}{8} \int \left (-\frac {3 \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4}+\frac {16 x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4}\right ) \, dx-\frac {9}{16} \text {Subst}\left (\int \frac {\left (1+x^2\right )^{2/3}}{x^2} \, dx,x,x^2\right ) \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {3}{4} \text {Subst}\left (\int \frac {1}{\sqrt [3]{1+x^2}} \, dx,x,x^2\right )-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx-\frac {\left (9 \sqrt {x^4}\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x^3}} \, dx,x,\sqrt [3]{1+x^4}\right )}{8 x^2} \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {3 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+\frac {\left (9 \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 )}{8 x^2}-\frac {\left (9 \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 )}{8 x^2} \\ & = \frac {9 \left (1+x^4\right )^{2/3}}{16 x^2}+\frac {9 x^2}{4 \left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )}-\frac {9 \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 )}{8 x^2 \sqrt {-\frac {1-\sqrt [3]{1+x^4}}{\left (1-\sqrt {3}-\sqrt [3]{1+x^4}\right )^2}}}+\frac {3\ 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 )}{2 \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 \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},-\frac {2}{3},-\frac {1}{4},-x^4\right )}{10 x^5}-\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{4},\frac {3}{4},-x^4\right )}{2 x}-\frac {9}{8} \int \frac {\left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx+6 \int \frac {x \left (1+x^4\right )^{2/3}}{4-x^3+4 x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.63 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=\frac {3}{320} \left (\frac {4 \left (1+x^4\right )^{2/3} \left (8+15 x^3+8 x^4\right )}{x^5}-10\ 2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2\ 2^{2/3} \sqrt [3]{1+x^4}}\right )+10\ 2^{2/3} \log \left (-x+2^{2/3} \sqrt [3]{1+x^4}\right )-5\ 2^{2/3} \log \left (x^2+2^{2/3} x \sqrt [3]{1+x^4}+2 \sqrt [3]{2} \left (1+x^4\right )^{2/3}\right )\right ) \]

[In]

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

[Out]

(3*((4*(1 + x^4)^(2/3)*(8 + 15*x^3 + 8*x^4))/x^5 - 10*2^(2/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^(2/3)*(1 + x
^4)^(1/3))] + 10*2^(2/3)*Log[-x + 2^(2/3)*(1 + x^4)^(1/3)] - 5*2^(2/3)*Log[x^2 + 2^(2/3)*x*(1 + x^4)^(1/3) + 2
*2^(1/3)*(1 + x^4)^(2/3)]))/320

Maple [A] (verified)

Time = 104.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.86

method	result	size
pseudoelliptic	\(\frac {\frac {3 x^{5} \left (2 \arctan \left (\frac {\sqrt {3}\, \left (x +2 \,2^{\frac {2}{3}} \left (x^{4}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) \sqrt {3}+2 \ln \left (\frac {-2^{\frac {1}{3}} x +2 \left (x^{4}+1\right )^{\frac {1}{3}}}{x}\right )-\ln \left (\frac {2^{\frac {2}{3}} x^{2}+2 \,2^{\frac {1}{3}} x \left (x^{4}+1\right )^{\frac {1}{3}}+4 \left (x^{4}+1\right )^{\frac {2}{3}}}{x^{2}}\right )\right ) 2^{\frac {2}{3}}}{64}+\frac {3 \left (x^{4}+1\right )^{\frac {2}{3}} \left (8 x^{4}+15 x^{3}+8\right )}{80}}{x^{5}}\)	\(125\)
risch	\(\text {Expression too large to display}\)	\(812\)
trager	\(\text {Expression too large to display}\)	\(996\)

[In]

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

[Out]

3/320*(5*x^5*(2*arctan(1/3*3^(1/2)*(x+2*2^(2/3)*(x^4+1)^(1/3))/x)*3^(1/2)+2*ln((-2^(1/3)*x+2*(x^4+1)^(1/3))/x)
-ln((2^(2/3)*x^2+2*2^(1/3)*x*(x^4+1)^(1/3)+4*(x^4+1)^(2/3))/x^2))*2^(2/3)+4*(x^4+1)^(2/3)*(8*x^4+15*x^3+8))/x^
5

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (110) = 220\).

