Integrand size = 76, antiderivative size = 19 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=x^2 \left (5+x^4\right ) \left (x+\log \left (\frac {1}{x^2}+2 x\right )\right ) \] Output:
x^2*(x+ln(2*x+1/x^2))*(x^4+5)
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.79 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=5 x^3+x^7+5 x^2 \log \left (\frac {1}{x^2}+2 x\right )+x^6 \log \left (\frac {1}{x^2}+2 x\right ) \] Input:
Integrate[(-10*x + 15*x^2 + 10*x^4 + 28*x^5 + 7*x^6 + 2*x^8 + 14*x^9 + (10 *x + 20*x^4 + 6*x^5 + 12*x^8)*Log[(1 + 2*x^3)/x^2])/(1 + 2*x^3),x]
Output:
5*x^3 + x^7 + 5*x^2*Log[x^(-2) + 2*x] + x^6*Log[x^(-2) + 2*x]
Leaf count is larger than twice the leaf count of optimal. \(129\) vs. \(2(19)=38\).
Time = 0.78 (sec) , antiderivative size = 129, normalized size of antiderivative = 6.79, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {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.
\(\displaystyle \int \frac {14 x^9+2 x^8+7 x^6+28 x^5+10 x^4+15 x^2+\left (12 x^8+6 x^5+20 x^4+10 x\right ) \log \left (\frac {2 x^3+1}{x^2}\right )-10 x}{2 x^3+1} \, dx\) |
\(\Big \downarrow \) 7276 |
\(\displaystyle \int \left (-\frac {10 x}{2 x^3+1}+\frac {14 x^9}{2 x^3+1}+\frac {2 x^8}{2 x^3+1}+\frac {7 x^6}{2 x^3+1}+\frac {28 x^5}{2 x^3+1}+\frac {10 x^4}{2 x^3+1}+\frac {15 x^2}{2 x^3+1}+2 \left (3 x^4+5\right ) x \log \left (\frac {2 x^3+1}{x^2}\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5 \sqrt {3} \arctan \left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3}}+\frac {5 \sqrt [3]{2} \arctan \left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {5 \arctan \left (\frac {1-2 \sqrt [3]{2} x}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3}}+x^7+5 x^3+5 x^2 \log \left (\frac {2 x^3+1}{x^2}\right )+x^6 \log \left (\frac {2 x^3+1}{x^2}\right )\) |
Input:
Int[(-10*x + 15*x^2 + 10*x^4 + 28*x^5 + 7*x^6 + 2*x^8 + 14*x^9 + (10*x + 2 0*x^4 + 6*x^5 + 12*x^8)*Log[(1 + 2*x^3)/x^2])/(1 + 2*x^3),x]
Output:
5*x^3 + x^7 + (5*ArcTan[(1 - 2*2^(1/3)*x)/Sqrt[3]])/(2^(2/3)*Sqrt[3]) + (5 *2^(1/3)*ArcTan[(1 - 2*2^(1/3)*x)/Sqrt[3]])/Sqrt[3] - (5*Sqrt[3]*ArcTan[(1 - 2*2^(1/3)*x)/Sqrt[3]])/2^(2/3) + 5*x^2*Log[(1 + 2*x^3)/x^2] + x^6*Log[( 1 + 2*x^3)/x^2]
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 = 2.32 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.68
method | result | size |
risch | \(\left (x^{6}+5 x^{2}\right ) \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+x^{7}+5 x^{3}\) | \(32\) |
default | \(x^{7}+5 x^{3}+5 x^{2} \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+\ln \left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}\) | \(43\) |
norman | \(x^{7}+5 x^{3}+5 x^{2} \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+\ln \left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}\) | \(43\) |
