Integrand size = 65, antiderivative size = 25 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=2 x+\frac {\left (1+x^2\right )^2 \log (5) \log \left (3-x^4\right )}{x^2} \] Output:
2*x+ln(-x^4+3)/x^2*ln(5)*(x^2+1)^2
Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(25)=50\).
Time = 0.14 (sec) , antiderivative size = 111, normalized size of antiderivative = 4.44 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=\frac {1}{3} \left (6 x+8 \sqrt {3} \text {arctanh}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)+\left (\sqrt {3} \log (625)+\log (15625)\right ) \log \left (\sqrt {3}-x^2\right )-\sqrt {3} \log (625) \log \left (\sqrt {3}+x^2\right )+\log (15625) \log \left (\sqrt {3}+x^2\right )+\frac {\log (125) \log \left (3-x^4\right )}{x^2}+x^2 \log (125) \log \left (3-x^4\right )\right ) \] Input:
Integrate[(-6*x^3 + 2*x^7 + (4*x^4 + 8*x^6 + 4*x^8)*Log[5] + (6 - 8*x^4 + 2*x^8)*Log[5]*Log[3 - x^4])/(-3*x^3 + x^7),x]
Output:
(6*x + 8*Sqrt[3]*ArcTanh[x^2/Sqrt[3]]*Log[5] + (Sqrt[3]*Log[625] + Log[156 25])*Log[Sqrt[3] - x^2] - Sqrt[3]*Log[625]*Log[Sqrt[3] + x^2] + Log[15625] *Log[Sqrt[3] + x^2] + (Log[125]*Log[3 - x^4])/x^2 + x^2*Log[125]*Log[3 - x ^4])/3
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(25)=50\).
Time = 1.18 (sec) , antiderivative size = 129, normalized size of antiderivative = 5.16, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.046, Rules used = {2026, 7276, 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.
\(\displaystyle \int \frac {2 x^7-6 x^3+\left (2 x^8-8 x^4+6\right ) \log (5) \log \left (3-x^4\right )+\left (4 x^8+8 x^6+4 x^4\right ) \log (5)}{x^7-3 x^3} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {2 x^7-6 x^3+\left (2 x^8-8 x^4+6\right ) \log (5) \log \left (3-x^4\right )+\left (4 x^8+8 x^6+4 x^4\right ) \log (5)}{x^3 \left (x^4-3\right )}dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (\frac {2 \left (x^5 \log (25)+x^4+4 x^3 \log (5)+x \log (25)-3\right )}{x^4-3}+\frac {2 (x-1) (x+1) \left (x^2+1\right ) \log (5) \log \left (3-x^4\right )}{x^3}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\left (4 \log ^2(5)+3 \log ^2(25)\right ) \text {arctanh}\left (\frac {x^2}{\sqrt {3}}\right )}{\sqrt {3} \log (25)}+2 \sqrt {3} \log (5) \text {arctanh}\left (\frac {x^2}{\sqrt {3}}\right )+\frac {2 \log (5) \text {arctanh}\left (\frac {x^2}{\sqrt {3}}\right )}{\sqrt {3}}+2 \log (5) \log \left (3-x^4\right )+x^2 \log (25)-2 x^2 \log (5)+x^2 \log (5) \log \left (3-x^4\right )+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+2 x\) |
Input:
Int[(-6*x^3 + 2*x^7 + (4*x^4 + 8*x^6 + 4*x^8)*Log[5] + (6 - 8*x^4 + 2*x^8) *Log[5]*Log[3 - x^4])/(-3*x^3 + x^7),x]
Output:
2*x - 2*x^2*Log[5] + (2*ArcTanh[x^2/Sqrt[3]]*Log[5])/Sqrt[3] + 2*Sqrt[3]*A rcTanh[x^2/Sqrt[3]]*Log[5] + x^2*Log[25] - (ArcTanh[x^2/Sqrt[3]]*(4*Log[5] ^2 + 3*Log[25]^2))/(Sqrt[3]*Log[25]) + 2*Log[5]*Log[3 - x^4] + (Log[5]*Log [3 - x^4])/x^2 + x^2*Log[5]*Log[3 - x^4]
