Integrand size = 65, antiderivative size = 22 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=4 \left (36 x^2 (-x+x \log (4))^2+\log (5)\right )^2 \] Output:
2*(ln(5)+16*x^2*(3*x*ln(2)-3/2*x)^2)*(2*ln(5)+32*x^2*(3*x*ln(2)-3/2*x)^2)
Time = 0.01 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.41 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=1152 (-1+\log (4))^2 \left (\frac {9}{2} x^8 (-1+\log (4))^2+\frac {1}{4} x^4 \log (5)\right ) \] Input:
Integrate[41472*x^7 - 165888*x^7*Log[4] + 248832*x^7*Log[4]^2 - 165888*x^7 *Log[4]^3 + 41472*x^7*Log[4]^4 + (1152*x^3 - 2304*x^3*Log[4] + 1152*x^3*Lo g[4]^2)*Log[5],x]
Output:
1152*(-1 + Log[4])^2*((9*x^8*(-1 + Log[4])^2)/2 + (x^4*Log[5])/4)
Time = 0.18 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {6, 6, 6, 6, 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.
\(\displaystyle \int \left (41472 x^7+41472 x^7 \log ^4(4)-165888 x^7 \log ^3(4)+248832 x^7 \log ^2(4)-165888 x^7 \log (4)+\log (5) \left (1152 x^3+1152 x^3 \log ^2(4)-2304 x^3 \log (4)\right )\right ) \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (41472 x^7 \log ^4(4)-165888 x^7 \log ^3(4)+248832 x^7 \log ^2(4)+x^7 (41472-165888 \log (4))+\log (5) \left (1152 x^3+1152 x^3 \log ^2(4)-2304 x^3 \log (4)\right )\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (41472 x^7 \log ^4(4)-165888 x^7 \log ^3(4)+x^7 \left (41472+248832 \log ^2(4)-165888 \log (4)\right )+\log (5) \left (1152 x^3+1152 x^3 \log ^2(4)-2304 x^3 \log (4)\right )\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (x^7 \left (41472+248832 \log ^2(4)-165888 \log (4)\right )+x^7 \left (41472 \log ^4(4)-165888 \log ^3(4)\right )+\log (5) \left (1152 x^3+1152 x^3 \log ^2(4)-2304 x^3 \log (4)\right )\right )dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \left (x^7 \left (41472+41472 \log ^4(4)-165888 \log ^3(4)+248832 \log ^2(4)-165888 \log (4)\right )+\log (5) \left (1152 x^3+1152 x^3 \log ^2(4)-2304 x^3 \log (4)\right )\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 5184 x^8 (1-\log (4))^4+288 x^4 (1-\log (4))^2 \log (5)\) |
Input:
Int[41472*x^7 - 165888*x^7*Log[4] + 248832*x^7*Log[4]^2 - 165888*x^7*Log[4 ]^3 + 41472*x^7*Log[4]^4 + (1152*x^3 - 2304*x^3*Log[4] + 1152*x^3*Log[4]^2 )*Log[5],x]
Output:
5184*x^8*(1 - Log[4])^4 + 288*x^4*(1 - Log[4])^2*Log[5]
Int[(u_.)*((v_.) + (a_.)*(Fx_) + (b_.)*(Fx_))^(p_.), x_Symbol] :> Int[u*(v + (a + b)*Fx)^p, x] /; FreeQ[{a, b}, x] && !FreeQ[Fx, x]
Time = 0.20 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.91
method | result | size |
gosper | \(288 \left (4 \ln \left (2\right )^{2}-4 \ln \left (2\right )+1\right ) \left (72 x^{4} \ln \left (2\right )^{2}-72 x^{4} \ln \left (2\right )+18 x^{4}+\ln \left (5\right )\right ) x^{4}\) | \(42\) |
