Integrand size = 77, antiderivative size = 27 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx=-x+\left (e^2-\frac {e^4}{\log ^2\left (-\frac {1}{e^4}+e^x\right )}\right )^2 \]
Time = 0.09 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx=-x+\frac {e^8}{\left (-4+\log \left (-1+e^{4+x}\right )\right )^4}-\frac {2 e^6}{\left (-4+\log \left (-1+e^{4+x}\right )\right )^2} \]
Integrate[(-4*E^(12 + x) + 4*E^(10 + x)*Log[(-1 + E^(4 + x))/E^4]^2 + (1 - E^(4 + x))*Log[(-1 + E^(4 + x))/E^4]^5)/((-1 + E^(4 + x))*Log[(-1 + E^(4 + x))/E^4]^5),x]
Time = 0.75 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.48, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.039, Rules used = {2720, 7293, 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.
\(\displaystyle \int \frac {-4 e^{x+12}+\left (1-e^{x+4}\right ) \log ^5\left (\frac {e^{x+4}-1}{e^4}\right )+4 e^{x+10} \log ^2\left (\frac {e^{x+4}-1}{e^4}\right )}{\left (e^{x+4}-1\right ) \log ^5\left (\frac {e^{x+4}-1}{e^4}\right )} \, dx\) |
\(\Big \downarrow \) 2720 |
\(\displaystyle \int \frac {e^{-x} \left (4 e^{x+12}-\left (\left (1-e^{x+4}\right ) \log ^5\left (e^x-\frac {1}{e^4}\right )\right )-4 e^{x+10} \log ^2\left (e^x-\frac {1}{e^4}\right )\right )}{\left (1-e^{x+4}\right ) \log ^5\left (e^x-\frac {1}{e^4}\right )}de^x\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-e^{-x}-\frac {4 e^{12}}{\left (e^{x+4}-1\right ) \log ^5\left (e^x-\frac {1}{e^4}\right )}+\frac {4 e^{10}}{\left (e^{x+4}-1\right ) \log ^3\left (e^x-\frac {1}{e^4}\right )}\right )de^x\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e^8}{\log ^4\left (e^x-\frac {1}{e^4}\right )}-\frac {2 e^6}{\log ^2\left (e^x-\frac {1}{e^4}\right )}-\log \left (e^x\right )\) |
Int[(-4*E^(12 + x) + 4*E^(10 + x)*Log[(-1 + E^(4 + x))/E^4]^2 + (1 - E^(4 + x))*Log[(-1 + E^(4 + x))/E^4]^5)/((-1 + E^(4 + x))*Log[(-1 + E^(4 + x))/ E^4]^5),x]
3.23.49.3.1 Defintions of rubi rules used
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Time = 0.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.37
method | result | size |
risch | \(-x +\frac {{\mathrm e}^{6} \left (-2 {\ln \left (\left ({\mathrm e}^{4+x}-1\right ) {\mathrm e}^{-4}\right )}^{2}+{\mathrm e}^{2}\right )}{{\ln \left (\left ({\mathrm e}^{4+x}-1\right ) {\mathrm e}^{-4}\right )}^{4}}\) | \(37\) |
derivativedivides | \(-\ln \left (\left ({\mathrm e}^{x}-{\mathrm e}^{-4}\right ) {\mathrm e}^{4}+1\right )-\frac {2 \,{\mathrm e}^{6}}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{-4}\right )^{2}}+\frac {{\mathrm e}^{8}}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{-4}\right )^{4}}\) | \(54\) |
default | \(-\ln \left (\left ({\mathrm e}^{x}-{\mathrm e}^{-4}\right ) {\mathrm e}^{4}+1\right )-\frac {2 \,{\mathrm e}^{6}}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{-4}\right )^{2}}+\frac {{\mathrm e}^{8}}{\ln \left ({\mathrm e}^{x}-{\mathrm e}^{-4}\right )^{4}}\) | \(54\) |
parallelrisch | \(\frac {-2 \,{\mathrm e}^{6} {\ln \left (\left ({\mathrm e}^{4} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-4}\right )}^{2}-x {\ln \left (\left ({\mathrm e}^{4} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-4}\right )}^{4}+{\mathrm e}^{8}}{{\ln \left (\left ({\mathrm e}^{4} {\mathrm e}^{x}-1\right ) {\mathrm e}^{-4}\right )}^{4}}\) | \(61\) |
