Integrand size = 142, antiderivative size = 29 \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=\frac {5}{4-x^4+\log \left (\frac {x}{-e^{2/3}+\frac {16}{x^2}}\right )} \] Output:
5/(4+ln(x/(16/x^2-exp(2/3)))-x^4)
Time = 0.05 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03 \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=\frac {5}{4-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )} \] Input:
Integrate[(240 - 320*x^4 + E^(2/3)*(-5*x^2 + 20*x^6))/(-256*x + 128*x^5 - 16*x^9 + E^(2/3)*(16*x^3 - 8*x^7 + x^11) + (-128*x + 32*x^5 + E^(2/3)*(8*x ^3 - 2*x^7))*Log[-(x^3/(-16 + E^(2/3)*x^2))] + (-16*x + E^(2/3)*x^3)*Log[- (x^3/(-16 + E^(2/3)*x^2))]^2),x]
Output:
5/(4 - x^4 + Log[x^3/(16 - E^(2/3)*x^2)])
Time = 0.68 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.028, Rules used = {7239, 27, 25, 7237}
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.
\(\displaystyle \int \frac {-320 x^4+e^{2/3} \left (20 x^6-5 x^2\right )+240}{-16 x^9+128 x^5+\left (e^{2/3} x^3-16 x\right ) \log ^2\left (-\frac {x^3}{e^{2/3} x^2-16}\right )+e^{2/3} \left (x^{11}-8 x^7+16 x^3\right )+\left (32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )-128 x\right ) \log \left (-\frac {x^3}{e^{2/3} x^2-16}\right )-256 x} \, dx\) |
\(\Big \downarrow \) 7239 |
\(\displaystyle \int \frac {5 \left (64 x^4-e^{2/3} \left (4 x^4-1\right ) x^2-48\right )}{x \left (16-e^{2/3} x^2\right ) \left (-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )+4\right )^2}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 5 \int -\frac {-64 x^4-e^{2/3} \left (1-4 x^4\right ) x^2+48}{x \left (16-e^{2/3} x^2\right ) \left (-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )+4\right )^2}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -5 \int \frac {-64 x^4-e^{2/3} \left (1-4 x^4\right ) x^2+48}{x \left (16-e^{2/3} x^2\right ) \left (-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )+4\right )^2}dx\) |
\(\Big \downarrow \) 7237 |
\(\displaystyle \frac {5}{-x^4+\log \left (\frac {x^3}{16-e^{2/3} x^2}\right )+4}\) |
Input:
Int[(240 - 320*x^4 + E^(2/3)*(-5*x^2 + 20*x^6))/(-256*x + 128*x^5 - 16*x^9 + E^(2/3)*(16*x^3 - 8*x^7 + x^11) + (-128*x + 32*x^5 + E^(2/3)*(8*x^3 - 2 *x^7))*Log[-(x^3/(-16 + E^(2/3)*x^2))] + (-16*x + E^(2/3)*x^3)*Log[-(x^3/( -16 + E^(2/3)*x^2))]^2),x]
Output:
5/(4 - x^4 + Log[x^3/(16 - E^(2/3)*x^2)])
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Si mp[q*(y^(m + 1)/(m + 1)), x] /; !FalseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]
Int[u_, x_Symbol] :> With[{v = SimplifyIntegrand[u, x]}, Int[v, x] /; Simpl erIntegrandQ[v, u, x]]
Time = 1.00 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97
method | result | size |
risch | \(-\frac {5}{x^{4}-\ln \left (-\frac {x^{3}}{x^{2} {\mathrm e}^{\frac {2}{3}}-16}\right )-4}\) | \(28\) |
parallelrisch | \(-\frac {5}{x^{4}-\ln \left (-\frac {x^{3}}{x^{2} {\mathrm e}^{\frac {2}{3}}-16}\right )-4}\) | \(28\) |
Input:
int(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*ln(-x^3/(x^ 2*exp(2/3)-16))^2+((-2*x^7+8*x^3)*exp(2/3)+32*x^5-128*x)*ln(-x^3/(x^2*exp( 2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x,method=_RET URNVERBOSE)
Output:
-5/(x^4-ln(-x^3/(x^2*exp(2/3)-16))-4)
Time = 0.09 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=-\frac {5}{x^{4} - \log \left (-\frac {x^{3}}{x^{2} e^{\frac {2}{3}} - 16}\right ) - 4} \] Input:
integrate(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(- x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/( x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x, al gorithm="fricas")
Output:
-5/(x^4 - log(-x^3/(x^2*e^(2/3) - 16)) - 4)
Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.76 \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=\frac {5}{- x^{4} + \log {\left (- \frac {x^{3}}{x^{2} e^{\frac {2}{3}} - 16} \right )} + 4} \] Input:
integrate(((20*x**6-5*x**2)*exp(2/3)-320*x**4+240)/((x**3*exp(2/3)-16*x)*l n(-x**3/(x**2*exp(2/3)-16))**2+((-2*x**7+8*x**3)*exp(2/3)+32*x**5-128*x)*l n(-x**3/(x**2*exp(2/3)-16))+(x**11-8*x**7+16*x**3)*exp(2/3)-16*x**9+128*x* *5-256*x),x)
Output:
5/(-x**4 + log(-x**3/(x**2*exp(2/3) - 16)) + 4)
Exception generated. \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(- x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/( x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x, al gorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is un defined.
