Integrand size = 92, antiderivative size = 26 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=\log \left (\frac {1}{4 \left (-3+\left (-4 x+\frac {x}{e^3}+\log \left (5 e^x\right )\right )^2\right )}\right ) \] Output:
ln(1/(4*(ln(5*exp(x))-4*x+x/exp(3))^2-12))
Leaf count is larger than twice the leaf count of optimal. \(74\) vs. \(2(26)=52\).
Time = 0.34 (sec) , antiderivative size = 74, normalized size of antiderivative = 2.85 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=-\frac {2 \left (-1+3 e^3\right ) \log \left (x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )\right )}{-2+6 e^3} \] Input:
Integrate[(-2*x + 14*E^3*x - 24*E^6*x + (-2*E^3 + 6*E^6)*Log[5*E^x])/(x^2 - 8*E^3*x^2 + E^6*(-3 + 16*x^2) + (2*E^3*x - 8*E^6*x)*Log[5*E^x] + E^6*Log [5*E^x]^2),x]
Output:
(-2*(-1 + 3*E^3)*Log[x^2 - 8*E^3*x^2 + E^6*(-3 + 16*x^2) + (2*E^3*x - 8*E^ 6*x)*Log[5*E^x] + E^6*Log[5*E^x]^2])/(-2 + 6*E^3)
Leaf count is larger than twice the leaf count of optimal. \(59\) vs. \(2(26)=52\).
Time = 0.53 (sec) , antiderivative size = 59, normalized size of antiderivative = 2.27, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.076, Rules used = {6, 6, 6, 7292, 27, 25, 7235}
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.
\(\displaystyle \int \frac {-24 e^6 x+14 e^3 x-2 x+\left (6 e^6-2 e^3\right ) \log \left (5 e^x\right )}{-8 e^3 x^2+x^2+e^6 \left (16 x^2-3\right )+e^6 \log ^2\left (5 e^x\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )} \, dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (14 e^3-2\right ) x-24 e^6 x+\left (6 e^6-2 e^3\right ) \log \left (5 e^x\right )}{-8 e^3 x^2+x^2+e^6 \left (16 x^2-3\right )+e^6 \log ^2\left (5 e^x\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (-2+14 e^3-24 e^6\right ) x+\left (6 e^6-2 e^3\right ) \log \left (5 e^x\right )}{-8 e^3 x^2+x^2+e^6 \left (16 x^2-3\right )+e^6 \log ^2\left (5 e^x\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )}dx\) |
\(\Big \downarrow \) 6 |
\(\displaystyle \int \frac {\left (-2+14 e^3-24 e^6\right ) x+\left (6 e^6-2 e^3\right ) \log \left (5 e^x\right )}{\left (1-8 e^3\right ) x^2+e^6 \left (16 x^2-3\right )+e^6 \log ^2\left (5 e^x\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )}dx\) |
\(\Big \downarrow \) 7292 |
\(\displaystyle \int \frac {2 \left (1-3 e^3\right ) \left (-\left (\left (1-4 e^3\right ) x\right )-e^3 \log \left (5 e^x\right )\right )}{\left (1-8 e^3\right ) x^2+e^6 \left (16 x^2-3\right )+e^6 \log ^2\left (5 e^x\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle 2 \left (1-3 e^3\right ) \int -\frac {\left (1-4 e^3\right ) x+e^3 \log \left (5 e^x\right )}{\left (1-8 e^3\right ) x^2+2 \left (e^3-4 e^6\right ) \log \left (5 e^x\right ) x+e^6 \log ^2\left (5 e^x\right )-e^6 \left (3-16 x^2\right )}dx\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -2 \left (1-3 e^3\right ) \int \frac {\left (1-4 e^3\right ) x+e^3 \log \left (5 e^x\right )}{\left (1-8 e^3\right ) x^2+2 e^3 \left (1-4 e^3\right ) \log \left (5 e^x\right ) x+e^6 \log ^2\left (5 e^x\right )-e^6 \left (3-16 x^2\right )}dx\) |
\(\Big \downarrow \) 7235 |
\(\displaystyle -\log \left (-\left (\left (1-8 e^3\right ) x^2\right )+e^6 \left (3-16 x^2\right )-e^6 \log ^2\left (5 e^x\right )-2 e^3 \left (1-4 e^3\right ) x \log \left (5 e^x\right )\right )\) |
Input:
Int[(-2*x + 14*E^3*x - 24*E^6*x + (-2*E^3 + 6*E^6)*Log[5*E^x])/(x^2 - 8*E^ 3*x^2 + E^6*(-3 + 16*x^2) + (2*E^3*x - 8*E^6*x)*Log[5*E^x] + E^6*Log[5*E^x ]^2),x]
Output:
-Log[-((1 - 8*E^3)*x^2) + E^6*(3 - 16*x^2) - 2*E^3*(1 - 4*E^3)*x*Log[5*E^x ] - E^6*Log[5*E^x]^2]
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]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*L og[RemoveContent[y, x]], x] /; !FalseQ[q]]
Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(24)=48\).
