3.9.97 \(\int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^3} \, dx\) [897]

Optimal. Leaf size=91 \[ -\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}-\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {98, 96, 95, 218, 212, 209} \begin {gather*} -\frac {1}{4} \text {ArcTan}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right )-\frac {(1-x)^{3/4} (x+1)^{5/4}}{2 x^2}-\frac {(1-x)^{3/4} \sqrt [4]{x+1}}{4 x}-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt [4]{x+1}}{\sqrt [4]{1-x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((1 - x)^(3/4)*(1 + x)^(1/4))/x - ((1 - x)^(3/4)*(1 + x)^(5/4))/(2*x^2) - ArcTan[(1 + x)^(1/4)/(1 - x)^(1
/4)]/4 - ArcTanh[(1 + x)^(1/4)/(1 - x)^(1/4)]/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 98

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(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] + Dist[(a*d*f*(m + 1)
+ b*c*f*(n + 1) + b*d*e*(p + 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, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || Sum
SimplerQ[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*} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^3} \, dx &=-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}+\frac {1}{4} \int \frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x} x^2} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}+\frac {1}{8} \int \frac {1}{\sqrt [4]{1-x} x (1+x)^{3/4}} \, dx\\ &=-\frac {(1-x)^{3/4} \sqrt [4]{1+x}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}+\frac {1}{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}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}-\frac {1}{4} \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \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}}{4 x}-\frac {(1-x)^{3/4} (1+x)^{5/4}}{2 x^2}-\frac {1}{4} \tan ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )-\frac {1}{4} \tanh ^{-1}\left (\frac {\sqrt [4]{1+x}}{\sqrt [4]{1-x}}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.27, size = 71, normalized size = 0.78 \begin {gather*} \frac {1}{4} \left (-\frac {(1-x)^{3/4} \sqrt [4]{1+x} (2+3 x)}{x^2}-\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 )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C] Result contains higher order function than in optimal. Order 9 vs. order 3.
time = 0.49, size = 388, normalized size = 4.26

method	result	size
risch	\(\frac {\left (1+x \right )^{\frac {1}{4}} \left (-1+x \right ) \left (2+3 x \right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{4 x^{2} \left (-\left (-1+x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}} \left (1-x \right )^{\frac {1}{4}}}+\frac {\left (\frac {\ln \left (\frac {\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}-\sqrt {-x^{4}-2 x^{3}+2 x +1}+2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x -x^{2}+\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}-2 x -1}{x \left (1+x \right )^{2}}\right )}{8}-\frac {\RootOf \left (\textit {\_Z}^{2}+1\right ) \ln \left (\frac {-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}\, x +\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {3}{4}}-\RootOf \left (\textit {\_Z}^{2}+1\right ) \sqrt {-x^{4}-2 x^{3}+2 x +1}-\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x^{2}+\RootOf \left (\textit {\_Z}^{2}+1\right ) x^{2}-2 \left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}} x +2 \RootOf \left (\textit {\_Z}^{2}+1\right ) x -\left (-x^{4}-2 x^{3}+2 x +1\right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+1\right )}{x \left (1+x \right )^{2}}\right )}{8}\right ) \left (\left (1-x \right ) \left (1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1+x \right )^{\frac {3}{4}} \left (1-x \right )^{\frac {1}{4}}}\)	\(388\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(1+x)^(1/4)*(-1+x)*(2+3*x)/x^2/(-(-1+x)*(1+x)^3)^(1/4)*((1-x)*(1+x)^3)^(1/4)/(1-x)^(1/4)+(1/8*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)-1/8*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)
)/x/(1+x)^2))/(1+x)^(3/4)*((1-x)*(1+x)^3)^(1/4)/(1-x)^(1/4)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.55, size = 106, normalized size = 1.16 \begin {gather*} \frac {2 \, x^{2} \arctan \left (\frac {{\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{x - 1}\right ) + x^{2} \log \left (\frac {x + {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - x^{2} \log \left (-\frac {x - {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}} - 1}{x - 1}\right ) - 2 \, {\left (3 \, x + 2\right )} {\left (x + 1\right )}^{\frac {1}{4}} {\left (-x + 1\right )}^{\frac {3}{4}}}{8 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(2*x^2*arctan((x + 1)^(1/4)*(-x + 1)^(3/4)/(x - 1)) + x^2*log((x + (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x -
1)) - x^2*log(-(x - (x + 1)^(1/4)*(-x + 1)^(3/4) - 1)/(x - 1)) - 2*(3*x + 2)*(x + 1)^(1/4)*(-x + 1)^(3/4))/x^2

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt [4]{x + 1}}{x^{3} \sqrt [4]{1 - x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (x+1\right )}^{1/4}}{x^3\,{\left (1-x\right )}^{1/4}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________