\(\int \frac {-3+x^4}{(1+x^4) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx\) [1254]

Optimal result

Integrand size = 31, antiderivative size = 91 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\sqrt {2} \arctan \left (\frac {\sqrt {2} \left (-3 x+4 x^4-3 x^5\right )^{3/4}}{3-4 x^3+3 x^4}\right )+\sqrt {2} \text {arctanh}\left (\frac {\sqrt {2} \left (-3 x+4 x^4-3 x^5\right )^{3/4}}{3-4 x^3+3 x^4}\right ) \]

[Out]

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

Rubi [F]

\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx \]

[In]

Int[(-3 + x^4)/((1 + x^4)*(-3*x + 4*x^4 - 3*x^5)^(1/4)),x]

[Out]

-(((-1)^(3/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/(((-1)^(1/16) - x)*(-3 + 4*x^12 -
3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4)) - ((-1)^(9/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/
4)*Defer[Subst][Defer[Int][1/(((-1)^(3/16) - x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4
- 3*x^5)^(1/4) - ((-1)^(15/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/(((-1)^(5/16) - x)
*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^(5/16)*x^(1/4)*(-3 + 4*x
^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/(((-1)^(7/16) - x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])
/(-3*x + 4*x^4 - 3*x^5)^(1/4) - ((-1)^(11/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/((-
(-1)^(9/16) - x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^(1/16)*x
^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/((-(-1)^(11/16) - x)*(-3 + 4*x^12 - 3*x^16)^(1/4))
, x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^(7/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst]
[Defer[Int][1/((-(-1)^(13/16) - x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4
) + ((-1)^(13/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/((-(-1)^(15/16) - x)*(-3 + 4*x^
12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + (4*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Def
er[Subst][Defer[Int][x^2/(-3 + 4*x^12 - 3*x^16)^(1/4), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) - ((-1)^(
3/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/(((-1)^(1/16) + x)*(-3 + 4*x^12 - 3*x^16)^(
1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) - ((-1)^(9/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[S
ubst][Defer[Int][1/(((-1)^(3/16) + x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(
1/4) - ((-1)^(15/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/(((-1)^(5/16) + x)*(-3 + 4*x
^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^(5/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4
)^(1/4)*Defer[Subst][Defer[Int][1/(((-1)^(7/16) + x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4
*x^4 - 3*x^5)^(1/4) - ((-1)^(11/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/((-(-1)^(9/16
) + x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^(1/16)*x^(1/4)*(-3
+ 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/((-(-1)^(11/16) + x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x
^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^(7/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int
][1/((-(-1)^(13/16) + x)*(-3 + 4*x^12 - 3*x^16)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4) + ((-1)^
(13/16)*x^(1/4)*(-3 + 4*x^3 - 3*x^4)^(1/4)*Defer[Subst][Defer[Int][1/((-(-1)^(15/16) + x)*(-3 + 4*x^12 - 3*x^1
6)^(1/4)), x], x, x^(1/4)])/(-3*x + 4*x^4 - 3*x^5)^(1/4)

Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {-3+x^4}{\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4} \left (1+x^4\right )} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \left (\frac {1}{\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}}-\frac {4}{\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4} \left (1+x^4\right )}\right ) \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4} \left (1+x^4\right )} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = -\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \left (\frac {i}{2 \sqrt [4]{x} \left (i-x^2\right ) \sqrt [4]{-3+4 x^3-3 x^4}}+\frac {i}{2 \sqrt [4]{x} \left (i+x^2\right ) \sqrt [4]{-3+4 x^3-3 x^4}}\right ) \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = -\frac {\left (2 i \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (i-x^2\right ) \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 i \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (i+x^2\right ) \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = -\frac {\left (2 i \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \left (-\frac {(-1)^{3/4}}{2 \left (\sqrt [4]{-1}-x\right ) \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}}-\frac {(-1)^{3/4}}{2 \sqrt [4]{x} \left (\sqrt [4]{-1}+x\right ) \sqrt [4]{-3+4 x^3-3 x^4}}\right ) \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 i \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \left (-\frac {\sqrt [4]{-1}}{2 \left (-(-1)^{3/4}-x\right ) \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}}-\frac {\sqrt [4]{-1}}{2 \sqrt [4]{x} \left (-(-1)^{3/4}+x\right ) \sqrt [4]{-3+4 x^3-3 x^4}}\right ) \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (\sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\left (\sqrt [4]{-1}-x\right ) \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (\sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (\sqrt [4]{-1}+x\right ) \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left ((-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\left (-(-1)^{3/4}-x\right ) \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left ((-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \int \frac {1}{\sqrt [4]{x} \left (-(-1)^{3/4}+x\right ) \sqrt [4]{-3+4 x^3-3 x^4}} \, dx}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (4 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (\sqrt [4]{-1}-x^4\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (4 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (\sqrt [4]{-1}+x^4\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-(-1)^{3/4}-x^4\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (-(-1)^{3/4}+x^4\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (4 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \left (\sqrt [8]{-1}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}-\frac {1}{2 \left (\sqrt [8]{-1}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (4 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 \left (-(-1)^{5/8}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}+\frac {1}{2 \left (-(-1)^{5/8}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (-\frac {1}{2 \left ((-1)^{3/8}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}+\frac {1}{2 \left ((-1)^{3/8}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (4 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (\frac {1}{2 \left (-(-1)^{7/8}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}-\frac {1}{2 \left (-(-1)^{7/8}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [8]{-1}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{5/8}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt [8]{-1}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{5/8}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{3/8}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{7/8}-x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left ((-1)^{3/8}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {1}{\left (-(-1)^{7/8}+x^2\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \frac {\left (4 \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt [4]{-3+4 x^{12}-3 x^{16}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{15/16}}{2 \left (\sqrt [16]{-1}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}-\frac {(-1)^{15/16}}{2 \left (\sqrt [16]{-1}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{11/16}}{2 \left ((-1)^{5/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}+\frac {(-1)^{11/16}}{2 \left ((-1)^{5/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{7/16}}{2 \left (-(-1)^{9/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}-\frac {(-1)^{7/16}}{2 \left (-(-1)^{9/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 \sqrt [4]{-1} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{3/16}}{2 \left (-(-1)^{13/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}+\frac {(-1)^{3/16}}{2 \left (-(-1)^{13/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{13/16}}{2 \left ((-1)^{3/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}-\frac {(-1)^{13/16}}{2 \left ((-1)^{3/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}-\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (\frac {(-1)^{9/16}}{2 \left ((-1)^{7/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}+\frac {(-1)^{9/16}}{2 \left ((-1)^{7/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (-\frac {(-1)^{5/16}}{2 \left (-(-1)^{11/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}-\frac {(-1)^{5/16}}{2 \left (-(-1)^{11/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}}+\frac {\left (2 (-1)^{3/4} \sqrt [4]{x} \sqrt [4]{-3+4 x^3-3 x^4}\right ) \text {Subst}\left (\int \left (\frac {\sqrt [16]{-1}}{2 \left (-(-1)^{15/16}-x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}+\frac {\sqrt [16]{-1}}{2 \left (-(-1)^{15/16}+x\right ) \sqrt [4]{-3+4 x^{12}-3 x^{16}}}\right ) \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 4.62 (sec) , antiderivative size = 142, normalized size of antiderivative = 1.56 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=-\frac {\sqrt [4]{x} \sqrt [4]{3-4 x^3+3 x^4} \left (\arctan \left (\frac {2 x^{3/4} \sqrt [4]{3-4 x^3+3 x^4}}{-2 x^{3/2}+\sqrt {3-4 x^3+3 x^4}}\right )+\text {arctanh}\left (\frac {2 x^{3/4} \sqrt [4]{3-4 x^3+3 x^4}}{2 x^{3/2}+\sqrt {3-4 x^3+3 x^4}}\right )\right )}{\sqrt [4]{-3 x+4 x^4-3 x^5}} \]

[In]

Integrate[(-3 + x^4)/((1 + x^4)*(-3*x + 4*x^4 - 3*x^5)^(1/4)),x]

[Out]

-((x^(1/4)*(3 - 4*x^3 + 3*x^4)^(1/4)*(ArcTan[(2*x^(3/4)*(3 - 4*x^3 + 3*x^4)^(1/4))/(-2*x^(3/2) + Sqrt[3 - 4*x^
3 + 3*x^4])] + ArcTanh[(2*x^(3/4)*(3 - 4*x^3 + 3*x^4)^(1/4))/(2*x^(3/2) + Sqrt[3 - 4*x^3 + 3*x^4])]))/(-3*x +
4*x^4 - 3*x^5)^(1/4))

Maple [A] (verified)

Time = 6.18 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.64

method	result	size
pseudoelliptic	\(\left (-\operatorname {arctanh}\left (\frac {\left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right )+\arctan \left (\frac {\left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {1}{4}} \sqrt {2}}{2 x}\right )\right ) \sqrt {2}\)	\(58\)
trager	\(-\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {3}{4}}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2\right ) \left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {1}{4}} x^{2}-3 x^{4}+4 \sqrt {-3 x^{5}+4 x^{4}-3 x}\, x +8 x^{3}-3}{x^{4}+1}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{4}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {-3 x^{5}+4 x^{4}-3 x}\, x -8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) x^{3}+4 \left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {3}{4}}-8 \left (-3 x^{5}+4 x^{4}-3 x \right )^{\frac {1}{4}} x^{2}+3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right )}{x^{4}+1}\right )}{2}\)	\(214\)

[In]

int((x^4-3)/(x^4+1)/(-3*x^5+4*x^4-3*x)^(1/4),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (79) = 158\).

