Integrand size = 13, antiderivative size = 89 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=\frac {3}{8} \left (-1+x^4\right )^{2/3}+\frac {1}{4} \sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )+\frac {1}{4} \log \left (1+\sqrt [3]{-1+x^4}\right )-\frac {1}{8} \log \left (1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right ) \] Output:
3/8*(x^4-1)^(2/3)-1/4*3^(1/2)*arctan(-1/3*3^(1/2)+2/3*(x^4-1)^(1/3)*3^(1/2 ))+1/4*ln(1+(x^4-1)^(1/3))-1/8*ln(1-(x^4-1)^(1/3)+(x^4-1)^(2/3))
Time = 0.05 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=\frac {1}{8} \left (3 \left (-1+x^4\right )^{2/3}+2 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{-1+x^4}}{\sqrt {3}}\right )+2 \log \left (1+\sqrt [3]{-1+x^4}\right )-\log \left (1-\sqrt [3]{-1+x^4}+\left (-1+x^4\right )^{2/3}\right )\right ) \] Input:
Integrate[(-1 + x^4)^(2/3)/x,x]
Output:
(3*(-1 + x^4)^(2/3) + 2*Sqrt[3]*ArcTan[(1 - 2*(-1 + x^4)^(1/3))/Sqrt[3]] + 2*Log[1 + (-1 + x^4)^(1/3)] - Log[1 - (-1 + x^4)^(1/3) + (-1 + x^4)^(2/3) ])/8
Time = 0.19 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.78, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {798, 60, 68, 16, 1083, 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.
\(\displaystyle \int \frac {\left (x^4-1\right )^{2/3}}{x} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle \frac {1}{4} \int \frac {\left (x^4-1\right )^{2/3}}{x^4}dx^4\) |
\(\Big \downarrow \) 60 |
\(\displaystyle \frac {1}{4} \left (\frac {3}{2} \left (x^4-1\right )^{2/3}-\int \frac {1}{x^4 \sqrt [3]{x^4-1}}dx^4\right )\) |
\(\Big \downarrow \) 68 |
\(\displaystyle \frac {1}{4} \left (\frac {3}{2} \int \frac {1}{\sqrt [3]{x^4-1}+1}d\sqrt [3]{x^4-1}-\frac {3}{2} \int \frac {1}{x^8-\sqrt [3]{x^4-1}+1}d\sqrt [3]{x^4-1}+\frac {3}{2} \left (x^4-1\right )^{2/3}-\frac {1}{2} \log \left (x^4\right )\right )\) |
\(\Big \downarrow \) 16 |
\(\displaystyle \frac {1}{4} \left (-\frac {3}{2} \int \frac {1}{x^8-\sqrt [3]{x^4-1}+1}d\sqrt [3]{x^4-1}+\frac {3}{2} \left (x^4-1\right )^{2/3}-\frac {1}{2} \log \left (x^4\right )+\frac {3}{2} \log \left (\sqrt [3]{x^4-1}+1\right )\right )\) |
\(\Big \downarrow \) 1083 |
\(\displaystyle \frac {1}{4} \left (3 \int \frac {1}{-x^8-3}d\left (2 \sqrt [3]{x^4-1}-1\right )+\frac {3}{2} \left (x^4-1\right )^{2/3}-\frac {1}{2} \log \left (x^4\right )+\frac {3}{2} \log \left (\sqrt [3]{x^4-1}+1\right )\right )\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {1}{4} \left (-\sqrt {3} \arctan \left (\frac {2 \sqrt [3]{x^4-1}-1}{\sqrt {3}}\right )+\frac {3}{2} \left (x^4-1\right )^{2/3}-\frac {\log \left (x^4\right )}{2}+\frac {3}{2} \log \left (\sqrt [3]{x^4-1}+1\right )\right )\) |
Input:
Int[(-1 + x^4)^(2/3)/x,x]
Output:
((3*(-1 + x^4)^(2/3))/2 - Sqrt[3]*ArcTan[(-1 + 2*(-1 + x^4)^(1/3))/Sqrt[3] ] - Log[x^4]/2 + (3*Log[1 + (-1 + x^4)^(1/3)])/2)/4
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_))^(1/3)), x_Symbol] :> With[ {q = Rt[-(b*c - a*d)/b, 3]}, Simp[Log[RemoveContent[a + b*x, x]]/(2*b*q), x ] + (Simp[3/(2*b) Subst[Int[1/(q^2 - q*x + x^2), x], x, (c + d*x)^(1/3)], x] - Simp[3/(2*b*q) Subst[Int[1/(q + x), x], x, (c + d*x)^(1/3)], x])] / ; FreeQ[{a, b, c, d}, x] && NegQ[(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[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[1/n Subst [Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Time = 2.77 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}-\frac {\ln \left (1-\left (x^{4}-1\right )^{\frac {1}{3}}+\left (x^{4}-1\right )^{\frac {2}{3}}\right )}{8}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 \left (x^{4}-1\right )^{\frac {1}{3}}-1\right ) \sqrt {3}}{3}\right )}{4}+\frac {\ln \left (1+\left (x^{4}-1\right )^{\frac {1}{3}}\right )}{4}\) | \(66\) |
