\(\int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^4} \, dx\) [508]

Optimal result

Integrand size = 20, antiderivative size = 114 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^4} \, dx=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{3 x^3}-\frac {5 (1-x)^{3/4} \sqrt [4]{1+x}}{12 x^2}-\frac {11 (1-x)^{3/4} \sqrt [4]{1+x}}{24 x}-\frac {3}{8} \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {3}{8} \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right ) \] Output:

-1/3*(1-x)^(3/4)*(1+x)^(1/4)/x^3-5/12*(1-x)^(3/4)*(1+x)^(1/4)/x^2-11/24*(1 
-x)^(3/4)*(1+x)^(1/4)/x-3/8*arctan((1+x)^(1/4)/(1-x)^(1/4))-3/8*arctanh((1 
+x)^(1/4)/(1-x)^(1/4))
                                                                                    
                                                                                    
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.67 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^4} \, dx=\frac {1}{24} \left (-\frac {(1-x)^{3/4} \sqrt [4]{1+x} \left (8+10 x+11 x^2\right )}{x^3}-9 \arctan \left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-9 \text {arctanh}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\right ) \] Input:

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

Output:

(-(((1 - x)^(3/4)*(1 + x)^(1/4)*(8 + 10*x + 11*x^2))/x^3) - 9*ArcTan[(1 + 
x)^(1/4)/(1 - x)^(1/4)] - 9*ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)])/24
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 125, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {110, 27, 168, 27, 168, 27, 104, 756, 216, 219}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

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

\(\Big \downarrow \) 110

\(\displaystyle \frac {1}{3} \int \frac {4 x+5}{2 \sqrt [4]{1-x} x^3 (x+1)^{3/4}}dx-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \int \frac {4 x+5}{\sqrt [4]{1-x} x^3 (x+1)^{3/4}}dx-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{6} \left (-\frac {1}{2} \int -\frac {10 x+11}{2 \sqrt [4]{1-x} x^2 (x+1)^{3/4}}dx-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \int \frac {10 x+11}{\sqrt [4]{1-x} x^2 (x+1)^{3/4}}dx-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 168

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (-\int -\frac {9}{2 \sqrt [4]{1-x} x (x+1)^{3/4}}dx-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (\frac {9}{2} \int \frac {1}{\sqrt [4]{1-x} x (x+1)^{3/4}}dx-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 104

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (18 \int \frac {1}{\frac {x+1}{1-x}-1}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 756

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (18 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {x+1}}{\sqrt {1-x}}}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {1}{2} \int \frac {1}{\frac {\sqrt {x+1}}{\sqrt {1-x}}+1}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 216

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (18 \left (-\frac {1}{2} \int \frac {1}{1-\frac {\sqrt {x+1}}{\sqrt {1-x}}}d\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {1}{6} \left (\frac {1}{4} \left (18 \left (-\frac {1}{2} \arctan \left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {1}{2} \text {arctanh}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )\right )-\frac {11 (1-x)^{3/4} \sqrt [4]{x+1}}{x}\right )-\frac {5 (1-x)^{3/4} \sqrt [4]{x+1}}{2 x^2}\right )-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{3 x^3}\)

Input:

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

Output:

-1/3*((1 - x)^(3/4)*(1 + x)^(1/4))/x^3 + ((-5*(1 - x)^(3/4)*(1 + x)^(1/4)) 
/(2*x^2) + ((-11*(1 - x)^(3/4)*(1 + x)^(1/4))/x + 18*(-1/2*ArcTan[(1 + x)^ 
(1/4)/(1 - x)^(1/4)] - ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)]/2))/4)/6
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[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] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 110

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

rule 168

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + 
 d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && ILtQ[m, -1]
 

rule 216

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

rule 219

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] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 756

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

Maple [C] (verified)

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

Time = 0.68 (sec) , antiderivative size = 393, normalized size of antiderivative = 3.45

method	result	size
risch	\(\frac {\left (1+x \right )^{\frac {1}{4}} \left (-1+x \right ) \left (11 x^{2}+10 x +8\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{24 x^{3} \left (-\left (-1+x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}+\frac {\left (\frac {3 \ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -x^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}-2 x -1}{\left (1+x \right )^{2} x}\right )}{16}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x -\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}+\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )}{\left (1+x \right )^{2} x}\right )}{16}\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1+x \right )^{\frac {3}{4}} \left (1-x \right )^{\frac {1}{4}}}\)	\(393\)

Input:

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

Output:

1/24*(1+x)^(1/4)*(-1+x)*(11*x^2+10*x+8)/x^3/(-(-1+x)*(1+x)^3)^(1/4)*((1-x) 
*(1+x)^3)^(1/4)/(1-x)^(1/4)+(3/16*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)/(1+x)^2/x)-3/ 
16*RootOf(_Z^2+1)*ln((-RootOf(_Z^2+1)*(-x^4-2*x^3+2*x+1)^(1/2)*x+(-x^4-2*x 
^3+2*x+1)^(3/4)-RootOf(_Z^2+1)*(-x^4-2*x^3+2*x+1)^(1/2)-(-x^4-2*x^3+2*x+1) 
^(1/4)*x^2+RootOf(_Z^2+1)*x^2-2*(-x^4-2*x^3+2*x+1)^(1/4)*x+2*RootOf(_Z^2+1 
)*x-(-x^4-2*x^3+2*x+1)^(1/4)+RootOf(_Z^2+1))/(1+x)^2/x))/(1+x)^(3/4)*((1-x 
)*(1+x)^3)^(1/4)/(1-x)^(1/4)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.94 \[ \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^4} \, dx=\frac {18 \, x^{3} \arctan \left (\frac {{\left (-x + 1\right )}^{\frac {1}{4}}}{{\left (x + 1\right )}^{\frac {1}{4}}}\right ) + 9 \, x^{3} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 9 \, x^{3} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (11 \, x^{2} + 10 \, x + 8\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{48 \, x^{3}} \] Input:

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

Output:

1/48*(18*x^3*arctan((-x + 1)^(1/4)/(x + 1)^(1/4)) + 9*x^3*log((x + (x + 1) 
^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - 9*x^3*log(-(x - (x + 1)^(1/4)*(-x + 
1)^(3/4) - 1)/(x - 1)) - 2*(11*x^2 + 10*x + 8)*(x + 1)^(1/4)*(-x + 1)^(3/4 
))/x^3
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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