Time = 50.24 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.51 \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\frac {1}{2} \, \sqrt {2} \arctan \left (\frac {2 \, \sqrt {2} {\left (-3 \, x^{5} + 4 \, x^{4} - 3 \, x\right )}^{\frac {3}{4}} x + \sqrt {2} {\left (-3 \, x^{5} + 4 \, x^{4} - 3 \, x\right )}^{\frac {1}{4}} {\left (3 \, x^{4} - 4 \, x^{3} + 3\right )}}{4 \, {\left (3 \, x^{5} - 4 \, x^{4} + 3 \, x\right )}}\right ) + \frac {1}{4} \, \sqrt {2} \log \left (\frac {9 \, x^{8} - 192 \, x^{7} + 256 \, x^{6} + 18 \, x^{4} - 192 \, x^{3} + 4 \, \sqrt {2} {\left (-3 \, x^{5} + 4 \, x^{4} - 3 \, x\right )}^{\frac {3}{4}} {\left (3 \, x^{4} - 16 \, x^{3} + 3\right )} + 8 \, \sqrt {2} {\left (9 \, x^{6} - 16 \, x^{5} + 9 \, x^{2}\right )} {\left (-3 \, x^{5} + 4 \, x^{4} - 3 \, x\right )}^{\frac {1}{4}} - 16 \, {\left (3 \, x^{5} - 8 \, x^{4} + 3 \, x\right )} \sqrt {-3 \, x^{5} + 4 \, x^{4} - 3 \, x} + 9}{x^{8} + 2 \, x^{4} + 1}\right ) \]

[In]

integrate((x^4-3)/(x^4+1)/(-3*x^5+4*x^4-3*x)^(1/4),x, algorithm="fricas")

[Out]

1/2*sqrt(2)*arctan(1/4*(2*sqrt(2)*(-3*x^5 + 4*x^4 - 3*x)^(3/4)*x + sqrt(2)*(-3*x^5 + 4*x^4 - 3*x)^(1/4)*(3*x^4
- 4*x^3 + 3))/(3*x^5 - 4*x^4 + 3*x)) + 1/4*sqrt(2)*log((9*x^8 - 192*x^7 + 256*x^6 + 18*x^4 - 192*x^3 + 4*sqrt
(2)*(-3*x^5 + 4*x^4 - 3*x)^(3/4)*(3*x^4 - 16*x^3 + 3) + 8*sqrt(2)*(9*x^6 - 16*x^5 + 9*x^2)*(-3*x^5 + 4*x^4 - 3
*x)^(1/4) - 16*(3*x^5 - 8*x^4 + 3*x)*sqrt(-3*x^5 + 4*x^4 - 3*x) + 9)/(x^8 + 2*x^4 + 1))

Sympy [F]

\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\int \frac {x^{4} - 3}{\sqrt [4]{- x \left (3 x^{4} - 4 x^{3} + 3\right )} \left (x^{4} + 1\right )}\, dx \]

[In]

integrate((x**4-3)/(x**4+1)/(-3*x**5+4*x**4-3*x)**(1/4),x)

[Out]

Integral((x**4 - 3)/((-x*(3*x**4 - 4*x**3 + 3))**(1/4)*(x**4 + 1)), x)

Maxima [F]

\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\int { \frac {x^{4} - 3}{{\left (-3 \, x^{5} + 4 \, x^{4} - 3 \, x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^4-3)/(x^4+1)/(-3*x^5+4*x^4-3*x)^(1/4),x, algorithm="maxima")

[Out]

integrate((x^4 - 3)/((-3*x^5 + 4*x^4 - 3*x)^(1/4)*(x^4 + 1)), x)

Giac [F]

\[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\int { \frac {x^{4} - 3}{{\left (-3 \, x^{5} + 4 \, x^{4} - 3 \, x\right )}^{\frac {1}{4}} {\left (x^{4} + 1\right )}} \,d x } \]

[In]

integrate((x^4-3)/(x^4+1)/(-3*x^5+4*x^4-3*x)^(1/4),x, algorithm="giac")

[Out]

integrate((x^4 - 3)/((-3*x^5 + 4*x^4 - 3*x)^(1/4)*(x^4 + 1)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {-3+x^4}{\left (1+x^4\right ) \sqrt [4]{-3 x+4 x^4-3 x^5}} \, dx=\int \frac {x^4-3}{\left (x^4+1\right )\,{\left (-3\,x^5+4\,x^4-3\,x\right )}^{1/4}} \,d x \]

[In]

int((x^4 - 3)/((x^4 + 1)*(4*x^4 - 3*x - 3*x^5)^(1/4)),x)

[Out]

int((x^4 - 3)/((x^4 + 1)*(4*x^4 - 3*x - 3*x^5)^(1/4)), x)