Integrand size = 38, antiderivative size = 21 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \] Output:
exp(ln(-139/12*ln(x)/x)/ln(5))
Time = 0.05 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \] Input:
Integrate[((139/12)^Log[5]^(-1)*(1 - Log[x])*(-(Log[x]/x))^Log[5]^(-1))/(x *Log[5]*Log[x]),x]
Output:
(139/12)^Log[5]^(-1)*(-(Log[x]/x))^Log[5]^(-1)
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.72 (sec) , antiderivative size = 111, normalized size of antiderivative = 5.29, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.184, Rules used = {27, 7270, 2813, 25, 27, 3039, 7111}
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.
\(\displaystyle \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} \int \frac {(1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (x)}dx}{\log (5)}\) |
\(\Big \downarrow \) 7270 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \int x^{-1-\frac {1}{\log (5)}} (1-\log (x)) \log ^{-1+\frac {1}{\log (5)}}(x)dx}{\log (5)}\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \left (\int -\frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \sqrt [\log (5)]{\log (5)}}{x}dx-\sqrt [\log (5)]{\log (5)} (1-\log (x)) \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )\right )}{\log (5)}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \left (\sqrt [\log (5)]{\log (5)} (1-\log (x)) \left (-\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )\right )-\int \frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right ) \sqrt [\log (5)]{\log (5)}}{x}dx\right )}{\log (5)}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \left (\sqrt [\log (5)]{\log (5)} (1-\log (x)) \left (-\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )\right )-\sqrt [\log (5)]{\log (5)} \int \frac {\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )}{x}dx\right )}{\log (5)}\) |
\(\Big \downarrow \) 3039 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \left (\sqrt [\log (5)]{\log (5)} (1-\log (x)) \left (-\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )\right )-\sqrt [\log (5)]{\log (5)} \int \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )d\log (x)\right )}{\log (5)}\) |
\(\Big \downarrow \) 7111 |
\(\displaystyle \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} x^{\frac {1}{\log (5)}} \log ^{-\frac {1}{\log (5)}}(x) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}} \left (\sqrt [\log (5)]{\log (5)} (1-\log (x)) \left (-\Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )\right )-\sqrt [\log (5)]{\log (5)} \left (\log (x) \Gamma \left (\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )-\log (5) \Gamma \left (1+\frac {1}{\log (5)},\frac {\log (x)}{\log (5)}\right )\right )\right )}{\log (5)}\) |
Input:
Int[((139/12)^Log[5]^(-1)*(1 - Log[x])*(-(Log[x]/x))^Log[5]^(-1))/(x*Log[5 ]*Log[x]),x]
Output:
((139/12)^Log[5]^(-1)*x^Log[5]^(-1)*(-(Log[x]/x))^Log[5]^(-1)*(-(Gamma[Log [5]^(-1), Log[x]/Log[5]]*Log[5]^Log[5]^(-1)*(1 - Log[x])) - Log[5]^Log[5]^ (-1)*(-(Gamma[1 + Log[5]^(-1), Log[x]/Log[5]]*Log[5]) + Gamma[Log[5]^(-1), Log[x]/Log[5]]*Log[x])))/(Log[5]*Log[x]^Log[5]^(-1))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_ .)]*(e_.))*((g_.)*(x_))^(m_.), x_Symbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Simp[(d + e*Log[f*x^r]) u, x] - Simp[e*r Int[Simp lifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] && !(EqQ[p, 1] && EqQ[a, 0] && NeQ[d, 0])
Int[u_, x_Symbol] :> With[{lst = FunctionOfLog[Cancel[x*u], x]}, Simp[1/lst [[3]] Subst[Int[lst[[1]], x], x, Log[lst[[2]]]], x] /; !FalseQ[lst]] /; NonsumQ[u]
Int[Gamma[n_, (a_.) + (b_.)*(x_)], x_Symbol] :> Simp[(a + b*x)*(Gamma[n, a + b*x]/b), x] - Simp[Gamma[n + 1, a + b*x]/b, x] /; FreeQ[{a, b, n}, x]
Int[(u_.)*((a_.)*(v_)^(m_.)*(w_)^(n_.))^(p_), x_Symbol] :> Simp[a^IntPart[p ]*((a*v^m*w^n)^FracPart[p]/(v^(m*FracPart[p])*w^(n*FracPart[p]))) Int[u*v ^(m*p)*w^(n*p), x], x] /; FreeQ[{a, m, n, p}, x] && !IntegerQ[p] && !Free Q[v, x] && !FreeQ[w, x]
Time = 1.36 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71
method | result | size |