Int[(Fx_.)*(Px_)^(p_.), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Int[x^(p *r)*ExpandToSum[Px/x^r, x]^p*Fx, x] /; IGtQ[r, 0]] /; PolyQ[Px, x] && Integ erQ[p] && !MonomialQ[Px, x] && (ILtQ[p, 0] || !PolyQ[u, x])
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.92 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36
method | result | size |
risch | \(\frac {\ln \left (5\right ) \left (x^{4}+1\right ) \ln \left (-x^{4}+3\right )}{x^{2}}+2 x +2 \ln \left (5\right ) \ln \left (x^{4}-3\right )\) | \(34\) |
default | \(2 x +2 \ln \left (5\right ) \left (\frac {x^{2} \ln \left (-x^{4}+3\right )}{2}+\frac {\ln \left (-x^{4}+3\right )}{2 x^{2}}+\ln \left (x^{4}-3\right )\right )\) | \(42\) |
norman | \(\frac {\ln \left (5\right ) \ln \left (-x^{4}+3\right )+2 \ln \left (5\right ) x^{2} \ln \left (-x^{4}+3\right )+\ln \left (5\right ) \ln \left (-x^{4}+3\right ) x^{4}+2 x^{3}}{x^{2}}\) | \(51\) |
parallelrisch | \(\frac {\ln \left (5\right ) \ln \left (-x^{4}+3\right )+2 \ln \left (5\right ) x^{2} \ln \left (-x^{4}+3\right )+\ln \left (5\right ) \ln \left (-x^{4}+3\right ) x^{4}+2 x^{3}}{x^{2}}\) | \(51\) |
parts | \(2 x +2 x^{2} \ln \left (5\right )+4 \ln \left (5\right ) \left (\frac {\ln \left (x^{4}-3\right )}{2}-\frac {2 \sqrt {3}\, \operatorname {arctanh}\left (\frac {x^{2} \sqrt {3}}{3}\right )}{3}\right )+2 \ln \left (5\right ) \left (\frac {x^{2} \ln \left (-x^{4}+3\right )}{2}-x^{2}+\frac {4 \sqrt {3}\, \operatorname {arctanh}\left (\frac {x^{2} \sqrt {3}}{3}\right )}{3}+\frac {\ln \left (-x^{4}+3\right )}{2 x^{2}}\right )\) | \(89\) |
Input:
int(((2*x^8-8*x^4+6)*ln(5)*ln(-x^4+3)+(4*x^8+8*x^6+4*x^4)*ln(5)+2*x^7-6*x^ 3)/(x^7-3*x^3),x,method=_RETURNVERBOSE)
Output:
ln(5)*(x^4+1)/x^2*ln(-x^4+3)+2*x+2*ln(5)*ln(x^4-3)
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.24 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=\frac {2 \, x^{3} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (5\right ) \log \left (-x^{4} + 3\right )}{x^{2}} \] Input:
integrate(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2 *x^7-6*x^3)/(x^7-3*x^3),x, algorithm="fricas")
Output:
(2*x^3 + (x^4 + 2*x^2 + 1)*log(5)*log(-x^4 + 3))/x^2
Time = 0.18 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.36 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=2 x + 2 \log {\left (5 \right )} \log {\left (x^{4} - 3 \right )} + \frac {\left (x^{4} \log {\left (5 \right )} + \log {\left (5 \right )}\right ) \log {\left (3 - x^{4} \right )}}{x^{2}} \] Input:
integrate(((2*x**8-8*x**4+6)*ln(5)*ln(-x**4+3)+(4*x**8+8*x**6+4*x**4)*ln(5 )+2*x**7-6*x**3)/(x**7-3*x**3),x)
Output:
2*x + 2*log(5)*log(x**4 - 3) + (x**4*log(5) + log(5))*log(3 - x**4)/x**2
Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (25) = 50\).
