\(\int \frac {(-3+x^4) (1-x^3+x^4)^{2/3}}{x^3 (1+x^3+x^4)} \, dx\) [2122]

Optimal result

Integrand size = 33, antiderivative size = 154 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\frac {3 \left (1-x^3+x^4\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3+x^4}}\right )+2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3+x^4}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3+x^4}-\sqrt [3]{2} \left (1-x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{2}} \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {3 \left (1-x^3+x^4\right )^{2/3}}{x^3}+\frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \frac {(3+4 x) \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx \\ & = -\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+\int \left (\frac {3 \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}+\frac {4 x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4}\right ) \, dx \\ & = -\left (3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{x^3} \, dx\right )+3 \int \frac {\left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx+4 \int \frac {x \left (1-x^3+x^4\right )^{2/3}}{1+x^3+x^4} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.60 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\frac {3 \left (1-x^3+x^4\right )^{2/3}}{2 x^2}+2^{2/3} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{-x+2^{2/3} \sqrt [3]{1-x^3+x^4}}\right )+2^{2/3} \log \left (2 x+2^{2/3} \sqrt [3]{1-x^3+x^4}\right )-\frac {\log \left (-2 x^2+2^{2/3} x \sqrt [3]{1-x^3+x^4}-\sqrt [3]{2} \left (1-x^3+x^4\right )^{2/3}\right )}{\sqrt [3]{2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 60.28 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.92

method	result	size
pseudoelliptic	\(\frac {2 \sqrt {3}\, 2^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3}\, \left (x -2^{\frac {2}{3}} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}}\right )}{3 x}\right ) x^{2}+2 \,2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {1}{3}} x +\left (x^{4}-x^{3}+1\right )^{\frac {1}{3}}}{x}\right ) x^{2}-2^{\frac {2}{3}} \ln \left (\frac {2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} \left (x^{4}-x^{3}+1\right )^{\frac {1}{3}} x +\left (x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{x^{2}}\right ) x^{2}+3 \left (x^{4}-x^{3}+1\right )^{\frac {2}{3}}}{2 x^{2}}\)	\(142\)
risch	\(\text {Expression too large to display}\)	\(1061\)
trager	\(\text {Expression too large to display}\)	\(1511\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (124) = 248\).

Time = 48.97 (sec) , antiderivative size = 413, normalized size of antiderivative = 2.68 \[ \int \frac {\left (-3+x^4\right ) \left (1-x^3+x^4\right )^{2/3}}{x^3 \left (1+x^3+x^4\right )} \, dx=\frac {2 \cdot 4^{\frac {1}{3}} \sqrt {3} x^{2} \arctan \left (\frac {3 \cdot 4^{\frac {2}{3}} \sqrt {3} {\left (x^{9} - 4 \, x^{8} - 5 \, x^{7} + 2 \, x^{5} - 4 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 6 \cdot 4^{\frac {1}{3}} \sqrt {3} {\left (x^{10} - 16 \, x^{9} + 19 \, x^{8} + 2 \, x^{6} - 16 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} - \sqrt {3} {\left (x^{12} - 33 \, x^{11} + 111 \, x^{10} - 71 \, x^{9} + 3 \, x^{8} - 66 \, x^{7} + 111 \, x^{6} + 3 \, x^{4} - 33 \, x^{3} + 1\right )}}{3 \, {\left (x^{12} + 3 \, x^{11} - 105 \, x^{10} + 109 \, x^{9} + 3 \, x^{8} + 6 \, x^{7} - 105 \, x^{6} + 3 \, x^{4} + 3 \, x^{3} + 1\right )}}\right ) + 2 \cdot 4^{\frac {1}{3}} x^{2} \log \left (-\frac {3 \cdot 4^{\frac {2}{3}} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}} x^{2} + 6 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} x + 4^{\frac {1}{3}} {\left (x^{4} + x^{3} + 1\right )}}{x^{4} + x^{3} + 1}\right ) - 4^{\frac {1}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{5} - 5 \, x^{4} + x\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}} - 4^{\frac {2}{3}} {\left (x^{8} - 16 \, x^{7} + 19 \, x^{6} + 2 \, x^{4} - 16 \, x^{3} + 1\right )} - 24 \, {\left (x^{6} - 2 \, x^{5} + x^{2}\right )} {\left (x^{4} - x^{3} + 1\right )}^{\frac {1}{3}}}{x^{8} + 2 \, x^{7} + x^{6} + 2 \, x^{4} + 2 \, x^{3} + 1}\right ) + 9 \, {\left (x^{4} - x^{3} + 1\right )}^{\frac {2}{3}}}{6 \, x^{2}} \]

[In]

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

[Out]

1/6*(2*4^(1/3)*sqrt(3)*x^2*arctan(1/3*(3*4^(2/3)*sqrt(3)*(x^9 - 4*x^8 - 5*x^7 + 2*x^5 - 4*x^4 + x)*(x^4 - x^3
+ 1)^(2/3) - 6*4^(1/3)*sqrt(3)*(x^10 - 16*x^9 + 19*x^8 + 2*x^6 - 16*x^5 + x^2)*(x^4 - x^3 + 1)^(1/3) - sqrt(3)
*(x^12 - 33*x^11 + 111*x^10 - 71*x^9 + 3*x^8 - 66*x^7 + 111*x^6 + 3*x^4 - 33*x^3 + 1))/(x^12 + 3*x^11 - 105*x^
10 + 109*x^9 + 3*x^8 + 6*x^7 - 105*x^6 + 3*x^4 + 3*x^3 + 1)) + 2*4^(1/3)*x^2*log(-(3*4^(2/3)*(x^4 - x^3 + 1)^(
1/3)*x^2 + 6*(x^4 - x^3 + 1)^(2/3)*x + 4^(1/3)*(x^4 + x^3 + 1))/(x^4 + x^3 + 1)) - 4^(1/3)*x^2*log(-(6*4^(1/3)
*(x^5 - 5*x^4 + x)*(x^4 - x^3 + 1)^(2/3) - 4^(2/3)*(x^8 - 16*x^7 + 19*x^6 + 2*x^4 - 16*x^3 + 1) - 24*(x^6 - 2*
x^5 + x^2)*(x^4 - x^3 + 1)^(1/3))/(x^8 + 2*x^7 + x^6 + 2*x^4 + 2*x^3 + 1)) + 9*(x^4 - x^3 + 1)^(2/3))/x^2

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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