Time = 0.33 (sec) , antiderivative size = 64, normalized size of antiderivative = 2.46
method | result | size |
norman | \(-\ln \left ({\mathrm e}^{6} \ln \left (5 \,{\mathrm e}^{x}\right )^{2}-8 \ln \left (5 \,{\mathrm e}^{x}\right ) x \,{\mathrm e}^{6}+16 x^{2} {\mathrm e}^{6}+2 \ln \left (5 \,{\mathrm e}^{x}\right ) x \,{\mathrm e}^{3}-8 x^{2} {\mathrm e}^{3}-3 \,{\mathrm e}^{6}+x^{2}\right )\) | \(64\) |
risch | \(-\ln \left (\ln \left (5\right )^{2}+2 \ln \left ({\mathrm e}^{x}\right ) \ln \left (5\right )+2 x \ln \left (5\right ) {\mathrm e}^{-3}-8 x \ln \left (5\right )-8 x \ln \left ({\mathrm e}^{x}\right )-8 x^{2} {\mathrm e}^{-3}+16 x^{2}+2 \ln \left ({\mathrm e}^{x}\right ) {\mathrm e}^{-3} x +{\mathrm e}^{-6} x^{2}-3+\ln \left ({\mathrm e}^{x}\right )^{2}\right )\) | \(66\) |
parallelrisch | \(-\ln \left (\frac {{\mathrm e}^{6} \ln \left (5 \,{\mathrm e}^{x}\right )^{2}-8 \ln \left (5 \,{\mathrm e}^{x}\right ) x \,{\mathrm e}^{6}+16 x^{2} {\mathrm e}^{6}+2 \ln \left (5 \,{\mathrm e}^{x}\right ) x \,{\mathrm e}^{3}-8 x^{2} {\mathrm e}^{3}-3 \,{\mathrm e}^{6}+x^{2}}{16 \,{\mathrm e}^{6}-8 \,{\mathrm e}^{3}+1}\right )\) | \(79\) |
default | \(-\frac {\left (6 \,{\mathrm e}^{3}-2\right ) \ln \left (9 x^{2} {\mathrm e}^{6}-6 \,{\mathrm e}^{6} x \left (\ln \left (5 \,{\mathrm e}^{x}\right )-x \right )+{\mathrm e}^{6} {\left (\ln \left (5 \,{\mathrm e}^{x}\right )-x \right )}^{2}-6 x^{2} {\mathrm e}^{3}+2 \,{\mathrm e}^{3} x \left (\ln \left (5 \,{\mathrm e}^{x}\right )-x \right )-3 \,{\mathrm e}^{6}+x^{2}\right )}{2 \left (3 \,{\mathrm e}^{3}-1\right )}\) | \(90\) |
Input:
int(((6*exp(3)^2-2*exp(3))*ln(5*exp(x))-24*x*exp(3)^2+14*x*exp(3)-2*x)/(ex p(3)^2*ln(5*exp(x))^2+(-8*x*exp(3)^2+2*x*exp(3))*ln(5*exp(x))+(16*x^2-3)*e xp(3)^2-8*x^2*exp(3)+x^2),x,method=_RETURNVERBOSE)
Output:
-ln(exp(3)^2*ln(5*exp(x))^2-8*ln(5*exp(x))*x*exp(3)^2+16*x^2*exp(3)^2+2*ln (5*exp(x))*x*exp(3)-8*x^2*exp(3)-3*exp(3)^2+x^2)
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.09 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=-\log \left (-6 \, x^{2} e^{3} + e^{6} \log \left (5\right )^{2} + x^{2} + 3 \, {\left (3 \, x^{2} - 1\right )} e^{6} - 2 \, {\left (3 \, x e^{6} - x e^{3}\right )} \log \left (5\right )\right ) \] Input:
integrate(((6*exp(3)^2-2*exp(3))*log(5*exp(x))-24*x*exp(3)^2+14*x*exp(3)-2 *x)/(exp(3)^2*log(5*exp(x))^2+(-8*x*exp(3)^2+2*x*exp(3))*log(5*exp(x))+(16 *x^2-3)*exp(3)^2-8*x^2*exp(3)+x^2),x, algorithm="fricas")
Output:
-log(-6*x^2*e^3 + e^6*log(5)^2 + x^2 + 3*(3*x^2 - 1)*e^6 - 2*(3*x*e^6 - x* e^3)*log(5))
Leaf count of result is larger than twice the leaf count of optimal. 49 vs. \(2 (22) = 44\).
