3.7.98 \(\int \frac {\sqrt [4]{1-x}}{1+x} \, dx\) [698]

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

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Rubi [A]
time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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 = {52, 65, 218, 212, 209} \begin {gather*} 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 {gather*}

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 52

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 65

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 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}}{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 \text {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 ) \text {Subst}\left (\int \frac {1}{\sqrt {2}-x^2} \, dx,x,\sqrt [4]{1-x}\right )-\left (2 \sqrt {2}\right ) \text {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*}

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Mathematica [A]
time = 0.06, size = 58, normalized size = 1.00 \begin {gather*} 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 {gather*}

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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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 4.03, size = 98, normalized size = 1.69 \begin {gather*} 4-2^{\frac {1}{4}} \text {Log}\left [1-\frac {2^{\frac {3}{4}} \left (-1+x\right )^{\frac {1}{4}} \text {exp\_polar}\left [\frac {5 I}{4} \text {Pi}\right ]}{2}\right ]-\left (-1+x\right )^{\frac {1}{4}}-I 2^{\frac {1}{4}} \text {Log}\left [1-\frac {2^{\frac {3}{4}} \left (-1+x\right )^{\frac {1}{4}} \text {exp\_polar}\left [\frac {3 I}{4} \text {Pi}\right ]}{2}\right ]+I 2^{\frac {1}{4}} \text {Log}\left [1-\frac {2^{\frac {3}{4}} \left (-1+x\right )^{\frac {1}{4}} \text {exp\_polar}\left [\frac {7 I}{4} \text {Pi}\right ]}{2}\right ]+2^{\frac {1}{4}} \text {Log}\left [1-\frac {2^{\frac {3}{4}} \left (-1+x\right )^{\frac {1}{4}} \text {exp\_polar}\left [\frac {I}{4} \text {Pi}\right ]}{2}\right ] \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-2 ^ (1 / 4) Log[1 - 2 ^ (3 / 4) (-1 + x) ^ (1 / 4) exp_polar[5 I / 4 Pi] / 2] - I 2 ^ (1 / 4) Log[1 - 2 ^ (3
/ 4) (-1 + x) ^ (1 / 4) exp_polar[3 I / 4 Pi] / 2] + I 2 ^ (1 / 4) Log[1 - 2 ^ (3 / 4) (-1 + x) ^ (1 / 4) exp_
polar[7 I / 4 Pi] / 2] + 4 -1 ^ (1 / 4) (-1 + x) ^ (1 / 4) + 2 ^ (1 / 4) Log[1 - 2 ^ (3 / 4) (-1 + x) ^ (1 / 4
) exp_polar[I / 4 Pi] / 2]

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Maple [A]
time = 1.21, size = 60, normalized size = 1.03

method	result	size
derivativedivides	\(4 \left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}} \left (\ln \left (\frac {\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right )\)	\(60\)
default	\(4 \left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}} \left (\ln \left (\frac {\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right )\)	\(60\)
trager	\(4 \left (1-x \right )^{\frac {1}{4}}+\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-x \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )+4 \left (1-x \right )^{\frac {3}{4}}-4 \sqrt {1-x}\, \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )-4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (1-x \right )^{\frac {1}{4}}+3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{1+x}\right )+\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {\RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x +4 \left (1-x \right )^{\frac {3}{4}}-4 \sqrt {1-x}\, \RootOf \left (\textit {\_Z}^{4}-2\right )+4 \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \left (1-x \right )^{\frac {1}{4}}-3 \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}}{1+x}\right )\)	\(204\)
risch	\(-\frac {4 \left (-1+x \right )}{\left (1-x \right )^{\frac {3}{4}}}+\frac {\left (\RootOf \left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3} x -2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{3}+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x +\RootOf \left (\textit {\_Z}^{4}-2\right ) x^{3}+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}-5 \RootOf \left (\textit {\_Z}^{4}-2\right ) x^{2}+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {3}{4}}+7 \RootOf \left (\textit {\_Z}^{4}-2\right ) x -3 \RootOf \left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )-\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )-2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2} x -\RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{3}-2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \RootOf \left (\textit {\_Z}^{4}-2\right )^{2}+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {3}{4}}+5 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}-7 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +3 \RootOf \left (\textit {\_Z}^{2}+\RootOf \left (\textit {\_Z}^{4}-2\right )^{2}\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )\right ) \left (-\left (-1+x \right )^{3}\right )^{\frac {1}{4}}}{\left (1-x \right )^{\frac {3}{4}}}\)	\(529\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.37, size = 61, normalized size = 1.05 \begin {gather*} -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 {gather*}

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 = 0.31, size = 82, normalized size = 1.41 \begin {gather*} 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 {gather*}

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] Result contains complex when optimal does not.
time = 1.28, size = 243, normalized size = 4.19 \begin {gather*} \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}} \cdot \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}} \cdot \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 {gather*}

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 = 0.01, size = 80, normalized size = 1.38 \begin {gather*} 2^{\frac {1}{4}} \ln \left |\left (-x+1\right )^{\frac {1}{4}}-2^{\frac {1}{4}}\right |-2^{\frac {1}{4}} \ln \left (\left (-x+1\right )^{\frac {1}{4}}+2^{\frac {1}{4}}\right )-2\cdot 2^{\frac {1}{4}} \arctan \left (\frac {\left (-x+1\right )^{\frac {1}{4}}}{2^{\frac {1}{4}}}\right )+4 \left (-x+1\right )^{\frac {1}{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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)

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Mupad [B]
time = 0.07, size = 46, normalized size = 0.79 \begin {gather*} 4\,{\left (1-x\right )}^{1/4}-2\,2^{1/4}\,\mathrm {atanh}\left (\frac {2^{3/4}\,{\left (1-x\right )}^{1/4}}{2}\right )-2\,2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (1-x\right )}^{1/4}}{2}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - x)^(1/4)/(x + 1),x)

[Out]

4*(1 - x)^(1/4) - 2*2^(1/4)*atanh((2^(3/4)*(1 - x)^(1/4))/2) - 2*2^(1/4)*atan((2^(3/4)*(1 - x)^(1/4))/2)

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