Integrand size = 15, antiderivative size = 58 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \] Output:
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))
Time = 0.06 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=4 \sqrt [4]{1-x}-2 \sqrt [4]{2} \arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )-2 \sqrt [4]{2} \text {arctanh}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right ) \] Input:
Integrate[(1 - x)^(1/4)/(1 + x),x]
Output:
4*(1 - x)^(1/4) - 2*2^(1/4)*ArcTan[(1 - x)^(1/4)/2^(1/4)] - 2*2^(1/4)*ArcT anh[(1 - x)^(1/4)/2^(1/4)]
Time = 0.17 (sec) , antiderivative size = 65, normalized size of antiderivative = 1.12, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {60, 73, 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.
| \(\displaystyle \int \frac {\sqrt [4]{1-x}}{x+1} \, dx\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle 2 \int \frac {1}{(1-x)^{3/4} (x+1)}dx+4 \sqrt [4]{1-x}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle 4 \sqrt [4]{1-x}-8 \int \frac {1}{x+1}d\sqrt [4]{1-x}\) |
| \(\Big \downarrow \) 756 |
| \(\displaystyle 4 \sqrt [4]{1-x}-8 \left (\frac {\int \frac {1}{\sqrt {2}-\sqrt {1-x}}d\sqrt [4]{1-x}}{2 \sqrt {2}}+\frac {\int \frac {1}{\sqrt {1-x}+\sqrt {2}}d\sqrt [4]{1-x}}{2 \sqrt {2}}\right )\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle 4 \sqrt [4]{1-x}-8 \left (\frac {\int \frac {1}{\sqrt {2}-\sqrt {1-x}}d\sqrt [4]{1-x}}{2 \sqrt {2}}+\frac {\arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle 4 \sqrt [4]{1-x}-8 \left (\frac {\arctan \left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{1-x}}{\sqrt [4]{2}}\right )}{2\ 2^{3/4}}\right )\) |
Input:
Int[(1 - x)^(1/4)/(1 + x),x]
Output:
4*(1 - x)^(1/4) - 8*(ArcTan[(1 - x)^(1/4)/2^(1/4)]/(2*2^(3/4)) + ArcTanh[( 1 - x)^(1/4)/2^(1/4)]/(2*2^(3/4)))
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] + Simp[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] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[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] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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])
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])
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]
Time = 1.33 (sec) , antiderivative size = 60, normalized size of antiderivative = 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 | \(4 \left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}} \left (\ln \left (\frac {\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}{\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\right )\) | \(60\) |
| pseudoelliptic | \(4 \left (1-x \right )^{\frac {1}{4}}-\ln \left (\frac {-\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}}{-\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}}\right ) 2^{\frac {1}{4}}-2 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )\) | \(66\) |
| trager | \(4 \left (1-x \right )^{\frac {1}{4}}-\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +4 \left (1-x \right )^{\frac {3}{4}}+4 \sqrt {1-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )-4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (1-x \right )^{\frac {1}{4}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )}{1+x}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x +4 \left (1-x \right )^{\frac {3}{4}}-4 \sqrt {1-x}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )+4 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \left (1-x \right )^{\frac {1}{4}}-3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}}{1+x}\right )\) | \(204\) |
| risch | \(-\frac {4 \left (-1+x \right )}{\left (1-x \right )^{\frac {3}{4}}}+\frac {\left (-\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) \ln \left (\frac {-2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3} x +2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{3}+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}-4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x -\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{3}+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {3}{4}}+2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}+5 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x^{2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right ) x +3 \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )}{\left (-1+x \right )^{2} \left (1+x \right )}\right )+\operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) \ln \left (\frac {-2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x +2 \sqrt {-x^{3}+3 x^{2}-3 x +1}\, \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )-2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x^{2}+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2} x +x^{3} \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right )+4 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {3}{4}}-2 \left (-x^{3}+3 x^{2}-3 x +1\right )^{\frac {1}{4}} \operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x^{2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {RootOf}\left (\textit {\_Z}^{4}-2\right )^{2}\right ) x -3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+\operatorname {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}}}\) | \(529\) |
Input:
int((1-x)^(1/4)/(1+x),x,method=_RETURNVERBOSE)
Output:
4*(1-x)^(1/4)-2^(1/4)*(ln(((1-x)^(1/4)+2^(1/4))/((1-x)^(1/4)-2^(1/4)))+2*a rctan(1/2*(1-x)^(1/4)*2^(3/4)))
Time = 0.09 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left (-2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \] Input:
integrate((1-x)^(1/4)/(1+x),x, algorithm="fricas")
Output:
-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(-2^(1/4) + (-x + 1)^(1/4)) + 4*(-x + 1)^(1/4)
Result contains complex when optimal does not.