Time = 0.29 (sec) , antiderivative size = 49, normalized size of antiderivative = 1.88 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=- \log {\left (x^{2} \left (- 6 e^{3} + 1 + 9 e^{6}\right ) + x \left (- 6 e^{6} \log {\left (5 \right )} + 2 e^{3} \log {\left (5 \right )}\right ) - 3 e^{6} + e^{6} \log {\left (5 \right )}^{2} \right )} \] Input:
integrate(((6*exp(3)**2-2*exp(3))*ln(5*exp(x))-24*x*exp(3)**2+14*x*exp(3)- 2*x)/(exp(3)**2*ln(5*exp(x))**2+(-8*x*exp(3)**2+2*x*exp(3))*ln(5*exp(x))+( 16*x**2-3)*exp(3)**2-8*x**2*exp(3)+x**2),x)
Output:
-log(x**2*(-6*exp(3) + 1 + 9*exp(6)) + x*(-6*exp(6)*log(5) + 2*exp(3)*log( 5)) - 3*exp(6) + exp(6)*log(5)**2)
Leaf count of result is larger than twice the leaf count of optimal. 1021 vs. \(2 (22) = 44\).
Time = 0.16 (sec) , antiderivative size = 1021, normalized size of antiderivative = 39.27 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=\text {Too large to display} \] Input:
integrate(((6*exp(3)^2-2*exp(3))*log(5*exp(x))-24*x*exp(3)^2+14*x*exp(3)-2 *x)/(exp(3)^2*log(5*exp(x))^2+(-8*x*exp(3)^2+2*x*exp(3))*log(5*exp(x))+(16 *x^2-3)*exp(3)^2-8*x^2*exp(3)+x^2),x, algorithm="maxima")
Output:
sqrt(3)*e^3*log(5*e^x)*log(-(sqrt(3)*(3*e^3 - 1)*e^3 - x*(9*e^6 - 6*e^3 + 1) + 3*e^6*log(5) - e^3*log(5))/(sqrt(3)*(3*e^3 - 1)*e^3 + x*(9*e^6 - 6*e^ 3 + 1) - 3*e^6*log(5) + e^3*log(5)))/(3*e^3 - 1) - 4*(sqrt(3)*log(5)*log(- (sqrt(3)*(3*e^3 - 1)*e^3 - x*(9*e^6 - 6*e^3 + 1) + 3*e^6*log(5) - e^3*log( 5))/(sqrt(3)*(3*e^3 - 1)*e^3 + x*(9*e^6 - 6*e^3 + 1) - 3*e^6*log(5) + e^3* log(5)))/(3*e^3 - 1)^2 + 3*log(x^2*(9*e^6 - 6*e^3 + 1) - 2*(3*e^6*log(5) - e^3*log(5))*x + (log(5)^2 - 3)*e^6)/(9*e^6 - 6*e^3 + 1))*e^6 + 7/3*(sqrt( 3)*log(5)*log(-(sqrt(3)*(3*e^3 - 1)*e^3 - x*(9*e^6 - 6*e^3 + 1) + 3*e^6*lo g(5) - e^3*log(5))/(sqrt(3)*(3*e^3 - 1)*e^3 + x*(9*e^6 - 6*e^3 + 1) - 3*e^ 6*log(5) + e^3*log(5)))/(3*e^3 - 1)^2 + 3*log(x^2*(9*e^6 - 6*e^3 + 1) - 2* (3*e^6*log(5) - e^3*log(5))*x + (log(5)^2 - 3)*e^6)/(9*e^6 - 6*e^3 + 1))*e ^3 - sqrt(3)*(x*log(-(sqrt(3)*(3*e^3 - 1)*e^3 - x*(9*e^6 - 6*e^3 + 1) + 3* e^6*log(5) - e^3*log(5))/(sqrt(3)*(3*e^3 - 1)*e^3 + x*(9*e^6 - 6*e^3 + 1) - 3*e^6*log(5) + e^3*log(5))) + (e^3*log(5) - sqrt(3)*e^3)*log(x*(3*e^3 - 1) - e^3*log(5) + sqrt(3)*e^3)/(3*e^3 - 1) - (e^3*log(5) + sqrt(3)*e^3)*lo g(x*(3*e^3 - 1) - e^3*log(5) - sqrt(3)*e^3)/(3*e^3 - 1))*e^3/(3*e^3 - 1) - 1/3*sqrt(3)*log(5*e^x)*log(-(sqrt(3)*(3*e^3 - 1)*e^3 - x*(9*e^6 - 6*e^3 + 1) + 3*e^6*log(5) - e^3*log(5))/(sqrt(3)*(3*e^3 - 1)*e^3 + x*(9*e^6 - 6*e ^3 + 1) - 3*e^6*log(5) + e^3*log(5)))/(3*e^3 - 1) + 1/3*sqrt(3)*(x*log(-(s qrt(3)*(3*e^3 - 1)*e^3 - x*(9*e^6 - 6*e^3 + 1) + 3*e^6*log(5) - e^3*log...