parts | \(x^{7}+5 x^{3}+5 x^{2} \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+\ln \left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}\) | \(43\) |
parallelrisch | \(x^{7}+\ln \left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}+5 x^{3}+5 x^{2} \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )-\frac {5}{4}\) | \(44\) |
orering | \(\frac {\left (144 x^{17}+132 x^{16}+144 x^{14}+948 x^{13}+840 x^{12}+36 x^{11}+594 x^{10}+1320 x^{9}+900 x^{8}-60 x^{7}+680 x^{6}+900 x^{5}-75 x^{4}+200 x^{3}-450 x^{2}-25\right ) \left (\left (12 x^{8}+6 x^{5}+20 x^{4}+10 x \right ) \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+14 x^{9}+2 x^{8}+7 x^{6}+28 x^{5}+10 x^{4}+15 x^{2}-10 x \right )}{6 x \left (84 x^{15}+72 x^{14}+84 x^{12}+178 x^{11}+80 x^{10}+21 x^{9}+124 x^{8}+620 x^{7}+200 x^{6}+40 x^{5}+260 x^{4}+350 x^{3}+75 x -100\right ) \left (2 x^{3}+1\right )}-\frac {\left (6 x^{14}+6 x^{13}+3 x^{11}+54 x^{10}+60 x^{9}+90 x^{6}+150 x^{5}-15 x^{4}+25 x^{3}-150 x^{2}-25\right ) \left (2 x^{3}+1\right ) \left (\frac {\left (96 x^{7}+30 x^{4}+80 x^{3}+10\right ) \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+\frac {\left (12 x^{8}+6 x^{5}+20 x^{4}+10 x \right ) \left (6-\frac {2 \left (2 x^{3}+1\right )}{x^{3}}\right ) x^{2}}{2 x^{3}+1}+126 x^{8}+16 x^{7}+42 x^{5}+140 x^{4}+40 x^{3}+30 x -10}{2 x^{3}+1}-\frac {6 \left (\left (12 x^{8}+6 x^{5}+20 x^{4}+10 x \right ) \ln \left (\frac {2 x^{3}+1}{x^{2}}\right )+14 x^{9}+2 x^{8}+7 x^{6}+28 x^{5}+10 x^{4}+15 x^{2}-10 x \right ) x^{2}}{\left (2 x^{3}+1\right )^{2}}\right )}{6 \left (84 x^{15}+72 x^{14}+84 x^{12}+178 x^{11}+80 x^{10}+21 x^{9}+124 x^{8}+620 x^{7}+200 x^{6}+40 x^{5}+260 x^{4}+350 x^{3}+75 x -100\right )}\) | \(551\) |
Input:
int(((12*x^8+6*x^5+20*x^4+10*x)*ln((2*x^3+1)/x^2)+14*x^9+2*x^8+7*x^6+28*x^ 5+10*x^4+15*x^2-10*x)/(2*x^3+1),x,method=_RETURNVERBOSE)
Output:
(x^6+5*x^2)*ln((2*x^3+1)/x^2)+x^7+5*x^3
Time = 0.07 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=x^{7} + 5 \, x^{3} + {\left (x^{6} + 5 \, x^{2}\right )} \log \left (\frac {2 \, x^{3} + 1}{x^{2}}\right ) \] Input:
integrate(((12*x^8+6*x^5+20*x^4+10*x)*log((2*x^3+1)/x^2)+14*x^9+2*x^8+7*x^ 6+28*x^5+10*x^4+15*x^2-10*x)/(2*x^3+1),x, algorithm="fricas")
Output:
x^7 + 5*x^3 + (x^6 + 5*x^2)*log((2*x^3 + 1)/x^2)
Time = 0.08 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.42 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=x^{7} + 5 x^{3} + \left (x^{6} + 5 x^{2}\right ) \log {\left (\frac {2 x^{3} + 1}{x^{2}} \right )} \] Input:
integrate(((12*x**8+6*x**5+20*x**4+10*x)*ln((2*x**3+1)/x**2)+14*x**9+2*x** 8+7*x**6+28*x**5+10*x**4+15*x**2-10*x)/(2*x**3+1),x)
Output:
x**7 + 5*x**3 + (x**6 + 5*x**2)*log((2*x**3 + 1)/x**2)
Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (19) = 38\).
