Integrand size = 77, antiderivative size = 21 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=x \left (-x^2+\frac {\log \left (9+x^5\right )}{\log (2)}\right )^2 \] Output:
x*(ln(x^5+9)/ln(2)-x^2)^2
Time = 5.03 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x \left (x^4 \log ^2(2)-x^2 \log (4) \log \left (9+x^5\right )+\log ^2\left (9+x^5\right )\right )}{\log ^2(2)} \] Input:
Integrate[(-10*x^7*Log[2] + (45*x^4 + 5*x^9)*Log[2]^2 + (10*x^5 + (-54*x^2 - 6*x^7)*Log[2])*Log[9 + x^5] + (9 + x^5)*Log[9 + x^5]^2)/((9 + x^5)*Log[ 2]^2),x]
Output:
(x*(x^4*Log[2]^2 - x^2*Log[4]*Log[9 + x^5] + Log[9 + x^5]^2))/Log[2]^2
Leaf count is larger than twice the leaf count of optimal. \(542\) vs. \(2(21)=42\).
Time = 8.99 (sec) , antiderivative size = 542, normalized size of antiderivative = 25.81, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.052, Rules used = {27, 25, 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.
\(\displaystyle \int \frac {-10 x^7 \log (2)+\left (x^5+9\right ) \log ^2\left (x^5+9\right )+\left (5 x^9+45 x^4\right ) \log ^2(2)+\left (10 x^5+\left (-6 x^7-54 x^2\right ) \log (2)\right ) \log \left (x^5+9\right )}{\left (x^5+9\right ) \log ^2(2)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {10 \log (2) x^7-\left (x^5+9\right ) \log ^2\left (x^5+9\right )-2 \left (5 x^5-3 \left (x^7+9 x^2\right ) \log (2)\right ) \log \left (x^5+9\right )-5 \left (x^9+9 x^4\right ) \log ^2(2)}{x^5+9}dx}{\log ^2(2)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {10 \log (2) x^7-\left (x^5+9\right ) \log ^2\left (x^5+9\right )-2 \left (5 x^5-3 \left (x^7+9 x^2\right ) \log (2)\right ) \log \left (x^5+9\right )-5 \left (x^9+9 x^4\right ) \log ^2(2)}{x^5+9}dx}{\log ^2(2)}\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle -\frac {\int \left (-\frac {5 \log (2) \left (\log (2) x^5-2 x^3+\log (512)\right ) x^4}{x^5+9}+\frac {2 \left (\log (8) x^5-5 x^3+27 \log (2)\right ) \log \left (x^5+9\right ) x^2}{x^5+9}-\log ^2\left (x^5+9\right )\right )dx}{\log ^2(2)}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {-\sqrt [5]{3} \sqrt {2 \left (5-\sqrt {5}\right )} \log (8) \arctan \left (\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} x}{3^{2/5}}+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )-\sqrt [5]{3} \sqrt {2 \left (5+\sqrt {5}\right )} \log (8) \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x}{3^{2/5}}\right )+3 \sqrt [5]{3} \sqrt {2 \left (5-\sqrt {5}\right )} \log (2) \arctan \left (\frac {2 \sqrt {\frac {2}{5+\sqrt {5}}} x}{3^{2/5}}+\sqrt {\frac {1}{5} \left (5-2 \sqrt {5}\right )}\right )+3 \sqrt [5]{3} \sqrt {2 \left (5+\sqrt {5}\right )} \log (2) \arctan \left (\sqrt {\frac {1}{5} \left (5+2 \sqrt {5}\right )}-\frac {\sqrt {\frac {2}{5} \left (5+\sqrt {5}\right )} x}{3^{2/5}}\right )+x^5 \left (-\log ^2(2)\right )-x \log ^2\left (x^5+9\right )-\frac {10}{9} x^3 \log (8)+\frac {10}{3} x^3 \log (2)-\frac {1}{2} \sqrt [5]{3} \left (1+\sqrt {5}\right ) \log (8) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+3^{4/5}\right )+\frac {3}{2} \sqrt [5]{3} \left (1+\sqrt {5}\right ) \log (2) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1-\sqrt {5}\right ) x+3^{4/5}\right )-\frac {1}{2} \sqrt [5]{3} \left (1-\sqrt {5}\right ) \log (8) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+3^{4/5}\right )+\frac {3}{2} \sqrt [5]{3} \left (1-\sqrt {5}\right ) \log (2) \log \left (x^2-\frac {1}{2} 3^{2/5} \left (1+\sqrt {5}\right ) x+3^{4/5}\right )+\frac {2}{3} x^3 \log (8) \log \left (x^5+9\right )+2 \sqrt [5]{3} \log (8) \log \left (x+3^{2/5}\right )-6 \sqrt [5]{3} \log (2) \log \left (x+3^{2/5}\right )}{\log ^2(2)}\) |
Input:
Int[(-10*x^7*Log[2] + (45*x^4 + 5*x^9)*Log[2]^2 + (10*x^5 + (-54*x^2 - 6*x ^7)*Log[2])*Log[9 + x^5] + (9 + x^5)*Log[9 + x^5]^2)/((9 + x^5)*Log[2]^2), x]
Output:
-(((10*x^3*Log[2])/3 + 3*3^(1/5)*Sqrt[2*(5 - Sqrt[5])]*ArcTan[Sqrt[(5 - 2* Sqrt[5])/5] + (2*Sqrt[2/(5 + Sqrt[5])]*x)/3^(2/5)]*Log[2] + 3*3^(1/5)*Sqrt [2*(5 + Sqrt[5])]*ArcTan[Sqrt[(5 + 2*Sqrt[5])/5] - (Sqrt[(2*(5 + Sqrt[5])) /5]*x)/3^(2/5)]*Log[2] - x^5*Log[2]^2 - (10*x^3*Log[8])/9 - 3^(1/5)*Sqrt[2 *(5 - Sqrt[5])]*ArcTan[Sqrt[(5 - 2*Sqrt[5])/5] + (2*Sqrt[2/(5 + Sqrt[5])]* x)/3^(2/5)]*Log[8] - 3^(1/5)*Sqrt[2*(5 + Sqrt[5])]*ArcTan[Sqrt[(5 + 2*Sqrt [5])/5] - (Sqrt[(2*(5 + Sqrt[5]))/5]*x)/3^(2/5)]*Log[8] - 6*3^(1/5)*Log[2] *Log[3^(2/5) + x] + 2*3^(1/5)*Log[8]*Log[3^(2/5) + x] + (3*3^(1/5)*(1 + Sq rt[5])*Log[2]*Log[3^(4/5) - (3^(2/5)*(1 - Sqrt[5])*x)/2 + x^2])/2 - (3^(1/ 5)*(1 + Sqrt[5])*Log[8]*Log[3^(4/5) - (3^(2/5)*(1 - Sqrt[5])*x)/2 + x^2])/ 2 + (3*3^(1/5)*(1 - Sqrt[5])*Log[2]*Log[3^(4/5) - (3^(2/5)*(1 + Sqrt[5])*x )/2 + x^2])/2 - (3^(1/5)*(1 - Sqrt[5])*Log[8]*Log[3^(4/5) - (3^(2/5)*(1 + Sqrt[5])*x)/2 + x^2])/2 + (2*x^3*Log[8]*Log[9 + x^5])/3 - x*Log[9 + x^5]^2 )/Log[2]^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_)/((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 = 4.19 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.62
method | result | size |
risch | \(x^{5}-\frac {2 x^{3} \ln \left (x^{5}+9\right )}{\ln \left (2\right )}+\frac {x \ln \left (x^{5}+9\right )^{2}}{\ln \left (2\right )^{2}}\) | \(34\) |
parallelrisch | \(\frac {x^{5} \ln \left (2\right )^{2}-2 \ln \left (2\right ) x^{3} \ln \left (x^{5}+9\right )+x \ln \left (x^{5}+9\right )^{2}-18 \ln \left (2\right )^{2}}{\ln \left (2\right )^{2}}\) | \(44\) |
Input:
int(((x^5+9)*ln(x^5+9)^2+((-6*x^7-54*x^2)*ln(2)+10*x^5)*ln(x^5+9)+(5*x^9+4 5*x^4)*ln(2)^2-10*x^7*ln(2))/(x^5+9)/ln(2)^2,x,method=_RETURNVERBOSE)
Output:
x^5-2/ln(2)*x^3*ln(x^5+9)+1/ln(2)^2*x*ln(x^5+9)^2
Time = 0.08 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x^{5} \log \left (2\right )^{2} - 2 \, x^{3} \log \left (2\right ) \log \left (x^{5} + 9\right ) + x \log \left (x^{5} + 9\right )^{2}}{\log \left (2\right )^{2}} \] Input:
integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9) +(5*x^9+45*x^4)*log(2)^2-10*x^7*log(2))/(x^5+9)/log(2)^2,x, algorithm="fri cas")
Output:
(x^5*log(2)^2 - 2*x^3*log(2)*log(x^5 + 9) + x*log(x^5 + 9)^2)/log(2)^2
Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (15) = 30\).
