\(\int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx\) [896]

Optimal result

Integrand size = 20, antiderivative size = 62 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{x}-\arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right ) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {96, 95, 218, 212, 209} \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx=-\arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{x} \]

[In]

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

[Out]

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

Rule 95

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 96

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(a + b*
x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Dist[n*((d*e - c*f)/((m + 1)*(b*e - a*f
))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[
m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] ||  !SumSimplerQ[p, 1]) && NeQ[m, -1]

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 218

Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]},
Dist[r/(2*a), Int[1/(r - s*x^2), x], x] + Dist[r/(2*a), Int[1/(r + s*x^2), x], x]] /; FreeQ[{a, b}, x] &&  !Gt
Q[a/b, 0]

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

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}+x \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )+x \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )}{x} \]

[In]

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

[Out]

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

Maple [C] (verified)

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

Time = 1.14 (sec) , antiderivative size = 383, normalized size of antiderivative = 6.18

method	result	size
risch	\(\frac {\left (-1+x \right ) \left (1+x \right )^{\frac {1}{4}} \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{x \left (-\left (-1+x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}+\frac {\left (\frac {\ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -x^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}-2 x -1}{x \left (1+x \right )^{2}}\right )}{2}+\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}}{x \left (1+x \right )^{2}}\right )}{2}\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1+x \right )^{\frac {3}{4}} \left (1-x \right )^{\frac {1}{4}}}\)	\(383\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.53 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx=\frac {2 \, x \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + x \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - x \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{2 \, x} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx=\int \frac {\sqrt [4]{x + 1}}{x^{2} \sqrt [4]{1 - x}}\, dx \]

[In]

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

[Out]

Integral((x + 1)**(1/4)/(x**2*(1 - x)**(1/4)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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