norman | \({\mathrm e}^{\frac {\ln \left (-\frac {139 \ln \left (x \right )}{12 x}\right )}{\ln \left (5\right )}}\) | \(15\) |
parallelrisch | \({\mathrm e}^{\frac {\ln \left (-\frac {139 \ln \left (x \right )}{12 x}\right )}{\ln \left (5\right )}}\) | \(15\) |
risch | \(\ln \left (x \right )^{\frac {1}{\ln \left (5\right )}} 139^{\frac {1}{\ln \left (5\right )}} \left (\frac {1}{4}\right )^{\frac {1}{\ln \left (5\right )}} \left (\frac {1}{3}\right )^{\frac {1}{\ln \left (5\right )}} x^{-\frac {1}{\ln \left (5\right )}} {\mathrm e}^{\frac {i \pi \left (\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2} \operatorname {csgn}\left (\frac {i}{x}\right )-\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right ) \operatorname {csgn}\left (\frac {i}{x}\right ) \operatorname {csgn}\left (i \ln \left (x \right )\right )+\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{3}+\operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2} \operatorname {csgn}\left (i \ln \left (x \right )\right )-2 \operatorname {csgn}\left (\frac {i \ln \left (x \right )}{x}\right )^{2}+2\right )}{2 \ln \left (5\right )}}\) | \(131\) |
Input:
int((1-ln(x))*exp(ln(-139/12*ln(x)/x)/ln(5))/x/ln(5)/ln(x),x,method=_RETUR NVERBOSE)
Output:
exp(ln(-139/12*ln(x)/x)/ln(5))
Time = 0.09 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\left (-\frac {139 \, \log \left (x\right )}{12 \, x}\right )^{\left (\frac {1}{\log \left (5\right )}\right )} \] Input:
integrate((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x, algorithm="fricas")
Output:
(-139/12*log(x)/x)^(1/log(5))
Time = 22.60 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.14 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\frac {139^{\frac {1}{\log {\left (5 \right )}}} \left (- \frac {\log {\left (x \right )}}{x}\right )^{\frac {1}{\log {\left (5 \right )}}}}{12^{\frac {1}{\log {\left (5 \right )}}}} \] Input:
integrate((1-ln(x))*exp(ln(-139/12*ln(x)/x)/ln(5))/x/ln(5)/ln(x),x)
Output:
139**(1/log(5))*(-log(x)/x)**(1/log(5))/12**(1/log(5))
Leaf count of result is larger than twice the leaf count of optimal. 45 vs. \(2 (12) = 24\).
Time = 0.17 (sec) , antiderivative size = 45, normalized size of antiderivative = 2.14 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\frac {139^{\left (\frac {1}{\log \left (5\right )}\right )} e^{\left (-\frac {\log \left (x\right )}{\log \left (5\right )} + \frac {\log \left (-\log \left (x\right )\right )}{\log \left (5\right )}\right )}}{3^{\left (\frac {1}{\log \left (5\right )}\right )} 2^{\frac {2}{\log \left (5\right )}}} \] Input:
integrate((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x, algorithm="maxima")
Output:
139^(1/log(5))*e^(-log(x)/log(5) + log(-log(x))/log(5))/(3^(1/log(5))*2^(2 /log(5)))
\[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=\int { -\frac {\left (-\frac {139 \, \log \left (x\right )}{12 \, x}\right )^{\left (\frac {1}{\log \left (5\right )}\right )} {\left (\log \left (x\right ) - 1\right )}}{x \log \left (5\right ) \log \left (x\right )} \,d x } \] Input:
integrate((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x, algorithm="giac")
Output:
integrate(-(-139/12*log(x)/x)^(1/log(5))*(log(x) - 1)/(x*log(5)*log(x)), x )
Time = 2.82 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.57 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx={\left (-\frac {139\,\ln \left (x\right )}{12\,x}\right )}^{\frac {1}{\ln \left (5\right )}} \] Input:
int(-((log(x) - 1)*(-(139*log(x))/(12*x))^(1/log(5)))/(x*log(5)*log(x)),x)
Output:
(-(139*log(x))/(12*x))^(1/log(5))
Time = 0.17 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.71 \[ \int \frac {\left (\frac {139}{12}\right )^{\frac {1}{\log (5)}} (1-\log (x)) \left (-\frac {\log (x)}{x}\right )^{\frac {1}{\log (5)}}}{x \log (5) \log (x)} \, dx=e^{\frac {\mathrm {log}\left (-\frac {139 \,\mathrm {log}\left (x \right )}{12 x}\right )}{\mathrm {log}\left (5\right )}} \] Input:
int((1-log(x))*exp(log(-139/12*log(x)/x)/log(5))/x/log(5)/log(x),x)
Output:
e**(log(( - 139*log(x))/(12*x))/log(5))