default | \(\frac {4 \left (2 \ln \left (2\right )-1\right )^{2} \left (144 x^{4} \ln \left (2\right )^{2}-144 x^{4} \ln \left (2\right )+36 x^{4}+\ln \left (5\right )\right )^{2}}{4 \ln \left (2\right )^{2}-4 \ln \left (2\right )+1}\) | \(51\) |
norman | \(\left (1152 \ln \left (5\right ) \ln \left (2\right )^{2}-1152 \ln \left (2\right ) \ln \left (5\right )+288 \ln \left (5\right )\right ) x^{4}+\left (82944 \ln \left (2\right )^{4}-165888 \ln \left (2\right )^{3}+124416 \ln \left (2\right )^{2}-41472 \ln \left (2\right )+5184\right ) x^{8}\) | \(53\) |
risch | \(82944 \ln \left (2\right )^{4} x^{8}-165888 \ln \left (2\right )^{3} x^{8}+124416 \ln \left (2\right )^{2} x^{8}-41472 \ln \left (2\right ) x^{8}+5184 x^{8}+1152 x^{4} \ln \left (5\right ) \ln \left (2\right )^{2}-1152 x^{4} \ln \left (5\right ) \ln \left (2\right )+288 x^{4} \ln \left (5\right )\) | \(68\) |
parallelrisch | \(82944 \ln \left (2\right )^{4} x^{8}-165888 \ln \left (2\right )^{3} x^{8}+124416 \ln \left (2\right )^{2} x^{8}-41472 \ln \left (2\right ) x^{8}+5184 x^{8}+1152 x^{4} \ln \left (5\right ) \ln \left (2\right )^{2}-1152 x^{4} \ln \left (5\right ) \ln \left (2\right )+288 x^{4} \ln \left (5\right )\) | \(68\) |
parts | \(82944 \ln \left (2\right )^{4} x^{8}-165888 \ln \left (2\right )^{3} x^{8}+124416 \ln \left (2\right )^{2} x^{8}-41472 \ln \left (2\right ) x^{8}+5184 x^{8}+1152 x^{4} \ln \left (5\right ) \ln \left (2\right )^{2}-1152 x^{4} \ln \left (5\right ) \ln \left (2\right )+288 x^{4} \ln \left (5\right )\) | \(68\) |
Input:
int((4608*x^3*ln(2)^2-4608*x^3*ln(2)+1152*x^3)*ln(5)+663552*x^7*ln(2)^4-13 27104*x^7*ln(2)^3+995328*x^7*ln(2)^2-331776*x^7*ln(2)+41472*x^7,x,method=_ RETURNVERBOSE)
Output:
288*(4*ln(2)^2-4*ln(2)+1)*(72*x^4*ln(2)^2-72*x^4*ln(2)+18*x^4+ln(5))*x^4
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
Time = 0.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.91 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=82944 \, x^{8} \log \left (2\right )^{4} - 165888 \, x^{8} \log \left (2\right )^{3} + 124416 \, x^{8} \log \left (2\right )^{2} - 41472 \, x^{8} \log \left (2\right ) + 5184 \, x^{8} + 288 \, {\left (4 \, x^{4} \log \left (2\right )^{2} - 4 \, x^{4} \log \left (2\right ) + x^{4}\right )} \log \left (5\right ) \] Input:
integrate((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*l og(2)^4-1327104*x^7*log(2)^3+995328*x^7*log(2)^2-331776*x^7*log(2)+41472*x ^7,x, algorithm="fricas")
Output:
82944*x^8*log(2)^4 - 165888*x^8*log(2)^3 + 124416*x^8*log(2)^2 - 41472*x^8 *log(2) + 5184*x^8 + 288*(4*x^4*log(2)^2 - 4*x^4*log(2) + x^4)*log(5)
Leaf count of result is larger than twice the leaf count of optimal. 56 vs. \(2 (24) = 48\).
Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.55 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=x^{8} \left (- 165888 \log {\left (2 \right )}^{3} - 41472 \log {\left (2 \right )} + 5184 + 82944 \log {\left (2 \right )}^{4} + 124416 \log {\left (2 \right )}^{2}\right ) + x^{4} \left (- 1152 \log {\left (2 \right )} \log {\left (5 \right )} + 288 \log {\left (5 \right )} + 1152 \log {\left (2 \right )}^{2} \log {\left (5 \right )}\right ) \] Input:
integrate((4608*x**3*ln(2)**2-4608*x**3*ln(2)+1152*x**3)*ln(5)+663552*x**7 *ln(2)**4-1327104*x**7*ln(2)**3+995328*x**7*ln(2)**2-331776*x**7*ln(2)+414 72*x**7,x)
Output:
x**8*(-165888*log(2)**3 - 41472*log(2) + 5184 + 82944*log(2)**4 + 124416*l og(2)**2) + x**4*(-1152*log(2)*log(5) + 288*log(5) + 1152*log(2)**2*log(5) )
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.91 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=82944 \, x^{8} \log \left (2\right )^{4} - 165888 \, x^{8} \log \left (2\right )^{3} + 124416 \, x^{8} \log \left (2\right )^{2} - 41472 \, x^{8} \log \left (2\right ) + 5184 \, x^{8} + 288 \, {\left (4 \, x^{4} \log \left (2\right )^{2} - 4 \, x^{4} \log \left (2\right ) + x^{4}\right )} \log \left (5\right ) \] Input:
integrate((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*l og(2)^4-1327104*x^7*log(2)^3+995328*x^7*log(2)^2-331776*x^7*log(2)+41472*x ^7,x, algorithm="maxima")
Output:
82944*x^8*log(2)^4 - 165888*x^8*log(2)^3 + 124416*x^8*log(2)^2 - 41472*x^8 *log(2) + 5184*x^8 + 288*(4*x^4*log(2)^2 - 4*x^4*log(2) + x^4)*log(5)
Leaf count of result is larger than twice the leaf count of optimal. 64 vs. \(2 (23) = 46\).
Time = 0.11 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.91 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=82944 \, x^{8} \log \left (2\right )^{4} - 165888 \, x^{8} \log \left (2\right )^{3} + 124416 \, x^{8} \log \left (2\right )^{2} - 41472 \, x^{8} \log \left (2\right ) + 5184 \, x^{8} + 288 \, {\left (4 \, x^{4} \log \left (2\right )^{2} - 4 \, x^{4} \log \left (2\right ) + x^{4}\right )} \log \left (5\right ) \] Input:
integrate((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*l og(2)^4-1327104*x^7*log(2)^3+995328*x^7*log(2)^2-331776*x^7*log(2)+41472*x ^7,x, algorithm="giac")
Output:
82944*x^8*log(2)^4 - 165888*x^8*log(2)^3 + 124416*x^8*log(2)^2 - 41472*x^8 *log(2) + 5184*x^8 + 288*(4*x^4*log(2)^2 - 4*x^4*log(2) + x^4)*log(5)
Time = 1.51 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.68 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=288\,x^4\,{\left (2\,\ln \left (2\right )-1\right )}^2\,\left (\ln \left (5\right )+72\,x^4\,{\ln \left (2\right )}^2-72\,x^4\,\ln \left (2\right )+18\,x^4\right ) \] Input:
int(995328*x^7*log(2)^2 - 1327104*x^7*log(2)^3 + 663552*x^7*log(2)^4 + log (5)*(4608*x^3*log(2)^2 - 4608*x^3*log(2) + 1152*x^3) - 331776*x^7*log(2) + 41472*x^7,x)
Output:
288*x^4*(2*log(2) - 1)^2*(log(5) + 72*x^4*log(2)^2 - 72*x^4*log(2) + 18*x^ 4)
Time = 0.19 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.77 \[ \int \left (41472 x^7-165888 x^7 \log (4)+248832 x^7 \log ^2(4)-165888 x^7 \log ^3(4)+41472 x^7 \log ^4(4)+\left (1152 x^3-2304 x^3 \log (4)+1152 x^3 \log ^2(4)\right ) \log (5)\right ) \, dx=288 x^{4} \left (4 \,\mathrm {log}\left (5\right ) \mathrm {log}\left (2\right )^{2}-4 \,\mathrm {log}\left (5\right ) \mathrm {log}\left (2\right )+\mathrm {log}\left (5\right )+288 \mathrm {log}\left (2\right )^{4} x^{4}-576 \mathrm {log}\left (2\right )^{3} x^{4}+432 \mathrm {log}\left (2\right )^{2} x^{4}-144 \,\mathrm {log}\left (2\right ) x^{4}+18 x^{4}\right ) \] Input:
int((4608*x^3*log(2)^2-4608*x^3*log(2)+1152*x^3)*log(5)+663552*x^7*log(2)^ 4-1327104*x^7*log(2)^3+995328*x^7*log(2)^2-331776*x^7*log(2)+41472*x^7,x)
Output:
288*x**4*(4*log(5)*log(2)**2 - 4*log(5)*log(2) + log(5) + 288*log(2)**4*x* *4 - 576*log(2)**3*x**4 + 432*log(2)**2*x**4 - 144*log(2)*x**4 + 18*x**4)