Time = 0.60 (sec) , antiderivative size = 136, normalized size of antiderivative = 7.16 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=x^{7} + 5 \, x^{3} + {\left (x^{6} + 5 \, x^{2}\right )} \log \left (2 \, x^{3} + 1\right ) + \frac {1}{4} \, {\left (5 \cdot 2^{\frac {1}{3}} - 1\right )} \log \left (2^{\frac {2}{3}} x^{2} - 2^{\frac {1}{3}} x + 1\right ) - \frac {1}{4} \, {\left (10 \cdot 2^{\frac {1}{3}} + 1\right )} \log \left (\frac {1}{2} \cdot 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x + 1\right )}\right ) - 2 \, {\left (x^{6} + 5 \, x^{2}\right )} \log \left (x\right ) - \frac {5}{4} \cdot 2^{\frac {1}{3}} \log \left (2^{\frac {2}{3}} x^{2} - 2^{\frac {1}{3}} x + 1\right ) + \frac {5}{2} \cdot 2^{\frac {1}{3}} \log \left (\frac {1}{2} \cdot 2^{\frac {2}{3}} {\left (2^{\frac {1}{3}} x + 1\right )}\right ) + \frac {1}{4} \, \log \left (2 \, x^{3} + 1\right ) \] Input:
integrate(((12*x^8+6*x^5+20*x^4+10*x)*log((2*x^3+1)/x^2)+14*x^9+2*x^8+7*x^ 6+28*x^5+10*x^4+15*x^2-10*x)/(2*x^3+1),x, algorithm="maxima")
Output:
x^7 + 5*x^3 + (x^6 + 5*x^2)*log(2*x^3 + 1) + 1/4*(5*2^(1/3) - 1)*log(2^(2/ 3)*x^2 - 2^(1/3)*x + 1) - 1/4*(10*2^(1/3) + 1)*log(1/2*2^(2/3)*(2^(1/3)*x + 1)) - 2*(x^6 + 5*x^2)*log(x) - 5/4*2^(1/3)*log(2^(2/3)*x^2 - 2^(1/3)*x + 1) + 5/2*2^(1/3)*log(1/2*2^(2/3)*(2^(1/3)*x + 1)) + 1/4*log(2*x^3 + 1)
Time = 0.16 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.63 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=x^{7} + 5 \, x^{3} + {\left (x^{6} + 5 \, x^{2}\right )} \log \left (\frac {2 \, x^{3} + 1}{x^{2}}\right ) \] Input:
integrate(((12*x^8+6*x^5+20*x^4+10*x)*log((2*x^3+1)/x^2)+14*x^9+2*x^8+7*x^ 6+28*x^5+10*x^4+15*x^2-10*x)/(2*x^3+1),x, algorithm="giac")
Output:
x^7 + 5*x^3 + (x^6 + 5*x^2)*log((2*x^3 + 1)/x^2)
Time = 1.45 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.21 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=x^2\,\left (x+\ln \left (\frac {2\,x^3+1}{x^2}\right )\right )\,\left (x^4+5\right ) \] Input:
int((log((2*x^3 + 1)/x^2)*(10*x + 20*x^4 + 6*x^5 + 12*x^8) - 10*x + 15*x^2 + 10*x^4 + 28*x^5 + 7*x^6 + 2*x^8 + 14*x^9)/(2*x^3 + 1),x)
Output:
x^2*(x + log((2*x^3 + 1)/x^2))*(x^4 + 5)
Time = 0.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 8.63 \[ \int \frac {-10 x+15 x^2+10 x^4+28 x^5+7 x^6+2 x^8+14 x^9+\left (10 x+20 x^4+6 x^5+12 x^8\right ) \log \left (\frac {1+2 x^3}{x^2}\right )}{1+2 x^3} \, dx=\frac {\left (2^{\frac {2}{3}} \mathrm {log}\left (2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} x +1\right )+2^{\frac {2}{3}} \mathrm {log}\left (2^{\frac {1}{3}} x +1\right )+4 \,2^{\frac {2}{3}} \mathrm {log}\left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{6}+20 \,2^{\frac {2}{3}} \mathrm {log}\left (\frac {2 x^{3}+1}{x^{2}}\right ) x^{2}-2^{\frac {2}{3}} \mathrm {log}\left (\frac {2 x^{3}+1}{x^{2}}\right )-2 \,2^{\frac {2}{3}} \mathrm {log}\left (x \right )+4 \,2^{\frac {2}{3}} x^{7}+20 \,2^{\frac {2}{3}} x^{3}-10 \,\mathrm {log}\left (2^{\frac {2}{3}} x^{2}-2^{\frac {1}{3}} x +1\right )-10 \,\mathrm {log}\left (2^{\frac {1}{3}} x +1\right )+10 \,\mathrm {log}\left (\frac {2 x^{3}+1}{x^{2}}\right )+20 \,\mathrm {log}\left (x \right )\right ) 2^{\frac {1}{3}}}{8} \] Input:
int(((12*x^8+6*x^5+20*x^4+10*x)*log((2*x^3+1)/x^2)+14*x^9+2*x^8+7*x^6+28*x ^5+10*x^4+15*x^2-10*x)/(2*x^3+1),x)
Output:
(2**(2/3)*log(2**(2/3)*x**2 - 2**(1/3)*x + 1) + 2**(2/3)*log(2**(1/3)*x + 1) + 4*2**(2/3)*log((2*x**3 + 1)/x**2)*x**6 + 20*2**(2/3)*log((2*x**3 + 1) /x**2)*x**2 - 2**(2/3)*log((2*x**3 + 1)/x**2) - 2*2**(2/3)*log(x) + 4*2**( 2/3)*x**7 + 20*2**(2/3)*x**3 - 10*log(2**(2/3)*x**2 - 2**(1/3)*x + 1) - 10 *log(2**(1/3)*x + 1) + 10*log((2*x**3 + 1)/x**2) + 20*log(x))/(4*2**(2/3))