Time = 70.79 (sec) , antiderivative size = 410, normalized size of antiderivative = 2.81 \[ \int \frac {\left (-3+x^4\right ) \left (1+x^4\right )^{2/3} \left (2+x^3+2 x^4\right )}{x^6 \left (4-x^3+4 x^4\right )} \, dx=-\frac {10 \, \sqrt {3} 2^{\frac {2}{3}} x^{5} \arctan \left (\frac {\sqrt {3} 2^{\frac {1}{6}} {\left (2^{\frac {5}{6}} {\left (64 \, x^{12} + 240 \, x^{11} + 48 \, x^{10} - x^{9} + 192 \, x^{8} + 480 \, x^{7} + 48 \, x^{6} + 192 \, x^{4} + 240 \, x^{3} + 64\right )} + 12 \, \sqrt {2} {\left (16 \, x^{10} + 28 \, x^{9} + x^{8} + 32 \, x^{6} + 28 \, x^{5} + 16 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}} + 48 \cdot 2^{\frac {1}{6}} {\left (8 \, x^{9} + 2 \, x^{8} - x^{7} + 16 \, x^{5} + 2 \, x^{4} + 8 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}\right )}}{6 \, {\left (64 \, x^{12} - 48 \, x^{11} - 96 \, x^{10} - x^{9} + 192 \, x^{8} - 96 \, x^{7} - 96 \, x^{6} + 192 \, x^{4} - 48 \, x^{3} + 64\right )}}\right ) - 10 \cdot 2^{\frac {2}{3}} x^{5} \log \left (\frac {6 \cdot 2^{\frac {1}{3}} {\left (x^{4} + 1\right )}^{\frac {1}{3}} x^{2} + 2^{\frac {2}{3}} {\left (4 \, x^{4} - x^{3} + 4\right )} - 12 \, {\left (x^{4} + 1\right )}^{\frac {2}{3}} x}{4 \, x^{4} - x^{3} + 4}\right ) + 5 \cdot 2^{\frac {2}{3}} x^{5} \log \left (\frac {12 \cdot 2^{\frac {2}{3}} {\left (2 \, x^{5} + x^{4} + 2 \, x\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (16 \, x^{8} + 28 \, x^{7} + x^{6} + 32 \, x^{4} + 28 \, x^{3} + 16\right )} + 6 \, {\left (8 \, x^{6} + x^{5} + 8 \, x^{2}\right )} {\left (x^{4} + 1\right )}^{\frac {1}{3}}}{16 \, x^{8} - 8 \, x^{7} + x^{6} + 32 \, x^{4} - 8 \, x^{3} + 16}\right ) - 12 \, {\left (8 \, x^{4} + 15 \, x^{3} + 8\right )} {\left (x^{4} + 1\right )}^{\frac {2}{3}}}{320 \, x^{5}} \]

[In]

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

[Out]

-1/320*(10*sqrt(3)*2^(2/3)*x^5*arctan(1/6*sqrt(3)*2^(1/6)*(2^(5/6)*(64*x^12 + 240*x^11 + 48*x^10 - x^9 + 192*x
^8 + 480*x^7 + 48*x^6 + 192*x^4 + 240*x^3 + 64) + 12*sqrt(2)*(16*x^10 + 28*x^9 + x^8 + 32*x^6 + 28*x^5 + 16*x^
2)*(x^4 + 1)^(1/3) + 48*2^(1/6)*(8*x^9 + 2*x^8 - x^7 + 16*x^5 + 2*x^4 + 8*x)*(x^4 + 1)^(2/3))/(64*x^12 - 48*x^
11 - 96*x^10 - x^9 + 192*x^8 - 96*x^7 - 96*x^6 + 192*x^4 - 48*x^3 + 64)) - 10*2^(2/3)*x^5*log((6*2^(1/3)*(x^4
+ 1)^(1/3)*x^2 + 2^(2/3)*(4*x^4 - x^3 + 4) - 12*(x^4 + 1)^(2/3)*x)/(4*x^4 - x^3 + 4)) + 5*2^(2/3)*x^5*log((12*
2^(2/3)*(2*x^5 + x^4 + 2*x)*(x^4 + 1)^(2/3) + 2^(1/3)*(16*x^8 + 28*x^7 + x^6 + 32*x^4 + 28*x^3 + 16) + 6*(8*x^
6 + x^5 + 8*x^2)*(x^4 + 1)^(1/3))/(16*x^8 - 8*x^7 + x^6 + 32*x^4 - 8*x^3 + 16)) - 12*(8*x^4 + 15*x^3 + 8)*(x^4
+ 1)^(2/3))/x^5

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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