Time = 1.23 (sec) , antiderivative size = 243, normalized size of antiderivative = 4.19 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=\frac {5 \sqrt [4]{-1} \sqrt [4]{x - 1} \Gamma \left (\frac {5}{4}\right )}{\Gamma \left (\frac {9}{4}\right )} + \frac {5 \sqrt [4]{-2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {5 \left (-1\right )^{\frac {3}{4}} \cdot \sqrt [4]{2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {3 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} - \frac {5 \sqrt [4]{-2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {5 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} + \frac {5 \left (-1\right )^{\frac {3}{4}} \cdot \sqrt [4]{2} e^{- \frac {i \pi }{4}} \log {\left (- \frac {2^{\frac {3}{4}} \sqrt [4]{x - 1} e^{\frac {7 i \pi }{4}}}{2} + 1 \right )} \Gamma \left (\frac {5}{4}\right )}{4 \Gamma \left (\frac {9}{4}\right )} \] Input:
integrate((1-x)**(1/4)/(1+x),x)
Output:
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)*exp_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))
Time = 0.12 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.05 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} - {\left (-x + 1\right )}^{\frac {1}{4}}}{2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \] Input:
integrate((1-x)^(1/4)/(1+x),x, algorithm="maxima")
Output:
-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)
Time = 0.16 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \cdot 2^{\frac {1}{4}} \arctan \left (\frac {1}{2} \cdot 2^{\frac {3}{4}} {\left (-x + 1\right )}^{\frac {1}{4}}\right ) - 2^{\frac {1}{4}} \log \left (2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}}\right ) + 2^{\frac {1}{4}} \log \left ({\left | -2^{\frac {1}{4}} + {\left (-x + 1\right )}^{\frac {1}{4}} \right |}\right ) + 4 \, {\left (-x + 1\right )}^{\frac {1}{4}} \] Input:
integrate((1-x)^(1/4)/(1+x),x, algorithm="giac")
Output:
-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)
Time = 0.04 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.79 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=4\,{\left (1-x\right )}^{1/4}-2\,2^{1/4}\,\mathrm {atanh}\left (\frac {2^{3/4}\,{\left (1-x\right )}^{1/4}}{2}\right )-2\,2^{1/4}\,\mathrm {atan}\left (\frac {2^{3/4}\,{\left (1-x\right )}^{1/4}}{2}\right ) \] Input:
int((1 - x)^(1/4)/(x + 1),x)
Output:
4*(1 - x)^(1/4) - 2*2^(1/4)*atanh((2^(3/4)*(1 - x)^(1/4))/2) - 2*2^(1/4)*a tan((2^(3/4)*(1 - x)^(1/4))/2)
Time = 0.17 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.09 \[ \int \frac {\sqrt [4]{1-x}}{1+x} \, dx=-2 \,2^{\frac {1}{4}} \mathit {atan} \left (\frac {\left (1-x \right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2}\right )+4 \left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}} \mathrm {log}\left (\left (1-x \right )^{\frac {1}{4}}+2^{\frac {1}{4}}\right )+2^{\frac {1}{4}} \mathrm {log}\left (\left (1-x \right )^{\frac {1}{4}}-2^{\frac {1}{4}}\right ) \] Input:
int((1-x)^(1/4)/(1+x),x)
Output:
- 2*2**(1/4)*atan(( - x + 1)**(1/4)/2**(1/4)) + 4*( - x + 1)**(1/4) - 2** (1/4)*log(( - x + 1)**(1/4) + 2**(1/4)) + 2**(1/4)*log(( - x + 1)**(1/4) - 2**(1/4))