Leaf count of result is larger than twice the leaf count of optimal. 47 vs. \(2 (22) = 44\).
Time = 0.13 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.81 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=-\log \left ({\left | 9 \, x^{2} e^{6} - 6 \, x^{2} e^{3} - 6 \, x e^{6} \log \left (5\right ) + 2 \, x e^{3} \log \left (5\right ) + e^{6} \log \left (5\right )^{2} + x^{2} - 3 \, e^{6} \right |}\right ) \] Input:
integrate(((6*exp(3)^2-2*exp(3))*log(5*exp(x))-24*x*exp(3)^2+14*x*exp(3)-2 *x)/(exp(3)^2*log(5*exp(x))^2+(-8*x*exp(3)^2+2*x*exp(3))*log(5*exp(x))+(16 *x^2-3)*exp(3)^2-8*x^2*exp(3)+x^2),x, algorithm="giac")
Output:
-log(abs(9*x^2*e^6 - 6*x^2*e^3 - 6*x*e^6*log(5) + 2*x*e^3*log(5) + e^6*log (5)^2 + x^2 - 3*e^6))
Time = 6.60 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.85 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=-\ln \left (3\,{\mathrm {e}}^6-{\mathrm {e}}^6\,{\ln \left (5\right )}^2+6\,x^2\,{\mathrm {e}}^3-9\,x^2\,{\mathrm {e}}^6-x^2+2\,x\,{\mathrm {e}}^3\,\ln \left (5\right )\,\left (3\,{\mathrm {e}}^3-1\right )\right ) \] Input:
int(-(2*x - 14*x*exp(3) + 24*x*exp(6) + log(5*exp(x))*(2*exp(3) - 6*exp(6) ))/(exp(6)*(16*x^2 - 3) - 8*x^2*exp(3) + exp(6)*log(5*exp(x))^2 + x^2 + lo g(5*exp(x))*(2*x*exp(3) - 8*x*exp(6))),x)
Output:
-log(3*exp(6) - exp(6)*log(5)^2 + 6*x^2*exp(3) - 9*x^2*exp(6) - x^2 + 2*x* exp(3)*log(5)*(3*exp(3) - 1))
Time = 0.15 (sec) , antiderivative size = 56, normalized size of antiderivative = 2.15 \[ \int \frac {-2 x+14 e^3 x-24 e^6 x+\left (-2 e^3+6 e^6\right ) \log \left (5 e^x\right )}{x^2-8 e^3 x^2+e^6 \left (-3+16 x^2\right )+\left (2 e^3 x-8 e^6 x\right ) \log \left (5 e^x\right )+e^6 \log ^2\left (5 e^x\right )} \, dx=-\mathrm {log}\left (-\sqrt {3}\, e^{3}+\mathrm {log}\left (5 e^{x}\right ) e^{3}-4 e^{3} x +x \right )-\mathrm {log}\left (\sqrt {3}\, e^{3}+\mathrm {log}\left (5 e^{x}\right ) e^{3}-4 e^{3} x +x \right ) \] Input:
int(((6*exp(3)^2-2*exp(3))*log(5*exp(x))-24*x*exp(3)^2+14*x*exp(3)-2*x)/(e xp(3)^2*log(5*exp(x))^2+(-8*x*exp(3)^2+2*x*exp(3))*log(5*exp(x))+(16*x^2-3 )*exp(3)^2-8*x^2*exp(3)+x^2),x)
Output:
- (log( - sqrt(3)*e**3 + log(5*e**x)*e**3 - 4*e**3*x + x) + log(sqrt(3)*e **3 + log(5*e**x)*e**3 - 4*e**3*x + x))