Integrand size = 18, antiderivative size = 33 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (160-95 x^3-26 x^6-39 x^9\right )}{440 x^{11}} \]
Time = 0.11 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\frac {\left (-1+x^3\right )^{2/3} \left (160-95 x^3-26 x^6-39 x^9\right )}{440 x^{11}} \]
Time = 0.18 (sec) , antiderivative size = 54, normalized size of antiderivative = 1.64, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {955, 803, 796}
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.
\(\displaystyle \int \frac {\left (x^3-4\right ) \left (x^3-1\right )^{2/3}}{x^{12}} \, dx\) |
\(\Big \downarrow \) 955 |
\(\displaystyle -\frac {13}{11} \int \frac {\left (x^3-1\right )^{2/3}}{x^9}dx-\frac {4 \left (x^3-1\right )^{5/3}}{11 x^{11}}\) |
\(\Big \downarrow \) 803 |
\(\displaystyle -\frac {13}{11} \left (\frac {3}{8} \int \frac {\left (x^3-1\right )^{2/3}}{x^6}dx+\frac {\left (x^3-1\right )^{5/3}}{8 x^8}\right )-\frac {4 \left (x^3-1\right )^{5/3}}{11 x^{11}}\) |
\(\Big \downarrow \) 796 |
\(\displaystyle -\frac {4 \left (x^3-1\right )^{5/3}}{11 x^{11}}-\frac {13}{11} \left (\frac {\left (x^3-1\right )^{5/3}}{8 x^8}+\frac {3 \left (x^3-1\right )^{5/3}}{40 x^5}\right )\) |
(-4*(-1 + x^3)^(5/3))/(11*x^11) - (13*((-1 + x^3)^(5/3)/(8*x^8) + (3*(-1 + x^3)^(5/3))/(40*x^5)))/11
3.4.94.3.1 Defintions of rubi rules used
Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c* x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*c*(m + 1))), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[x^(m + 1)*(( a + b*x^n)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + n*(p + 1) + 1)/(a*(m + 1 ))) Int[x^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, m, n, p}, x] && I LtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[m, -1]
Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n _)), x_Symbol] :> Simp[c*(e*x)^(m + 1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] + Simp[(a*d*(m + 1) - b*c*(m + n*(p + 1) + 1))/(a*e^n*(m + 1)) Int[(e *x)^(m + n)*(a + b*x^n)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[b* c - a*d, 0] && (IntegerQ[n] || GtQ[e, 0]) && ((GtQ[n, 0] && LtQ[m, -1]) || (LtQ[n, 0] && GtQ[m + n, -1])) && !ILtQ[p, -1]
Time = 0.87 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.76
method | result | size |
pseudoelliptic | \(-\frac {\left (39 x^{6}+65 x^{3}+160\right ) \left (x^{3}-1\right )^{\frac {5}{3}}}{440 x^{11}}\) | \(25\) |
trager | \(-\frac {\left (39 x^{9}+26 x^{6}+95 x^{3}-160\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{440 x^{11}}\) | \(30\) |
gosper | \(-\frac {\left (x -1\right ) \left (x^{2}+x +1\right ) \left (39 x^{6}+65 x^{3}+160\right ) \left (x^{3}-1\right )^{\frac {2}{3}}}{440 x^{11}}\) | \(34\) |
risch | \(-\frac {39 x^{12}-13 x^{9}+69 x^{6}-255 x^{3}+160}{440 x^{11} \left (x^{3}-1\right )^{\frac {1}{3}}}\) | \(35\) |
meijerg | \(\frac {4 \operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {9}{20} x^{9}-\frac {3}{10} x^{6}-\frac {1}{4} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{11 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{11}}-\frac {\operatorname {signum}\left (x^{3}-1\right )^{\frac {2}{3}} \left (-\frac {3}{5} x^{6}-\frac {2}{5} x^{3}+1\right ) \left (-x^{3}+1\right )^{\frac {2}{3}}}{8 {\left (-\operatorname {signum}\left (x^{3}-1\right )\right )}^{\frac {2}{3}} x^{8}}\) | \(95\) |
Time = 0.24 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=-\frac {{\left (39 \, x^{9} + 26 \, x^{6} + 95 \, x^{3} - 160\right )} {\left (x^{3} - 1\right )}^{\frac {2}{3}}}{440 \, x^{11}} \]
Result contains complex when optimal does not.