meijerg | \(-\frac {\sqrt {3}\, \Gamma \left (\frac {2}{3}\right ) \operatorname {signum}\left (x^{4}-1\right )^{\frac {2}{3}} \left (\frac {2 \pi \sqrt {3}\, x^{4} \operatorname {hypergeom}\left (\left [\frac {1}{3}, 1, 1\right ], \left [2, 2\right ], x^{4}\right )}{3 \Gamma \left (\frac {2}{3}\right )}-\frac {\left (\frac {3}{2}-\frac {\pi \sqrt {3}}{6}-\frac {3 \ln \left (3\right )}{2}+4 \ln \left (x \right )+i \pi \right ) \pi \sqrt {3}}{\Gamma \left (\frac {2}{3}\right )}\right )}{12 \pi {\left (-\operatorname {signum}\left (x^{4}-1\right )\right )}^{\frac {2}{3}}}\) | \(84\) |
trager | \(\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}-\frac {\ln \left (\frac {117171 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-440973 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-493850 x^{4}-560547 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-1874736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+560547 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}+666879 \left (x^{4}-1\right )^{\frac {2}{3}}-1185459 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-666879 \left (x^{4}-1\right )^{\frac {1}{3}}+458575}{x^{4}}\right )}{4}-\frac {3 \ln \left (\frac {117171 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}-440973 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-493850 x^{4}-560547 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-1874736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}+560547 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}+666879 \left (x^{4}-1\right )^{\frac {2}{3}}-1185459 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-666879 \left (x^{4}-1\right )^{\frac {1}{3}}+458575}{x^{4}}\right ) \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )}{4}+\frac {3 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \ln \left (\frac {117171 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2} x^{4}+519087 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) x^{4}-333840 x^{4}+560547 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {2}{3}}-1874736 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )^{2}-560547 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right ) \left (x^{4}-1\right )^{\frac {1}{3}}+853728 \left (x^{4}-1\right )^{\frac {2}{3}}-64365 \operatorname {RootOf}\left (9 \textit {\_Z}^{2}+3 \textit {\_Z} +1\right )-853728 \left (x^{4}-1\right )^{\frac {1}{3}}+645424}{x^{4}}\right )}{4}\) | \(435\) |
Input:
int((x^4-1)^(2/3)/x,x,method=_RETURNVERBOSE)
Output:
3/8*(x^4-1)^(2/3)-1/8*ln(1-(x^4-1)^(1/3)+(x^4-1)^(2/3))-1/4*3^(1/2)*arctan (1/3*(2*(x^4-1)^(1/3)-1)*3^(1/2))+1/4*ln(1+(x^4-1)^(1/3))
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.75 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {2}{3} \, \sqrt {3} {\left (x^{4} - 1\right )}^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) + \frac {3}{8} \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} - \frac {1}{8} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{4} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:
integrate((x^4-1)^(2/3)/x,x, algorithm="fricas")
Output:
-1/4*sqrt(3)*arctan(2/3*sqrt(3)*(x^4 - 1)^(1/3) - 1/3*sqrt(3)) + 3/8*(x^4 - 1)^(2/3) - 1/8*log((x^4 - 1)^(2/3) - (x^4 - 1)^(1/3) + 1) + 1/4*log((x^4 - 1)^(1/3) + 1)
Result contains complex when optimal does not.
