\(\int \frac {(-4+x^3) (1-x^3+x^4)}{x^2 (-1+x^3)^{3/4} (-1+x^3+x^4)} \, dx\) [1318]

Optimal result

Integrand size = 38, antiderivative size = 95 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \]

[Out]

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

Rubi [F]

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

[In]

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

[Out]

(-4*(1 - x^3)^(3/4)*Hypergeometric2F1[-1/3, 3/4, 2/3, x^3])/(x*(-1 + x^3)^(3/4)) - (2*x*(1 - x^3)^(3/4)*Hyperg
eometric2F1[1/3, 3/4, 4/3, x^3])/(-1 + x^3)^(3/4) + (x^2*(1 - x^3)^(3/4)*Hypergeometric2F1[2/3, 3/4, 5/3, x^3]
)/(2*(-1 + x^3)^(3/4)) + 2*Defer[Int][1/((-1 + x^3)^(3/4)*(1 - x^3 - x^4)), x] + 2*Defer[Int][x/((-1 + x^3)^(3
/4)*(-1 + x^3 + x^4)), x] - 8*Defer[Int][x^2/((-1 + x^3)^(3/4)*(-1 + x^3 + x^4)), x] + 2*Defer[Int][x^3/((-1 +
x^3)^(3/4)*(-1 + x^3 + x^4)), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {2}{\left (-1+x^3\right )^{3/4}}+\frac {4}{x^2 \left (-1+x^3\right )^{3/4}}+\frac {x}{\left (-1+x^3\right )^{3/4}}+\frac {2 \left (-1+x-4 x^2+x^3\right )}{\left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )}\right ) \, dx \\ & = -\left (2 \int \frac {1}{\left (-1+x^3\right )^{3/4}} \, dx\right )+2 \int \frac {-1+x-4 x^2+x^3}{\left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx+4 \int \frac {1}{x^2 \left (-1+x^3\right )^{3/4}} \, dx+\int \frac {x}{\left (-1+x^3\right )^{3/4}} \, dx \\ & = 2 \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 {\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}}-\frac {\left (2 \left (1-x^3\right )^{3/4}\right ) \int \frac {1}{\left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}}+\frac {\left (4 \left (1-x^3\right )^{3/4}\right ) \int \frac {1}{x^2 \left (1-x^3\right )^{3/4}} \, dx}{\left (-1+x^3\right )^{3/4}} \\ & = -\frac {4 \left (1-x^3\right )^{3/4} \operatorname {Hypergeometric2F1}\left (-\frac {1}{3},\frac {3}{4},\frac {2}{3},x^3\right )}{x \left (-1+x^3\right )^{3/4}}-\frac {2 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}}+2 \int \frac {1}{\left (-1+x^3\right )^{3/4} \left (1-x^3-x^4\right )} \, dx+2 \int \frac {x}{\left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx+2 \int \frac {x^3}{\left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx-8 \int \frac {x^2}{\left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 3.40 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {4 \sqrt [4]{-1+x^3}}{x}+2 \sqrt {2} \arctan \left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{-x^2+\sqrt {-1+x^3}}\right )-2 \sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} x \sqrt [4]{-1+x^3}}{x^2+\sqrt {-1+x^3}}\right ) \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 7.16 (sec) , antiderivative size = 219, normalized size of antiderivative = 2.31

method	result	size
trager	\(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (-\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}+2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{3}-2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{2}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3}}{x^{4}+x^{3}-1}\right )-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {2 \sqrt {x^{3}-1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{2}-2 \left (x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{4}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}+2 \left (x^{3}-1\right )^{\frac {3}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )}{x^{4}+x^{3}-1}\right )\)	\(219\)
risch	\(\frac {4 \left (x^{3}-1\right )^{\frac {1}{4}}}{x}+\frac {\left (-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}+2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{7}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-4 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) x^{3}-2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{5}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+2 \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{2}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x -1\right )^{2} \left (x^{2}+x +1\right )^{2} \left (x^{4}+x^{3}-1\right )}\right )+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{10}-\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{9}+2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{3} \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {3}{4}} x^{3}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{7}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{7}+3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{6}-2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{5}+\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x^{4}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2} x^{3}+2 \sqrt {x^{9}-3 x^{6}+3 x^{3}-1}\, x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right ) \left (x^{9}-3 x^{6}+3 x^{3}-1\right )^{\frac {1}{4}} x +\operatorname {RootOf}\left (\textit {\_Z}^{4}+1\right )^{2}}{\left (x -1\right )^{2} \left (x^{2}+x +1\right )^{2} \left (x^{4}+x^{3}-1\right )}\right )\right ) {\left (\left (x^{3}-1\right )^{3}\right )}^{\frac {1}{4}}}{\left (x^{3}-1\right )^{\frac {3}{4}}}\)	\(589\)

[In]

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

[Out]

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

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 15.43 (sec) , antiderivative size = 295, normalized size of antiderivative = 3.11 \[ \int \frac {\left (-4+x^3\right ) \left (1-x^3+x^4\right )}{x^2 \left (-1+x^3\right )^{3/4} \left (-1+x^3+x^4\right )} \, dx=\frac {-\left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {i \, x^{4} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + i}{x^{4} + x^{3} - 1}\right ) + \left (i + 1\right ) \, \sqrt {2} x \log \left (\frac {i \, x^{4} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} - i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x + i}{x^{4} + x^{3} - 1}\right ) + \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-i \, x^{4} - \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} + \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - i}{x^{4} + x^{3} - 1}\right ) - \left (i - 1\right ) \, \sqrt {2} x \log \left (\frac {-i \, x^{4} + \left (i - 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {1}{4}} x^{3} + i \, x^{3} + 2 \, \sqrt {x^{3} - 1} x^{2} - \left (i + 1\right ) \, \sqrt {2} {\left (x^{3} - 1\right )}^{\frac {3}{4}} x - i}{x^{4} + x^{3} - 1}\right ) + 8 \, {\left (x^{3} - 1\right )}^{\frac {1}{4}}}{2 \, x} \]

[In]

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

[Out]

1/2*(-(I + 1)*sqrt(2)*x*log((I*x^4 + (I + 1)*sqrt(2)*(x^3 - 1)^(1/4)*x^3 - I*x^3 + 2*sqrt(x^3 - 1)*x^2 - (I -
1)*sqrt(2)*(x^3 - 1)^(3/4)*x + I)/(x^4 + x^3 - 1)) + (I + 1)*sqrt(2)*x*log((I*x^4 - (I + 1)*sqrt(2)*(x^3 - 1)^
(1/4)*x^3 - I*x^3 + 2*sqrt(x^3 - 1)*x^2 + (I - 1)*sqrt(2)*(x^3 - 1)^(3/4)*x + I)/(x^4 + x^3 - 1)) + (I - 1)*sq
rt(2)*x*log((-I*x^4 - (I - 1)*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + I*x^3 + 2*sqrt(x^3 - 1)*x^2 + (I + 1)*sqrt(2)*(x^3
- 1)^(3/4)*x - I)/(x^4 + x^3 - 1)) - (I - 1)*sqrt(2)*x*log((-I*x^4 + (I - 1)*sqrt(2)*(x^3 - 1)^(1/4)*x^3 + I*
x^3 + 2*sqrt(x^3 - 1)*x^2 - (I + 1)*sqrt(2)*(x^3 - 1)^(3/4)*x - I)/(x^4 + x^3 - 1)) + 8*(x^3 - 1)^(1/4))/x

Sympy [F(-1)]

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

[In]

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

[Out]

Timed out

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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