Time = 1.68 (sec) , antiderivative size = 571, normalized size of antiderivative = 17.30 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=- 4 \left (\begin {cases} \frac {2 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {4 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {10 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {40 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{- \frac {i \pi }{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{9} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\- \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} - \frac {4 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {10 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{6} \Gamma \left (- \frac {2}{3}\right )} + \frac {40 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {11}{3}\right )}{27 x^{9} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases}\right ) + \begin {cases} \frac {3 x^{6} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {x^{3} \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} + \frac {5 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{9} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{6} \Gamma \left (- \frac {2}{3}\right )} - \frac {7 \left (-1 + \frac {1}{x^{3}}\right )^{\frac {2}{3}} e^{\frac {2 i \pi }{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right ) - 9 x^{3} \Gamma \left (- \frac {2}{3}\right )} & \text {for}\: \frac {1}{\left |{x^{3}}\right |} > 1 \\\frac {\left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{3 \Gamma \left (- \frac {2}{3}\right )} + \frac {2 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{3} \Gamma \left (- \frac {2}{3}\right )} - \frac {5 \left (1 - \frac {1}{x^{3}}\right )^{\frac {2}{3}} \Gamma \left (- \frac {8}{3}\right )}{9 x^{6} \Gamma \left (- \frac {2}{3}\right )} & \text {otherwise} \end {cases} \]
-4*Piecewise((2*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(3*gamma(- 2/3)) + 4*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(9*x**3*gamma(-2 /3)) + 10*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(27*x**6*gamma(- 2/3)) - 40*(-1 + x**(-3))**(2/3)*exp(-I*pi/3)*gamma(-11/3)/(27*x**9*gamma( -2/3)), 1/Abs(x**3) > 1), (-2*(1 - 1/x**3)**(2/3)*gamma(-11/3)/(3*gamma(-2 /3)) - 4*(1 - 1/x**3)**(2/3)*gamma(-11/3)/(9*x**3*gamma(-2/3)) - 10*(1 - 1 /x**3)**(2/3)*gamma(-11/3)/(27*x**6*gamma(-2/3)) + 40*(1 - 1/x**3)**(2/3)* gamma(-11/3)/(27*x**9*gamma(-2/3)), True)) + Piecewise((3*x**6*(-1 + x**(- 3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2 /3)) - x**3*(-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma( -2/3) - 9*x**3*gamma(-2/3)) + 5*(-1 + x**(-3))**(2/3)*exp(2*I*pi/3)*gamma( -8/3)/(9*x**9*gamma(-2/3) - 9*x**6*gamma(-2/3)) - 7*(-1 + x**(-3))**(2/3)* exp(2*I*pi/3)*gamma(-8/3)/(9*x**6*gamma(-2/3) - 9*x**3*gamma(-2/3)), 1/Abs (x**3) > 1), ((1 - 1/x**3)**(2/3)*gamma(-8/3)/(3*gamma(-2/3)) + 2*(1 - 1/x **3)**(2/3)*gamma(-8/3)/(9*x**3*gamma(-2/3)) - 5*(1 - 1/x**3)**(2/3)*gamma (-8/3)/(9*x**6*gamma(-2/3)), True))
Time = 0.20 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.12 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=-\frac {3 \, {\left (x^{3} - 1\right )}^{\frac {5}{3}}}{5 \, x^{5}} + \frac {7 \, {\left (x^{3} - 1\right )}^{\frac {8}{3}}}{8 \, x^{8}} - \frac {4 \, {\left (x^{3} - 1\right )}^{\frac {11}{3}}}{11 \, x^{11}} \]
\[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\int { \frac {{\left (x^{3} - 1\right )}^{\frac {2}{3}} {\left (x^{3} - 4\right )}}{x^{12}} \,d x } \]
Time = 5.38 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.48 \[ \int \frac {\left (-4+x^3\right ) \left (-1+x^3\right )^{2/3}}{x^{12}} \, dx=\frac {4\,{\left (x^3-1\right )}^{2/3}}{11\,x^{11}}-\frac {13\,{\left (x^3-1\right )}^{2/3}}{220\,x^5}-\frac {19\,{\left (x^3-1\right )}^{2/3}}{88\,x^8}-\frac {39\,{\left (x^3-1\right )}^{2/3}}{440\,x^2} \]