Integrand size = 32, antiderivative size = 133 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\frac {\left (1+x^6\right )^{2/3}}{4 x^2}-\frac {\arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^6}}\right )}{2\ 2^{2/3} \sqrt {3}}+\frac {\log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^6}\right )}{6\ 2^{2/3}}-\frac {\log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^6}+2^{2/3} \left (1+x^6\right )^{2/3}\right )}{12\ 2^{2/3}} \] Output:
1/4*(x^6+1)^(2/3)/x^2-1/12*arctan(3^(1/2)*x/(x+2*2^(1/3)*(x^6+1)^(1/3)))*2 ^(1/3)*3^(1/2)+1/12*ln(-x+2^(1/3)*(x^6+1)^(1/3))*2^(1/3)-1/24*ln(x^2+2^(1/ 3)*x*(x^6+1)^(1/3)+2^(2/3)*(x^6+1)^(2/3))*2^(1/3)
Time = 1.12 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.97 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\frac {1}{24} \left (\frac {6 \left (1+x^6\right )^{2/3}}{x^2}-2 \sqrt [3]{2} \sqrt {3} \arctan \left (\frac {\sqrt {3} x}{x+2 \sqrt [3]{2} \sqrt [3]{1+x^6}}\right )+2 \sqrt [3]{2} \log \left (-x+\sqrt [3]{2} \sqrt [3]{1+x^6}\right )-\sqrt [3]{2} \log \left (x^2+\sqrt [3]{2} x \sqrt [3]{1+x^6}+2^{2/3} \left (1+x^6\right )^{2/3}\right )\right ) \] Input:
Integrate[((-1 + x^6)*(1 + x^6)^(2/3))/(x^3*(2 - x^3 + 2*x^6)),x]
Output:
((6*(1 + x^6)^(2/3))/x^2 - 2*2^(1/3)*Sqrt[3]*ArcTan[(Sqrt[3]*x)/(x + 2*2^( 1/3)*(1 + x^6)^(1/3))] + 2*2^(1/3)*Log[-x + 2^(1/3)*(1 + x^6)^(1/3)] - 2^( 1/3)*Log[x^2 + 2^(1/3)*x*(1 + x^6)^(1/3) + 2^(2/3)*(1 + x^6)^(2/3)])/24
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.94 (sec) , antiderivative size = 241, normalized size of antiderivative = 1.81, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {7279, 2009}
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.
\(\displaystyle \int \frac {\left (x^6-1\right ) \left (x^6+1\right )^{2/3}}{x^3 \left (2 x^6-x^3+2\right )} \, dx\) |
\(\Big \downarrow \) 7279 |
\(\displaystyle \int \left (\frac {\left (4 x^3-1\right ) \left (x^6+1\right )^{2/3}}{2 \left (2 x^6-x^3+2\right )}-\frac {\left (x^6+1\right )^{2/3}}{2 x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\left (-\sqrt {15}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7-i \sqrt {15}},-x^6\right )}{\sqrt {15}+7 i}+\frac {\left (\sqrt {15}+i\right ) x \operatorname {AppellF1}\left (\frac {1}{6},1,-\frac {2}{3},\frac {7}{6},-\frac {8 x^6}{7+i \sqrt {15}},-x^6\right )}{-\sqrt {15}+7 i}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {8 x^6}{7-i \sqrt {15}}\right )}{7-i \sqrt {15}}+\frac {x^4 \operatorname {AppellF1}\left (\frac {2}{3},-\frac {2}{3},1,\frac {5}{3},-x^6,-\frac {8 x^6}{7+i \sqrt {15}}\right )}{7+i \sqrt {15}}+\frac {\operatorname {Hypergeometric2F1}\left (-\frac {2}{3},-\frac {1}{3},\frac {2}{3},-x^6\right )}{4 x^2}\) |