Time = 0.16 (sec) , antiderivative size = 113, normalized size of antiderivative = 4.52 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx={\left (2 \, x^{2} - \sqrt {3} \log \left (x^{2} + \sqrt {3}\right ) + \sqrt {3} \log \left (x^{2} - \sqrt {3}\right )\right )} \log \left (5\right ) + {\left (\sqrt {3} \log \left (x^{2} + \sqrt {3}\right ) - \sqrt {3} \log \left (x^{2} - \sqrt {3}\right )\right )} \log \left (5\right ) + 2 \, \log \left (5\right ) \log \left (x^{4} - 3\right ) + 2 \, x - \frac {2 \, x^{4} \log \left (5\right ) - {\left (x^{4} \log \left (5\right ) + \log \left (5\right )\right )} \log \left (-x^{4} + 3\right )}{x^{2}} \] Input:
integrate(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2 *x^7-6*x^3)/(x^7-3*x^3),x, algorithm="maxima")
Output:
(2*x^2 - sqrt(3)*log(x^2 + sqrt(3)) + sqrt(3)*log(x^2 - sqrt(3)))*log(5) + (sqrt(3)*log(x^2 + sqrt(3)) - sqrt(3)*log(x^2 - sqrt(3)))*log(5) + 2*log( 5)*log(x^4 - 3) + 2*x - (2*x^4*log(5) - (x^4*log(5) + log(5))*log(-x^4 + 3 ))/x^2
Time = 0.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.44 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=2 \, \log \left (5\right ) \log \left (x^{4} - 3\right ) + {\left (x^{2} \log \left (5\right ) + \frac {\log \left (5\right )}{x^{2}}\right )} \log \left (-x^{4} + 3\right ) + 2 \, x \] Input:
integrate(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2 *x^7-6*x^3)/(x^7-3*x^3),x, algorithm="giac")
Output:
2*log(5)*log(x^4 - 3) + (x^2*log(5) + log(5)/x^2)*log(-x^4 + 3) + 2*x
Time = 0.32 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.40 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=2\,x+2\,\ln \left (5\right )\,\ln \left (x^4-3\right )+\frac {\ln \left (3-x^4\right )\,\left (\ln \left (5\right )\,x^4+\ln \left (5\right )\right )}{x^2} \] Input:
int(-(log(5)*(4*x^4 + 8*x^6 + 4*x^8) - 6*x^3 + 2*x^7 + log(3 - x^4)*log(5) *(2*x^8 - 8*x^4 + 6))/(3*x^3 - x^7),x)
Output:
2*x + 2*log(5)*log(x^4 - 3) + (log(3 - x^4)*(log(5) + x^4*log(5)))/x^2
Time = 0.19 (sec) , antiderivative size = 145, normalized size of antiderivative = 5.80 \[ \int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{-3 x^3+x^7} \, dx=\frac {4 \sqrt {3}\, \mathrm {log}\left (3^{\frac {1}{4}}+x \right ) \mathrm {log}\left (5\right ) x^{2}-4 \sqrt {3}\, \mathrm {log}\left (-x^{4}+3\right ) \mathrm {log}\left (5\right ) x^{2}+4 \sqrt {3}\, \mathrm {log}\left (-3^{\frac {1}{4}}+x \right ) \mathrm {log}\left (5\right ) x^{2}+4 \sqrt {3}\, \mathrm {log}\left (\sqrt {3}+x^{2}\right ) \mathrm {log}\left (5\right ) x^{2}+6 \,\mathrm {log}\left (3^{\frac {1}{4}}+x \right ) \mathrm {log}\left (5\right ) x^{2}+3 \,\mathrm {log}\left (-x^{4}+3\right ) \mathrm {log}\left (5\right ) x^{4}+3 \,\mathrm {log}\left (-x^{4}+3\right ) \mathrm {log}\left (5\right )+6 \,\mathrm {log}\left (-3^{\frac {1}{4}}+x \right ) \mathrm {log}\left (5\right ) x^{2}+6 \,\mathrm {log}\left (\sqrt {3}+x^{2}\right ) \mathrm {log}\left (5\right ) x^{2}+6 x^{3}}{3 x^{2}} \] Input:
int(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2*x^7-6 *x^3)/(x^7-3*x^3),x)
Output:
(4*sqrt(3)*log(3**(1/4) + x)*log(5)*x**2 - 4*sqrt(3)*log( - x**4 + 3)*log( 5)*x**2 + 4*sqrt(3)*log( - 3**(1/4) + x)*log(5)*x**2 + 4*sqrt(3)*log(sqrt( 3) + x**2)*log(5)*x**2 + 6*log(3**(1/4) + x)*log(5)*x**2 + 3*log( - x**4 + 3)*log(5)*x**4 + 3*log( - x**4 + 3)*log(5) + 6*log( - 3**(1/4) + x)*log(5 )*x**2 + 6*log(sqrt(3) + x**2)*log(5)*x**2 + 6*x**3)/(3*x**2)