Integrand size = 15, antiderivative size = 84 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=3 \sqrt [3]{1-x}-\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x}}{\sqrt {3}}\right )+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2^{2/3}}-\frac {\log (1+x)}{2^{2/3}} \] Output:
3*(1-x)^(1/3)-2^(1/3)*3^(1/2)*arctan(1/3*(1+2^(2/3)*(1-x)^(1/3))*3^(1/2))+ 3/2*ln(2^(1/3)-(1-x)^(1/3))*2^(1/3)-1/2*ln(1+x)*2^(1/3)
Time = 0.10 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.33 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=3 \sqrt [3]{1-x}-\sqrt [3]{2} \sqrt {3} \arctan \left (\frac {1+2^{2/3} \sqrt [3]{1-x}}{\sqrt {3}}\right )+\sqrt [3]{2} \log \left (-2+2^{2/3} \sqrt [3]{1-x}\right )-\frac {\log \left (2+2^{2/3} \sqrt [3]{1-x}+\sqrt [3]{2} (1-x)^{2/3}\right )}{2^{2/3}} \] Input:
Integrate[(1 - x)^(1/3)/(1 + x),x]
Output:
3*(1 - x)^(1/3) - 2^(1/3)*Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x)^(1/3))/Sqrt[ 3]] + 2^(1/3)*Log[-2 + 2^(2/3)*(1 - x)^(1/3)] - Log[2 + 2^(2/3)*(1 - x)^(1 /3) + 2^(1/3)*(1 - x)^(2/3)]/2^(2/3)
Time = 0.20 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {60, 69, 16, 1082, 217}
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.
\(\displaystyle \int \frac {\sqrt [3]{1-x}}{x+1} \, dx\) |
\(\Big \downarrow \) 60 |
\(\displaystyle 2 \int \frac {1}{(1-x)^{2/3} (x+1)}dx+3 \sqrt [3]{1-x}\) |
\(\Big \downarrow \) 69 |
\(\displaystyle 2 \left (-\frac {3 \int \frac {1}{\sqrt [3]{2}-\sqrt [3]{1-x}}d\sqrt [3]{1-x}}{2\ 2^{2/3}}-\frac {3 \int \frac {1}{(1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}+2^{2/3}}d\sqrt [3]{1-x}}{2 \sqrt [3]{2}}-\frac {\log (x+1)}{2\ 2^{2/3}}\right )+3 \sqrt [3]{1-x}\) |
\(\Big \downarrow \) 16 |
\(\displaystyle 2 \left (-\frac {3 \int \frac {1}{(1-x)^{2/3}+\sqrt [3]{2} \sqrt [3]{1-x}+2^{2/3}}d\sqrt [3]{1-x}}{2 \sqrt [3]{2}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2\ 2^{2/3}}-\frac {\log (x+1)}{2\ 2^{2/3}}\right )+3 \sqrt [3]{1-x}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle 2 \left (\frac {3 \int \frac {1}{-(1-x)^{2/3}-3}d\left (2^{2/3} \sqrt [3]{1-x}+1\right )}{2^{2/3}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2\ 2^{2/3}}-\frac {\log (x+1)}{2\ 2^{2/3}}\right )+3 \sqrt [3]{1-x}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle 2 \left (-\frac {\sqrt {3} \arctan \left (\frac {2^{2/3} \sqrt [3]{1-x}+1}{\sqrt {3}}\right )}{2^{2/3}}+\frac {3 \log \left (\sqrt [3]{2}-\sqrt [3]{1-x}\right )}{2\ 2^{2/3}}-\frac {\log (x+1)}{2\ 2^{2/3}}\right )+3 \sqrt [3]{1-x}\) |
Input:
Int[(1 - x)^(1/3)/(1 + x),x]
Output:
3*(1 - x)^(1/3) + 2*(-((Sqrt[3]*ArcTan[(1 + 2^(2/3)*(1 - x)^(1/3))/Sqrt[3] ])/2^(2/3)) + (3*Log[2^(1/3) - (1 - x)^(1/3)])/(2*2^(2/3)) - Log[1 + x]/(2 *2^(2/3)))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
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[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(2/3)), x_Symbol] :> With[ {q = Rt[(b*c - a*d)/b, 3]}, Simp[-Log[RemoveContent[a + b*x, x]]/(2*b*q^2), x] + (-Simp[3/(2*b*q) Subst[Int[1/(q^2 + q*x + x^2), x], x, (c + d*x)^(1 /3)], x] - Simp[3/(2*b*q^2) Subst[Int[1/(q - x), x], x, (c + d*x)^(1/3)], x])] /; FreeQ[{a, b, c, d}, x] && PosQ[(b*c - a*d)/b]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Time = 2.20 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.00
method | result | size |