Time = 0.62 (sec) , antiderivative size = 39, normalized size of antiderivative = 0.44 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=- \frac {x^{\frac {8}{3}} \Gamma \left (- \frac {2}{3}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {2}{3}, - \frac {2}{3} \\ \frac {1}{3} \end {matrix}\middle | {\frac {e^{2 i \pi }}{x^{4}}} \right )}}{4 \Gamma \left (\frac {1}{3}\right )} \] Input:
integrate((x**4-1)**(2/3)/x,x)
Output:
-x**(8/3)*gamma(-2/3)*hyper((-2/3, -2/3), (1/3,), exp_polar(2*I*pi)/x**4)/ (4*gamma(1/3))
Time = 0.13 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.73 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {3}{8} \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} - \frac {1}{8} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{4} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) \] Input:
integrate((x^4-1)^(2/3)/x,x, algorithm="maxima")
Output:
-1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 - 1)^(1/3) - 1)) + 3/8*(x^4 - 1)^( 2/3) - 1/8*log((x^4 - 1)^(2/3) - (x^4 - 1)^(1/3) + 1) + 1/4*log((x^4 - 1)^ (1/3) + 1)
Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.74 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=-\frac {1}{4} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, {\left (x^{4} - 1\right )}^{\frac {1}{3}} - 1\right )}\right ) + \frac {3}{8} \, {\left (x^{4} - 1\right )}^{\frac {2}{3}} - \frac {1}{8} \, \log \left ({\left (x^{4} - 1\right )}^{\frac {2}{3}} - {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1\right ) + \frac {1}{4} \, \log \left ({\left | {\left (x^{4} - 1\right )}^{\frac {1}{3}} + 1 \right |}\right ) \] Input:
integrate((x^4-1)^(2/3)/x,x, algorithm="giac")
Output:
-1/4*sqrt(3)*arctan(1/3*sqrt(3)*(2*(x^4 - 1)^(1/3) - 1)) + 3/8*(x^4 - 1)^( 2/3) - 1/8*log((x^4 - 1)^(2/3) - (x^4 - 1)^(1/3) + 1) + 1/4*log(abs((x^4 - 1)^(1/3) + 1))
Time = 8.86 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=\frac {\ln \left (\frac {9\,{\left (x^4-1\right )}^{1/3}}{16}+\frac {9}{16}\right )}{4}+\ln \left (9\,{\left (-\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}^2+\frac {9\,{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (-\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )-\ln \left (9\,{\left (\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )}^2+\frac {9\,{\left (x^4-1\right )}^{1/3}}{16}\right )\,\left (\frac {1}{8}+\frac {\sqrt {3}\,1{}\mathrm {i}}{8}\right )+\frac {3\,{\left (x^4-1\right )}^{2/3}}{8} \] Input:
int((x^4 - 1)^(2/3)/x,x)
Output:
log((9*(x^4 - 1)^(1/3))/16 + 9/16)/4 + log(9*((3^(1/2)*1i)/8 - 1/8)^2 + (9 *(x^4 - 1)^(1/3))/16)*((3^(1/2)*1i)/8 - 1/8) - log(9*((3^(1/2)*1i)/8 + 1/8 )^2 + (9*(x^4 - 1)^(1/3))/16)*((3^(1/2)*1i)/8 + 1/8) + (3*(x^4 - 1)^(2/3)) /8
\[ \int \frac {\left (-1+x^4\right )^{2/3}}{x} \, dx=\frac {3 \left (x^{4}-1\right )^{\frac {2}{3}}}{8}-\left (\int \frac {\left (x^{4}-1\right )^{\frac {2}{3}}}{x^{5}-x}d x \right ) \] Input:
int((x^4-1)^(2/3)/x,x)
Output:
(3*(x**4 - 1)**(2/3) - 8*int((x**4 - 1)**(2/3)/(x**5 - x),x))/8