Time = 0.10 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.52 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=x^{5} - \frac {2 x^{3} \log {\left (x^{5} + 9 \right )}}{\log {\left (2 \right )}} + \frac {x \log {\left (x^{5} + 9 \right )}^{2}}{\log {\left (2 \right )}^{2}} \] Input:
integrate(((x**5+9)*ln(x**5+9)**2+((-6*x**7-54*x**2)*ln(2)+10*x**5)*ln(x** 5+9)+(5*x**9+45*x**4)*ln(2)**2-10*x**7*ln(2))/(x**5+9)/ln(2)**2,x)
Output:
x**5 - 2*x**3*log(x**5 + 9)/log(2) + x*log(x**5 + 9)**2/log(2)**2
Exception generated. \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9) +(5*x^9+45*x^4)*log(2)^2-10*x^7*log(2))/(x^5+9)/log(2)^2,x, algorithm="max ima")
Output:
Exception raised: RuntimeError >> ECL says: sign: argument cannot be imagi nary; found sqrt(sqrt(5)-5)
Time = 0.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.76 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x^{5} \log \left (2\right )^{2} - 2 \, x^{3} \log \left (2\right ) \log \left (x^{5} + 9\right ) + x \log \left (x^{5} + 9\right )^{2}}{\log \left (2\right )^{2}} \] Input:
integrate(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9) +(5*x^9+45*x^4)*log(2)^2-10*x^7*log(2))/(x^5+9)/log(2)^2,x, algorithm="gia c")
Output:
(x^5*log(2)^2 - 2*x^3*log(2)*log(x^5 + 9) + x*log(x^5 + 9)^2)/log(2)^2
Time = 0.40 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.05 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x\,{\left (\ln \left (x^5+9\right )-x^2\,\ln \left (2\right )\right )}^2}{{\ln \left (2\right )}^2} \] Input:
int(-(log(x^5 + 9)*(log(2)*(54*x^2 + 6*x^7) - 10*x^5) - log(x^5 + 9)^2*(x^ 5 + 9) + 10*x^7*log(2) - log(2)^2*(45*x^4 + 5*x^9))/(log(2)^2*(x^5 + 9)),x )
Output:
(x*(log(x^5 + 9) - x^2*log(2))^2)/log(2)^2
Time = 0.19 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.71 \[ \int \frac {-10 x^7 \log (2)+\left (45 x^4+5 x^9\right ) \log ^2(2)+\left (10 x^5+\left (-54 x^2-6 x^7\right ) \log (2)\right ) \log \left (9+x^5\right )+\left (9+x^5\right ) \log ^2\left (9+x^5\right )}{\left (9+x^5\right ) \log ^2(2)} \, dx=\frac {x \left (\mathrm {log}\left (x^{5}+9\right )^{2}-2 \,\mathrm {log}\left (x^{5}+9\right ) \mathrm {log}\left (2\right ) x^{2}+\mathrm {log}\left (2\right )^{2} x^{4}\right )}{\mathrm {log}\left (2\right )^{2}} \] Input:
int(((x^5+9)*log(x^5+9)^2+((-6*x^7-54*x^2)*log(2)+10*x^5)*log(x^5+9)+(5*x^ 9+45*x^4)*log(2)^2-10*x^7*log(2))/(x^5+9)/log(2)^2,x)
Output:
(x*(log(x**5 + 9)**2 - 2*log(x**5 + 9)*log(2)*x**2 + log(2)**2*x**4))/log( 2)**2