int(((-exp(4)*exp(x)+1)*ln((exp(4)*exp(x)-1)/exp(4))^5+4*exp(2)^3*exp(4)*e xp(x)*ln((exp(4)*exp(x)-1)/exp(4))^2-4*exp(2)^4*exp(4)*exp(x))/(exp(4)*exp (x)-1)/ln((exp(4)*exp(x)-1)/exp(4))^5,x,method=_RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 61, normalized size of antiderivative = 2.26 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx=-\frac {x \log \left (-{\left (e^{8} - e^{\left (x + 12\right )}\right )} e^{\left (-12\right )}\right )^{4} + 2 \, e^{6} \log \left (-{\left (e^{8} - e^{\left (x + 12\right )}\right )} e^{\left (-12\right )}\right )^{2} - e^{8}}{\log \left (-{\left (e^{8} - e^{\left (x + 12\right )}\right )} e^{\left (-12\right )}\right )^{4}} \]
integrate(((-exp(4)*exp(x)+1)*log((exp(4)*exp(x)-1)/exp(4))^5+4*exp(2)^3*e xp(4)*exp(x)*log((exp(4)*exp(x)-1)/exp(4))^2-4*exp(2)^4*exp(4)*exp(x))/(ex p(4)*exp(x)-1)/log((exp(4)*exp(x)-1)/exp(4))^5,x, algorithm=\
-(x*log(-(e^8 - e^(x + 12))*e^(-12))^4 + 2*e^6*log(-(e^8 - e^(x + 12))*e^( -12))^2 - e^8)/log(-(e^8 - e^(x + 12))*e^(-12))^4
Time = 0.11 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.44 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx=- x + \frac {- 2 e^{6} \log {\left (\frac {e^{4} e^{x} - 1}{e^{4}} \right )}^{2} + e^{8}}{\log {\left (\frac {e^{4} e^{x} - 1}{e^{4}} \right )}^{4}} \]
integrate(((-exp(4)*exp(x)+1)*ln((exp(4)*exp(x)-1)/exp(4))**5+4*exp(2)**3* exp(4)*exp(x)*ln((exp(4)*exp(x)-1)/exp(4))**2-4*exp(2)**4*exp(4)*exp(x))/( exp(4)*exp(x)-1)/ln((exp(4)*exp(x)-1)/exp(4))**5,x)
-x + (-2*exp(6)*log((exp(4)*exp(x) - 1)*exp(-4))**2 + exp(8))/log((exp(4)* exp(x) - 1)*exp(-4))**4
Leaf count of result is larger than twice the leaf count of optimal. 408 vs. \(2 (24) = 48\).
Time = 0.25 (sec) , antiderivative size = 408, normalized size of antiderivative = 15.11 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx =\text {Too large to display} \]
integrate(((-exp(4)*exp(x)+1)*log((exp(4)*exp(x)-1)/exp(4))^5+4*exp(2)^3*e xp(4)*exp(x)*log((exp(4)*exp(x)-1)/exp(4))^2-4*exp(2)^4*exp(4)*exp(x))/(ex p(4)*exp(x)-1)/log((exp(4)*exp(x)-1)/exp(4))^5,x, algorithm=\
1/4*log(-e^(-4) + e^x)^5/(log(e^(x + 4) - 1)^4 - 16*log(e^(x + 4) - 1)^3 + 96*log(e^(x + 4) - 1)^2 - 256*log(e^(x + 4) - 1) + 256) + 5/12*log(-e^(-4 ) + e^x)^4/(log(e^(x + 4) - 1)^3 - 12*log(e^(x + 4) - 1)^2 + 48*log(e^(x + 4) - 1) - 64) - 1/3*(2*log(-e^(-4) + e^x)/(log(e^(x + 4) - 1)^3 - 12*log( e^(x + 4) - 1)^2 + 48*log(e^(x + 4) - 1) - 64) + 1/(log(e^(x + 4) - 1)^2 - 8*log(e^(x + 4) - 1) + 16))*e^6 + 5/6*(e^4*log(-e^(-4) + e^x)^3/(log(e^(x + 4) - 1)^2 - 8*log(e^(x + 4) - 1) + 16) - 6*e^4*log(-e^(-4) + e^x)*log(l og(e^(x + 4) - 1) - 4) + 6*(log(-e^(-4) + e^x)*log(log(e^(x + 4) - 1) - 4) - log(e^(x + 4) - 1))*e^4 + 3*e^4*log(-e^(-4) + e^x)^2/(log(e^(x + 4) - 1 ) - 4))*e^(-4) - e^6*log(-e^(-4) + e^x)^2/(log(e^(x + 4) - 1)^4 - 16*log(e ^(x + 4) - 1)^3 + 96*log(e^(x + 4) - 1)^2 - 256*log(e^(x + 4) - 1) + 256) - x + e^8/(log(e^(x + 4) - 1)^4 - 16*log(e^(x + 4) - 1)^3 + 96*log(e^(x + 4) - 1)^2 - 256*log(e^(x + 4) - 1) + 256) + log(e^(x + 4) - 1) - 4
Leaf count of result is larger than twice the leaf count of optimal. 147 vs. \(2 (24) = 48\).