Input:
Int[((-1 + x^6)*(1 + x^6)^(2/3))/(x^3*(2 - x^3 + 2*x^6)),x]
Output:
((I - Sqrt[15])*x*AppellF1[1/6, 1, -2/3, 7/6, (-8*x^6)/(7 - I*Sqrt[15]), - x^6])/(7*I + Sqrt[15]) + ((I + Sqrt[15])*x*AppellF1[1/6, 1, -2/3, 7/6, (-8 *x^6)/(7 + I*Sqrt[15]), -x^6])/(7*I - Sqrt[15]) + (x^4*AppellF1[2/3, -2/3, 1, 5/3, -x^6, (-8*x^6)/(7 - I*Sqrt[15])])/(7 - I*Sqrt[15]) + (x^4*AppellF 1[2/3, -2/3, 1, 5/3, -x^6, (-8*x^6)/(7 + I*Sqrt[15])])/(7 + I*Sqrt[15]) + Hypergeometric2F1[-2/3, -1/3, 2/3, -x^6]/(4*x^2)
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
\[\int \frac {\left (x^{6}-1\right ) \left (x^{6}+1\right )^{\frac {2}{3}}}{x^{3} \left (2 x^{6}-x^{3}+2\right )}d x\]
Input:
int((x^6-1)*(x^6+1)^(2/3)/x^3/(2*x^6-x^3+2),x)
Output:
int((x^6-1)*(x^6+1)^(2/3)/x^3/(2*x^6-x^3+2),x)
Leaf count of result is larger than twice the leaf count of optimal. 343 vs. \(2 (97) = 194\).
Time = 90.28 (sec) , antiderivative size = 343, normalized size of antiderivative = 2.58 \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=-\frac {12 \cdot 4^{\frac {1}{6}} \sqrt {\frac {1}{3}} x^{2} \arctan \left (\frac {4^{\frac {1}{6}} \sqrt {\frac {1}{3}} {\left (12 \cdot 4^{\frac {2}{3}} {\left (2 \, x^{13} + x^{10} + 3 \, x^{7} + x^{4} + 2 \, x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (8 \, x^{18} + 60 \, x^{15} + 48 \, x^{12} + 119 \, x^{9} + 48 \, x^{6} + 60 \, x^{3} + 8\right )} + 12 \, {\left (4 \, x^{14} + 14 \, x^{11} + 9 \, x^{8} + 14 \, x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}\right )}}{2 \, {\left (8 \, x^{18} - 12 \, x^{15} - 24 \, x^{12} - 25 \, x^{9} - 24 \, x^{6} - 12 \, x^{3} + 8\right )}}\right ) - 2 \cdot 4^{\frac {2}{3}} x^{2} \log \left (-\frac {6 \cdot 4^{\frac {1}{3}} {\left (x^{6} + 1\right )}^{\frac {1}{3}} x^{2} + 4^{\frac {2}{3}} {\left (2 \, x^{6} - x^{3} + 2\right )} - 12 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}} x}{2 \, x^{6} - x^{3} + 2}\right ) + 4^{\frac {2}{3}} x^{2} \log \left (\frac {6 \cdot 4^{\frac {2}{3}} {\left (x^{7} + x^{4} + x\right )} {\left (x^{6} + 1\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (4 \, x^{12} + 14 \, x^{9} + 9 \, x^{6} + 14 \, x^{3} + 4\right )} + 6 \, {\left (4 \, x^{8} + x^{5} + 4 \, x^{2}\right )} {\left (x^{6} + 1\right )}^{\frac {1}{3}}}{4 \, x^{12} - 4 \, x^{9} + 9 \, x^{6} - 4 \, x^{3} + 4}\right ) - 36 \, {\left (x^{6} + 1\right )}^{\frac {2}{3}}}{144 \, x^{2}} \] Input:
integrate((x^6-1)*(x^6+1)^(2/3)/x^3/(2*x^6-x^3+2),x, algorithm="fricas")
Output:
-1/144*(12*4^(1/6)*sqrt(1/3)*x^2*arctan(1/2*4^(1/6)*sqrt(1/3)*(12*4^(2/3)* (2*x^13 + x^10 + 3*x^7 + x^4 + 2*x)*(x^6 + 1)^(2/3) + 4^(1/3)*(8*x^18 + 60 *x^15 + 48*x^12 + 119*x^9 + 48*x^6 + 60*x^3 + 8) + 12*(4*x^14 + 14*x^11 + 9*x^8 + 14*x^5 + 4*x^2)*(x^6 + 1)^(1/3))/(8*x^18 - 12*x^15 - 24*x^12 - 25* x^9 - 24*x^6 - 12*x^3 + 8)) - 2*4^(2/3)*x^2*log(-(6*4^(1/3)*(x^6 + 1)^(1/3 )*x^2 + 4^(2/3)*(2*x^6 - x^3 + 2) - 12*(x^6 + 1)^(2/3)*x)/(2*x^6 - x^3 + 2 )) + 4^(2/3)*x^2*log((6*4^(2/3)*(x^7 + x^4 + x)*(x^6 + 1)^(2/3) + 4^(1/3)* (4*x^12 + 14*x^9 + 9*x^6 + 14*x^3 + 4) + 6*(4*x^8 + x^5 + 4*x^2)*(x^6 + 1) ^(1/3))/(4*x^12 - 4*x^9 + 9*x^6 - 4*x^3 + 4)) - 36*(x^6 + 1)^(2/3))/x^2
Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\text {Timed out} \] Input:
integrate((x**6-1)*(x**6+1)**(2/3)/x**3/(2*x**6-x**3+2),x)
Output:
Timed out
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\int { \frac {{\left (x^{6} + 1\right )}^{\frac {2}{3}} {\left (x^{6} - 1\right )}}{{\left (2 \, x^{6} - x^{3} + 2\right )} x^{3}} \,d x } \] Input:
integrate((x^6-1)*(x^6+1)^(2/3)/x^3/(2*x^6-x^3+2),x, algorithm="maxima")
Output:
integrate((x^6 + 1)^(2/3)*(x^6 - 1)/((2*x^6 - x^3 + 2)*x^3), x)
Exception generated. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((x^6-1)*(x^6+1)^(2/3)/x^3/(2*x^6-x^3+2),x, algorithm="giac")
Output:
Exception raised: RuntimeError >> an error occurred running a Giac command :INPUT:sage2OUTPUT:Invalid _EXT in replace_ext Error: Bad Argument Valuein tegrate((2*sageVARx^6-sageVARx^3+2)^-1*((sageVARx^6+1)^(1/3))^2*(sageVARx^ 6-1)/sageVAR
Timed out. \[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\int \frac {\left (x^6-1\right )\,{\left (x^6+1\right )}^{2/3}}{x^3\,\left (2\,x^6-x^3+2\right )} \,d x \] Input:
int(((x^6 - 1)*(x^6 + 1)^(2/3))/(x^3*(2*x^6 - x^3 + 2)),x)
Output:
int(((x^6 - 1)*(x^6 + 1)^(2/3))/(x^3*(2*x^6 - x^3 + 2)), x)
\[ \int \frac {\left (-1+x^6\right ) \left (1+x^6\right )^{2/3}}{x^3 \left (2-x^3+2 x^6\right )} \, dx=\frac {\left (x^{6}+1\right )^{\frac {2}{3}}-2 \left (\int \frac {\left (x^{6}+1\right )^{\frac {2}{3}}}{2 x^{12}-x^{9}+4 x^{6}-x^{3}+2}d x \right ) x^{2}+2 \left (\int \frac {\left (x^{6}+1\right )^{\frac {2}{3}} x^{6}}{2 x^{12}-x^{9}+4 x^{6}-x^{3}+2}d x \right ) x^{2}}{4 x^{2}} \] Input:
int((x^6-1)*(x^6+1)^(2/3)/x^3/(2*x^6-x^3+2),x)
Output:
((x**6 + 1)**(2/3) - 2*int((x**6 + 1)**(2/3)/(2*x**12 - x**9 + 4*x**6 - x* *3 + 2),x)*x**2 + 2*int(((x**6 + 1)**(2/3)*x**6)/(2*x**12 - x**9 + 4*x**6 - x**3 + 2),x)*x**2)/(4*x**2)