derivativedivides | \(3 \left (1-x \right )^{\frac {1}{3}}+2^{\frac {1}{3}} \ln \left (\left (1-x \right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\frac {2^{\frac {1}{3}} \ln \left (\left (1-x \right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{2}-2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )\) | \(84\) |
default | \(3 \left (1-x \right )^{\frac {1}{3}}+2^{\frac {1}{3}} \ln \left (\left (1-x \right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\frac {2^{\frac {1}{3}} \ln \left (\left (1-x \right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{2}-2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )\) | \(84\) |
pseudoelliptic | \(3 \left (1-x \right )^{\frac {1}{3}}+2^{\frac {1}{3}} \ln \left (\left (1-x \right )^{\frac {1}{3}}-2^{\frac {1}{3}}\right )-\frac {2^{\frac {1}{3}} \ln \left (\left (1-x \right )^{\frac {2}{3}}+2^{\frac {1}{3}} \left (1-x \right )^{\frac {1}{3}}+2^{\frac {2}{3}}\right )}{2}-2^{\frac {1}{3}} \sqrt {3}\, \arctan \left (\frac {\left (1+2^{\frac {2}{3}} \left (1-x \right )^{\frac {1}{3}}\right ) \sqrt {3}}{3}\right )\) | \(84\) |
trager | \(\text {Expression too large to display}\) | \(866\) |
risch | \(\text {Expression too large to display}\) | \(970\) |
Input:
int((1-x)^(1/3)/(1+x),x,method=_RETURNVERBOSE)
Output:
3*(1-x)^(1/3)+2^(1/3)*ln((1-x)^(1/3)-2^(1/3))-1/2*2^(1/3)*ln((1-x)^(2/3)+2 ^(1/3)*(1-x)^(1/3)+2^(2/3))-2^(1/3)*3^(1/2)*arctan(1/3*(1+2^(2/3)*(1-x)^(1 /3))*3^(1/2))
Time = 0.10 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=-\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{3} \, \sqrt {3} 2^{\frac {2}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-x + 1\right )}^{\frac {1}{3}} \] Input:
integrate((1-x)^(1/3)/(1+x),x, algorithm="fricas")
Output:
-sqrt(3)*2^(1/3)*arctan(1/3*sqrt(3)*2^(2/3)*(-x + 1)^(1/3) + 1/3*sqrt(3)) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x + 1)^(1/3) + (-x + 1)^(2/3)) + 2^( 1/3)*log(-2^(1/3) + (-x + 1)^(1/3)) + 3*(-x + 1)^(1/3)
Result contains complex when optimal does not.
Time = 1.11 (sec) , antiderivative size = 170, normalized size of antiderivative = 2.02 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=\frac {4 \sqrt [3]{-1} \sqrt [3]{x - 1} \Gamma \left (\frac {4}{3}\right )}{\Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{-2} e^{- \frac {i \pi }{3}} \log {\left (- \frac {2^{\frac {2}{3}} \sqrt [3]{x - 1} e^{\frac {i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} - \frac {4 \sqrt [3]{-2} \log {\left (- \frac {2^{\frac {2}{3}} \sqrt [3]{x - 1} e^{i \pi }}{2} + 1 \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} + \frac {4 \sqrt [3]{-2} e^{\frac {i \pi }{3}} \log {\left (- \frac {2^{\frac {2}{3}} \sqrt [3]{x - 1} e^{\frac {5 i \pi }{3}}}{2} + 1 \right )} \Gamma \left (\frac {4}{3}\right )}{3 \Gamma \left (\frac {7}{3}\right )} \] Input:
integrate((1-x)**(1/3)/(1+x),x)
Output:
4*(-1)**(1/3)*(x - 1)**(1/3)*gamma(4/3)/gamma(7/3) + 4*(-2)**(1/3)*exp(-I* pi/3)*log(-2**(2/3)*(x - 1)**(1/3)*exp_polar(I*pi/3)/2 + 1)*gamma(4/3)/(3* gamma(7/3)) - 4*(-2)**(1/3)*log(-2**(2/3)*(x - 1)**(1/3)*exp_polar(I*pi)/2 + 1)*gamma(4/3)/(3*gamma(7/3)) + 4*(-2)**(1/3)*exp(I*pi/3)*log(-2**(2/3)* (x - 1)**(1/3)*exp_polar(5*I*pi/3)/2 + 1)*gamma(4/3)/(3*gamma(7/3))
Time = 0.14 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.02 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=-\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left (-2^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {1}{3}}\right ) + 3 \, {\left (-x + 1\right )}^{\frac {1}{3}} \] Input:
integrate((1-x)^(1/3)/(1+x),x, algorithm="maxima")
Output:
-sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x + 1)^(1/3))) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x + 1)^(1/3) + (-x + 1)^(2/3)) + 2^( 1/3)*log(-2^(1/3) + (-x + 1)^(1/3)) + 3*(-x + 1)^(1/3)
Time = 0.14 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=-\sqrt {3} 2^{\frac {1}{3}} \arctan \left (\frac {1}{6} \, \sqrt {3} 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} + 2 \, {\left (-x + 1\right )}^{\frac {1}{3}}\right )}\right ) - \frac {1}{2} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} + 2^{\frac {1}{3}} {\left (-x + 1\right )}^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {2}{3}}\right ) + 2^{\frac {1}{3}} \log \left ({\left | -2^{\frac {1}{3}} + {\left (-x + 1\right )}^{\frac {1}{3}} \right |}\right ) + 3 \, {\left (-x + 1\right )}^{\frac {1}{3}} \] Input:
integrate((1-x)^(1/3)/(1+x),x, algorithm="giac")
Output:
-sqrt(3)*2^(1/3)*arctan(1/6*sqrt(3)*2^(2/3)*(2^(1/3) + 2*(-x + 1)^(1/3))) - 1/2*2^(1/3)*log(2^(2/3) + 2^(1/3)*(-x + 1)^(1/3) + (-x + 1)^(2/3)) + 2^( 1/3)*log(abs(-2^(1/3) + (-x + 1)^(1/3))) + 3*(-x + 1)^(1/3)
Time = 0.04 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.24 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=2^{1/3}\,\ln \left (18\,{\left (1-x\right )}^{1/3}-18\,2^{1/3}\right )+3\,{\left (1-x\right )}^{1/3}+\frac {2^{1/3}\,\ln \left (18\,{\left (1-x\right )}^{1/3}-9\,2^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {2^{1/3}\,\ln \left (18\,{\left (1-x\right )}^{1/3}+9\,2^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )\right )\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \] Input:
int((1 - x)^(1/3)/(x + 1),x)
Output:
2^(1/3)*log(18*(1 - x)^(1/3) - 18*2^(1/3)) + 3*(1 - x)^(1/3) + (2^(1/3)*lo g(18*(1 - x)^(1/3) - 9*2^(1/3)*(3^(1/2)*1i - 1))*(3^(1/2)*1i - 1))/2 - (2^ (1/3)*log(18*(1 - x)^(1/3) + 9*2^(1/3)*(3^(1/2)*1i + 1))*(3^(1/2)*1i + 1)) /2
Time = 0.17 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt [3]{1-x}}{1+x} \, dx=\frac {\left (2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2 \left (1-x \right )^{\frac {1}{6}}+2^{\frac {1}{6}}\right ) 2^{\frac {5}{6}}}{2 \sqrt {3}}\right )-2 \sqrt {3}\, \mathit {atan} \left (\frac {\left (2 \left (1-x \right )^{\frac {1}{6}}-2^{\frac {1}{6}}\right ) 2^{\frac {5}{6}}}{2 \sqrt {3}}\right )+3 \left (1-x \right )^{\frac {1}{3}} 2^{\frac {2}{3}}+2 \,\mathrm {log}\left (\left (1-x \right )^{\frac {1}{6}}+2^{\frac {1}{6}}\right )+2 \,\mathrm {log}\left (\left (1-x \right )^{\frac {1}{6}}-2^{\frac {1}{6}}\right )-\mathrm {log}\left (-\left (1-x \right )^{\frac {1}{6}} 2^{\frac {1}{6}}+\left (1-x \right )^{\frac {1}{3}}+2^{\frac {1}{3}}\right )-\mathrm {log}\left (\left (1-x \right )^{\frac {1}{6}} 2^{\frac {1}{6}}+\left (1-x \right )^{\frac {1}{3}}+2^{\frac {1}{3}}\right )\right ) 2^{\frac {1}{3}}}{2} \] Input:
int((1-x)^(1/3)/(1+x),x)
Output:
(2*sqrt(3)*atan((2*( - x + 1)**(1/6) + 2**(1/6))/(2**(1/6)*sqrt(3))) - 2*s qrt(3)*atan((2*( - x + 1)**(1/6) - 2**(1/6))/(2**(1/6)*sqrt(3))) + 3*( - x + 1)**(1/3)*2**(2/3) + 2*log(( - x + 1)**(1/6) + 2**(1/6)) + 2*log(( - x + 1)**(1/6) - 2**(1/6)) - log( - ( - x + 1)**(1/6)*2**(1/6) + ( - x + 1)** (1/3) + 2**(1/3)) - log(( - x + 1)**(1/6)*2**(1/6) + ( - x + 1)**(1/3) + 2 **(1/3)))/2**(2/3)