Time = 0.35 (sec) , antiderivative size = 147, normalized size of antiderivative = 5.44 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx=-\frac {\log \left (e^{\left (x + 4\right )} - 1\right )^{4} \log \left (e^{\left (x + 4\right )}\right ) - 16 \, \log \left (e^{\left (x + 4\right )} - 1\right )^{3} \log \left (e^{\left (x + 4\right )}\right ) + 2 \, e^{6} \log \left (e^{\left (x + 4\right )} - 1\right )^{2} + 96 \, \log \left (e^{\left (x + 4\right )} - 1\right )^{2} \log \left (e^{\left (x + 4\right )}\right ) - 16 \, e^{6} \log \left (e^{\left (x + 4\right )} - 1\right ) - 256 \, \log \left (e^{\left (x + 4\right )} - 1\right ) \log \left (e^{\left (x + 4\right )}\right ) - e^{8} + 32 \, e^{6} + 256 \, \log \left (e^{\left (x + 4\right )}\right )}{\log \left (e^{\left (x + 4\right )} - 1\right )^{4} - 16 \, \log \left (e^{\left (x + 4\right )} - 1\right )^{3} + 96 \, \log \left (e^{\left (x + 4\right )} - 1\right )^{2} - 256 \, \log \left (e^{\left (x + 4\right )} - 1\right ) + 256} \]
integrate(((-exp(4)*exp(x)+1)*log((exp(4)*exp(x)-1)/exp(4))^5+4*exp(2)^3*e xp(4)*exp(x)*log((exp(4)*exp(x)-1)/exp(4))^2-4*exp(2)^4*exp(4)*exp(x))/(ex p(4)*exp(x)-1)/log((exp(4)*exp(x)-1)/exp(4))^5,x, algorithm=\
-(log(e^(x + 4) - 1)^4*log(e^(x + 4)) - 16*log(e^(x + 4) - 1)^3*log(e^(x + 4)) + 2*e^6*log(e^(x + 4) - 1)^2 + 96*log(e^(x + 4) - 1)^2*log(e^(x + 4)) - 16*e^6*log(e^(x + 4) - 1) - 256*log(e^(x + 4) - 1)*log(e^(x + 4)) - e^8 + 32*e^6 + 256*log(e^(x + 4)))/(log(e^(x + 4) - 1)^4 - 16*log(e^(x + 4) - 1)^3 + 96*log(e^(x + 4) - 1)^2 - 256*log(e^(x + 4) - 1) + 256)
Time = 0.18 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.59 \[ \int \frac {-4 e^{12+x}+4 e^{10+x} \log ^2\left (\frac {-1+e^{4+x}}{e^4}\right )+\left (1-e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )}{\left (-1+e^{4+x}\right ) \log ^5\left (\frac {-1+e^{4+x}}{e^4}\right )} \, dx=-\frac {x\,{\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^{-4}\right )}^4+2\,{\mathrm {e}}^6\,{\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^{-4}\right )}^2-{\mathrm {e}}^8}{{\ln \left ({\mathrm {e}}^x-{\mathrm {e}}^{-4}\right )}^4} \]
int(-(4*exp(12)*exp(x) + log(exp(-4)*(exp(4)*exp(x) - 1))^5*(exp(4)*exp(x) - 1) - 4*log(exp(-4)*(exp(4)*exp(x) - 1))^2*exp(10)*exp(x))/(log(exp(-4)* (exp(4)*exp(x) - 1))^5*(exp(4)*exp(x) - 1)),x)