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

Optimal. Leaf size=58 \[ 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.0209206, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {50, 63, 212, 206, 203} \[ 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Int[(1 - x)^(1/4)/(1 + x),x]

[Out]

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

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 212

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] &&
!GtQ[a/b, 0]

Rule 206

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

Rule 203

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

Rubi steps

\begin{align*} \int \frac{\sqrt [4]{1-x}}{1+x} \, dx &=4 \sqrt [4]{1-x}+2 \int \frac{1}{(1-x)^{3/4} (1+x)} \, dx\\ &=4 \sqrt [4]{1-x}-8 \operatorname{Subst}\left (\int \frac{1}{2-x^4} \, dx,x,\sqrt [4]{1-x}\right )\\ &=4 \sqrt [4]{1-x}-\left (2 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}-x^2} \, dx,x,\sqrt [4]{1-x}\right )-\left (2 \sqrt{2}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{2}+x^2} \, dx,x,\sqrt [4]{1-x}\right )\\ &=4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )\\ \end{align*}

Mathematica [A]  time = 0.0158726, size = 58, normalized size = 1. \[ 4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \tan ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \tanh ^{-1}\left (\frac{\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - x)^(1/4)/(1 + x),x]

[Out]

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

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Maple [A]  time = 0.007, size = 62, normalized size = 1.1 \begin{align*} 4\,\sqrt [4]{1-x}-2\,\sqrt [4]{2}\arctan \left ( 1/2\,\sqrt [4]{1-x}{2}^{3/4} \right ) -\sqrt [4]{2}\ln \left ({ \left ( \sqrt [4]{1-x}+\sqrt [4]{2} \right ) \left ( \sqrt [4]{1-x}-\sqrt [4]{2} \right ) ^{-1}} \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.54982, size = 82, normalized size = 1.41 \begin{align*} -2 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 2^{\frac{1}{4}} \log \left (-\frac{2^{\frac{1}{4}} -{\left (-x + 1\right )}^{\frac{1}{4}}}{2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}}\right ) + 4 \,{\left (-x + 1\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.66822, size = 255, normalized size = 4.4 \begin{align*} 4 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}} \sqrt{\sqrt{2} + \sqrt{-x + 1}} - \frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}\right ) - 2^{\frac{1}{4}} \log \left (2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 2^{\frac{1}{4}} \log \left (-2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 4 \,{\left (-x + 1\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 2.20202, size = 243, normalized size = 4.19 \begin{align*} \frac{5 \sqrt [4]{-1} \sqrt [4]{x - 1} \Gamma \left (\frac{5}{4}\right )}{\Gamma \left (\frac{9}{4}\right )} + \frac{5 \sqrt [4]{-2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \left (-1\right )^{\frac{3}{4}} \sqrt [4]{2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{3 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} - \frac{5 \sqrt [4]{-2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{5 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} + \frac{5 \left (-1\right )^{\frac{3}{4}} \sqrt [4]{2} e^{- \frac{i \pi }{4}} \log{\left (- \frac{2^{\frac{3}{4}} \sqrt [4]{x - 1} e^{\frac{7 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac{5}{4}\right )}{4 \Gamma \left (\frac{9}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5*(-1)**(1/4)*(x - 1)**(1/4)*gamma(5/4)/gamma(9/4) + 5*(-2)**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*(x - 1)**(1/4)*e
xp_polar(I*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4)) - 5*(-1)**(3/4)*2**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*(x - 1)*
*(1/4)*exp_polar(3*I*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4)) - 5*(-2)**(1/4)*exp(-I*pi/4)*log(-2**(3/4)*(x - 1)
**(1/4)*exp_polar(5*I*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4)) + 5*(-1)**(3/4)*2**(1/4)*exp(-I*pi/4)*log(-2**(3/
4)*(x - 1)**(1/4)*exp_polar(7*I*pi/4)/2 + 1)*gamma(5/4)/(4*gamma(9/4))

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Giac [A]  time = 1.09497, size = 86, normalized size = 1.48 \begin{align*} -2 \cdot 2^{\frac{1}{4}} \arctan \left (\frac{1}{2} \cdot 2^{\frac{3}{4}}{\left (-x + 1\right )}^{\frac{1}{4}}\right ) - 2^{\frac{1}{4}} \log \left (2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}}\right ) + 2^{\frac{1}{4}} \log \left ({\left | -2^{\frac{1}{4}} +{\left (-x + 1\right )}^{\frac{1}{4}} \right |}\right ) + 4 \,{\left (-x + 1\right )}^{\frac{1}{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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