Integrand size = 66, antiderivative size = 26 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=\frac {\log (16)}{x \left (-1+2 \left (4+x-x^2\right )\right ) \log \left (x^4\right )} \]
Time = 0.39 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=-\frac {\log (16)}{x \left (-7-2 x+2 x^2\right ) \log \left (x^4\right )} \]
Integrate[((-28 - 8*x + 8*x^2)*Log[16] + (-7 - 4*x + 6*x^2)*Log[16]*Log[x^ 4])/((49*x^2 + 28*x^3 - 24*x^4 - 8*x^5 + 4*x^6)*Log[x^4]^2),x]
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.
\(\displaystyle \int \frac {\left (8 x^2-8 x-28\right ) \log (16)+\left (6 x^2-4 x-7\right ) \log (16) \log \left (x^4\right )}{\left (4 x^6-8 x^5-24 x^4+28 x^3+49 x^2\right ) \log ^2\left (x^4\right )} \, dx\) |
\(\Big \downarrow \) 2026 |
\(\displaystyle \int \frac {\left (8 x^2-8 x-28\right ) \log (16)+\left (6 x^2-4 x-7\right ) \log (16) \log \left (x^4\right )}{x^2 \left (4 x^4-8 x^3-24 x^2+28 x+49\right ) \log ^2\left (x^4\right )}dx\) |
\(\Big \downarrow \) 2463 |
\(\displaystyle \int \left (\frac {2 \left (\left (8 x^2-8 x-28\right ) \log (16)+\left (6 x^2-4 x-7\right ) \log (16) \log \left (x^4\right )\right )}{15 \sqrt {15} x^2 \left (4 x+2 \sqrt {15}-2\right ) \log ^2\left (x^4\right )}+\frac {2 \left (\left (8 x^2-8 x-28\right ) \log (16)+\left (6 x^2-4 x-7\right ) \log (16) \log \left (x^4\right )\right )}{15 \sqrt {15} \left (-4 x+2 \sqrt {15}+2\right ) x^2 \log ^2\left (x^4\right )}+\frac {4 \left (\left (8 x^2-8 x-28\right ) \log (16)+\left (6 x^2-4 x-7\right ) \log (16) \log \left (x^4\right )\right )}{15 \left (-4 x+2 \sqrt {15}+2\right )^2 x^2 \log ^2\left (x^4\right )}+\frac {4 \left (\left (8 x^2-8 x-28\right ) \log (16)+\left (6 x^2-4 x-7\right ) \log (16) \log \left (x^4\right )\right )}{15 x^2 \left (4 x+2 \sqrt {15}-2\right )^2 \log ^2\left (x^4\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {4}{15} \log (16) \int \frac {-x-\frac {7}{1+\sqrt {15}}}{\left (-2 x+\sqrt {15}+1\right ) x^2 \log ^2\left (x^4\right )}dx+\frac {4}{15} \log (16) \int \frac {x-\frac {7}{-1+\sqrt {15}}}{x^2 \left (2 x+\sqrt {15}-1\right ) \log ^2\left (x^4\right )}dx+\frac {1}{15} \log (16) \int \frac {6 x^2-4 x-7}{\left (-2 x+\sqrt {15}+1\right )^2 x^2 \log \left (x^4\right )}dx+\frac {\log (16) \int \frac {6 x^2-4 x-7}{\left (-2 x+\sqrt {15}+1\right ) x^2 \log \left (x^4\right )}dx}{15 \sqrt {15}}+\frac {1}{15} \log (16) \int \frac {6 x^2-4 x-7}{x^2 \left (2 x+\sqrt {15}-1\right )^2 \log \left (x^4\right )}dx+\frac {\log (16) \int \frac {6 x^2-4 x-7}{x^2 \left (2 x+\sqrt {15}-1\right ) \log \left (x^4\right )}dx}{15 \sqrt {15}}+\frac {7 \sqrt [4]{x^4} \log (16) \operatorname {ExpIntegralEi}\left (-\frac {1}{4} \log \left (x^4\right )\right )}{60 \sqrt {15} \left (1+\sqrt {15}\right ) x}-\frac {7 \sqrt [4]{x^4} \log (16) \operatorname {ExpIntegralEi}\left (-\frac {1}{4} \log \left (x^4\right )\right )}{60 \sqrt {15} \left (1-\sqrt {15}\right ) x}+\frac {7 \log (16)}{15 \sqrt {15} \left (1+\sqrt {15}\right ) x \log \left (x^4\right )}-\frac {7 \log (16)}{15 \sqrt {15} \left (1-\sqrt {15}\right ) x \log \left (x^4\right )}\) |