Time = 0.15 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=-\frac {5}{x^{4} - \log \left (-\frac {x^{3}}{x^{2} e^{\frac {2}{3}} - 16}\right ) - 4} \] Input:
integrate(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(- x^3/(x^2*exp(2/3)-16))^2+((-2*x^7+8*x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/( x^2*exp(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x, al gorithm="giac")
Output:
-5/(x^4 - log(-x^3/(x^2*e^(2/3) - 16)) - 4)
Time = 9.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.93 \[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx=\frac {5}{\ln \left (-\frac {x^3}{x^2\,{\mathrm {e}}^{2/3}-16}\right )-x^4+4} \] Input:
int((exp(2/3)*(5*x^2 - 20*x^6) + 320*x^4 - 240)/(256*x - log(-x^3/(x^2*exp (2/3) - 16))*(exp(2/3)*(8*x^3 - 2*x^7) - 128*x + 32*x^5) - exp(2/3)*(16*x^ 3 - 8*x^7 + x^11) + log(-x^3/(x^2*exp(2/3) - 16))^2*(16*x - x^3*exp(2/3)) - 128*x^5 + 16*x^9),x)
Output:
5/(log(-x^3/(x^2*exp(2/3) - 16)) - x^4 + 4)
\[ \int \frac {240-320 x^4+e^{2/3} \left (-5 x^2+20 x^6\right )}{-256 x+128 x^5-16 x^9+e^{2/3} \left (16 x^3-8 x^7+x^{11}\right )+\left (-128 x+32 x^5+e^{2/3} \left (8 x^3-2 x^7\right )\right ) \log \left (-\frac {x^3}{-16+e^{2/3} x^2}\right )+\left (-16 x+e^{2/3} x^3\right ) \log ^2\left (-\frac {x^3}{-16+e^{2/3} x^2}\right )} \, dx =\text {Too large to display} \] Input:
int(((20*x^6-5*x^2)*exp(2/3)-320*x^4+240)/((x^3*exp(2/3)-16*x)*log(-x^3/(x ^2*exp(2/3)-16))^2+((-2*x^7+8*x^3)*exp(2/3)+32*x^5-128*x)*log(-x^3/(x^2*ex p(2/3)-16))+(x^11-8*x^7+16*x^3)*exp(2/3)-16*x^9+128*x^5-256*x),x)
Output:
5*(4*e**(2/3)*int(x**5/(e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))**2*x* *2 - 2*e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))*x**6 + 8*e**(2/3)*log( ( - x**3)/(e**(2/3)*x**2 - 16))*x**2 + e**(2/3)*x**10 - 8*e**(2/3)*x**6 + 16*e**(2/3)*x**2 - 16*log(( - x**3)/(e**(2/3)*x**2 - 16))**2 + 32*log(( - x**3)/(e**(2/3)*x**2 - 16))*x**4 - 128*log(( - x**3)/(e**(2/3)*x**2 - 16)) - 16*x**8 + 128*x**4 - 256),x) - e**(2/3)*int(x/(e**(2/3)*log(( - x**3)/( e**(2/3)*x**2 - 16))**2*x**2 - 2*e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 1 6))*x**6 + 8*e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))*x**2 + e**(2/3)* x**10 - 8*e**(2/3)*x**6 + 16*e**(2/3)*x**2 - 16*log(( - x**3)/(e**(2/3)*x* *2 - 16))**2 + 32*log(( - x**3)/(e**(2/3)*x**2 - 16))*x**4 - 128*log(( - x **3)/(e**(2/3)*x**2 - 16)) - 16*x**8 + 128*x**4 - 256),x) - 64*int(x**3/(e **(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))**2*x**2 - 2*e**(2/3)*log(( - x **3)/(e**(2/3)*x**2 - 16))*x**6 + 8*e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))*x**2 + e**(2/3)*x**10 - 8*e**(2/3)*x**6 + 16*e**(2/3)*x**2 - 16*log (( - x**3)/(e**(2/3)*x**2 - 16))**2 + 32*log(( - x**3)/(e**(2/3)*x**2 - 16 ))*x**4 - 128*log(( - x**3)/(e**(2/3)*x**2 - 16)) - 16*x**8 + 128*x**4 - 2 56),x) + 48*int(1/(e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))**2*x**3 - 2*e**(2/3)*log(( - x**3)/(e**(2/3)*x**2 - 16))*x**7 + 8*e**(2/3)*log(( - x **3)/(e**(2/3)*x**2 - 16))*x**3 + e**(2/3)*x**11 - 8*e**(2/3)*x**7 + 16*e* *(2/3)*x**3 - 16*log(( - x**3)/(e**(2/3)*x**2 - 16))**2*x + 32*log(( - ...