\(\int \frac {x^2 (-4+x^3)}{(-1+x^3)^{3/4} (1-x^3+x^4)} \, dx\) [351]

Optimal result

Integrand size = 30, antiderivative size = 29 \[ \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt [4]{-1+x^3}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(x*(1 - x^3)^(3/4)*Hypergeometric2F1[1/3, 3/4, 4/3, x^3])/(-1 + x^3)^(3/4) + (x^2*(1 - x^3)^(3/4)*Hypergeometr
ic2F1[2/3, 3/4, 5/3, x^3])/(2*(-1 + x^3)^(3/4)) - Defer[Int][1/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x] - Defer[
Int][x/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x] - 4*Defer[Int][x^2/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x] + Defe
r[Int][x^3/((-1 + x^3)^(3/4)*(1 - x^3 + x^4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {1}{\left (-1+x^3\right )^{3/4}}+\frac {x}{\left (-1+x^3\right )^{3/4}}-\frac {1+x+4 x^2-x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}\right ) \, dx \\ & = \int \frac {1}{\left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx-\int \frac {1+x+4 x^2-x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx \\ & = \frac {\left (1-x^3\right )^{3/4} \int \frac {1}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\frac {\left (1-x^3\right )^{3/4} \int \frac {x}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}-\int \left (\frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}+\frac {x}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}+\frac {4 x^2}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}-\frac {x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )}\right ) \, dx \\ & = \frac {x \left (1-x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},x^3\right )}{\left (-1+x^3\right )^{3/4}}+\frac {x^2 \left (1-x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {2}{3},\frac {3}{4},\frac {5}{3},x^3\right )}{2 \left (-1+x^3\right )^{3/4}}-4 \int \frac {x^2}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx-\int \frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx-\int \frac {x}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx+\int \frac {x^3}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 2.27 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00 \[ \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx=2 \arctan \left (\frac {x}{\sqrt [4]{-1+x^3}}\right )-2 \text {arctanh}\left (\frac {x}{\sqrt [4]{-1+x^3}}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 0.97 (sec) , antiderivative size = 153, normalized size of antiderivative = 5.28

method	result	size
trager	\(\ln \left (\frac {2 \left (x^{3}-1\right )^{\frac {3}{4}} x -2 x^{2} \sqrt {x^{3}-1}+2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}-x^{4}-x^{3}+1}{x^{4}-x^{3}+1}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{3}-2 \left (x^{3}-1\right )^{\frac {3}{4}} x +2 \left (x^{3}-1\right )^{\frac {1}{4}} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{x^{4}-x^{3}+1}\right )\)	\(153\)

[In]

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

[Out]

ln((2*(x^3-1)^(3/4)*x-2*x^2*(x^3-1)^(1/2)+2*(x^3-1)^(1/4)*x^3-x^4-x^3+1)/(x^4-x^3+1))+RootOf(_Z^2+1)*ln(-(2*Ro
otOf(_Z^2+1)*(x^3-1)^(1/2)*x^2-RootOf(_Z^2+1)*x^4-RootOf(_Z^2+1)*x^3-2*(x^3-1)^(3/4)*x+2*(x^3-1)^(1/4)*x^3+Roo
tOf(_Z^2+1))/(x^4-x^3+1))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.66 \[ \int \frac {x^2 \left (-4+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (1-x^3+x^4\right )} \, dx=-2 \, \arctan \left (\frac {{\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) - \log \left (\frac {x + {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) + \log \left (-\frac {x - {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{x}\right ) \]

[In]

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

[Out]

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

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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