Int[((-28 - 8*x + 8*x^2)*Log[16] + (-7 - 4*x + 6*x^2)*Log[16]*Log[x^4])/(( 49*x^2 + 28*x^3 - 24*x^4 - 8*x^5 + 4*x^6)*Log[x^4]^2),x]
3.28.39.3.1 Defintions of rubi rules used
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_.)*(Px_)^(p_), x_Symbol] :> With[{Qx = Factor[Px]}, Int[ExpandIntegr and[u, Qx^p, x], x] /; !SumQ[NonfreeFactors[Qx, x]]] /; PolyQ[Px, x] && Gt Q[Expon[Px, x], 2] && !BinomialQ[Px, x] && !TrinomialQ[Px, x] && ILtQ[p, 0]
Time = 7.23 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00
method | result | size |
norman | \(-\frac {4 \ln \left (2\right )}{x \left (2 x^{2}-2 x -7\right ) \ln \left (x^{4}\right )}\) | \(26\) |
risch | \(-\frac {4 \ln \left (2\right )}{x \left (2 x^{2}-2 x -7\right ) \ln \left (x^{4}\right )}\) | \(26\) |
parallelrisch | \(-\frac {4 \ln \left (2\right )}{x \left (2 x^{2}-2 x -7\right ) \ln \left (x^{4}\right )}\) | \(26\) |
int((4*(6*x^2-4*x-7)*ln(2)*ln(x^4)+4*(8*x^2-8*x-28)*ln(2))/(4*x^6-8*x^5-24 *x^4+28*x^3+49*x^2)/ln(x^4)^2,x,method=_RETURNVERBOSE)
Time = 0.25 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.00 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=-\frac {4 \, \log \left (2\right )}{{\left (2 \, x^{3} - 2 \, x^{2} - 7 \, x\right )} \log \left (x^{4}\right )} \]
integrate((4*(6*x^2-4*x-7)*log(2)*log(x^4)+4*(8*x^2-8*x-28)*log(2))/(4*x^6 -8*x^5-24*x^4+28*x^3+49*x^2)/log(x^4)^2,x, algorithm=\
Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=- \frac {4 \log {\left (2 \right )}}{\left (2 x^{3} - 2 x^{2} - 7 x\right ) \log {\left (x^{4} \right )}} \]
integrate((4*(6*x**2-4*x-7)*ln(2)*ln(x**4)+4*(8*x**2-8*x-28)*ln(2))/(4*x** 6-8*x**5-24*x**4+28*x**3+49*x**2)/ln(x**4)**2,x)
Time = 0.32 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.92 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=-\frac {\log \left (2\right )}{{\left (2 \, x^{3} - 2 \, x^{2} - 7 \, x\right )} \log \left (x\right )} \]
integrate((4*(6*x^2-4*x-7)*log(2)*log(x^4)+4*(8*x^2-8*x-28)*log(2))/(4*x^6 -8*x^5-24*x^4+28*x^3+49*x^2)/log(x^4)^2,x, algorithm=\
Time = 0.28 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.23 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=-\frac {4 \, \log \left (2\right )}{2 \, x^{3} \log \left (x^{4}\right ) - 2 \, x^{2} \log \left (x^{4}\right ) - 7 \, x \log \left (x^{4}\right )} \]
integrate((4*(6*x^2-4*x-7)*log(2)*log(x^4)+4*(8*x^2-8*x-28)*log(2))/(4*x^6 -8*x^5-24*x^4+28*x^3+49*x^2)/log(x^4)^2,x, algorithm=\
Time = 9.52 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.96 \[ \int \frac {\left (-28-8 x+8 x^2\right ) \log (16)+\left (-7-4 x+6 x^2\right ) \log (16) \log \left (x^4\right )}{\left (49 x^2+28 x^3-24 x^4-8 x^5+4 x^6\right ) \log ^2\left (x^4\right )} \, dx=\frac {4\,\ln \left (2\right )}{x\,\ln \left (x^4\right )\,\left (-2\,x^2+2\,x+7\right )} \]