Integrand size = 62, antiderivative size = 17 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {(28+\log (2))^4}{4096 x^4 \log ^4(x)} \]
Time = 0.03 (sec) , antiderivative size = 17, normalized size of antiderivative = 1.00 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {(28+\log (2))^4}{4096 x^4 \log ^4(x)} \]
Integrate[(-614656 - 87808*Log[2] - 4704*Log[2]^2 - 112*Log[2]^3 - Log[2]^ 4 + (-614656 - 87808*Log[2] - 4704*Log[2]^2 - 112*Log[2]^3 - Log[2]^4)*Log [x])/(1024*x^5*Log[x]^5),x]
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 0.78 (sec) , antiderivative size = 197, normalized size of antiderivative = 11.59, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {27, 25, 2813, 27, 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.
\(\displaystyle \int \frac {\left (-614656-\log ^4(2)-112 \log ^3(2)-4704 \log ^2(2)-87808 \log (2)\right ) \log (x)-614656-\log ^4(2)-112 \log ^3(2)-4704 \log ^2(2)-87808 \log (2)}{1024 x^5 \log ^5(x)} \, dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\int -\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{x^5 \log ^5(x)}dx}{1024}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{x^5 \log ^5(x)}dx}{1024}\) |
\(\Big \downarrow \) 2813 |
\(\displaystyle \frac {(28+\log (2))^4 \int -\frac {-128 x^4 \operatorname {ExpIntegralEi}(-4 \log (x)) \log ^4(x)-32 \log ^3(x)+8 \log ^2(x)-4 \log (x)+3}{12 x^5 \log ^4(x)}dx-\frac {32}{3} \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right ) \operatorname {ExpIntegralEi}(-4 \log (x))+\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{4 x^4 \log ^4(x)}-\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{3 x^4 \log ^3(x)}+\frac {2 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log ^2(x)}-\frac {8 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log (x)}}{1024}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {-\frac {1}{12} (28+\log (2))^4 \int \frac {-128 x^4 \operatorname {ExpIntegralEi}(-4 \log (x)) \log ^4(x)-32 \log ^3(x)+8 \log ^2(x)-4 \log (x)+3}{x^5 \log ^4(x)}dx-\frac {32}{3} \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right ) \operatorname {ExpIntegralEi}(-4 \log (x))+\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{4 x^4 \log ^4(x)}-\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{3 x^4 \log ^3(x)}+\frac {2 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log ^2(x)}-\frac {8 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log (x)}}{1024}\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \frac {-\frac {1}{12} (28+\log (2))^4 \int \left (-\frac {128 \operatorname {ExpIntegralEi}(-4 \log (x))}{x}-\frac {32}{x^5 \log (x)}+\frac {8}{x^5 \log ^2(x)}-\frac {4}{x^5 \log ^3(x)}+\frac {3}{x^5 \log ^4(x)}\right )dx-\frac {32}{3} \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right ) \operatorname {ExpIntegralEi}(-4 \log (x))+\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{4 x^4 \log ^4(x)}-\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{3 x^4 \log ^3(x)}+\frac {2 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log ^2(x)}-\frac {8 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log (x)}}{1024}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {1}{12} (28+\log (2))^4 \left (-128 \log (x) \operatorname {ExpIntegralEi}(-4 \log (x))-128 \operatorname {ExpIntegralEi}(-4 \log (x))-\frac {32}{x^4}-\frac {1}{x^4 \log ^3(x)}+\frac {4}{x^4 \log ^2(x)}-\frac {24}{x^4 \log (x)}\right )-\frac {32}{3} \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right ) \operatorname {ExpIntegralEi}(-4 \log (x))+\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{4 x^4 \log ^4(x)}-\frac {(28+\log (2))^4 \log (x)+(28+\log (2))^4}{3 x^4 \log ^3(x)}+\frac {2 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log ^2(x)}-\frac {8 \left ((28+\log (2))^4 \log (x)+(28+\log (2))^4\right )}{3 x^4 \log (x)}}{1024}\) |
Int[(-614656 - 87808*Log[2] - 4704*Log[2]^2 - 112*Log[2]^3 - Log[2]^4 + (- 614656 - 87808*Log[2] - 4704*Log[2]^2 - 112*Log[2]^3 - Log[2]^4)*Log[x])/( 1024*x^5*Log[x]^5),x]
(-1/12*((28 + Log[2])^4*(-32/x^4 - 128*ExpIntegralEi[-4*Log[x]] - 1/(x^4*L og[x]^3) + 4/(x^4*Log[x]^2) - 24/(x^4*Log[x]) - 128*ExpIntegralEi[-4*Log[x ]]*Log[x])) - (32*ExpIntegralEi[-4*Log[x]]*((28 + Log[2])^4 + (28 + Log[2] )^4*Log[x]))/3 + ((28 + Log[2])^4 + (28 + Log[2])^4*Log[x])/(4*x^4*Log[x]^ 4) - ((28 + Log[2])^4 + (28 + Log[2])^4*Log[x])/(3*x^4*Log[x]^3) + (2*((28 + Log[2])^4 + (28 + Log[2])^4*Log[x]))/(3*x^4*Log[x]^2) - (8*((28 + Log[2 ])^4 + (28 + Log[2])^4*Log[x]))/(3*x^4*Log[x]))/1024
3.1.38.3.1 Defintions of rubi rules used
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])
Leaf count of result is larger than twice the leaf count of optimal. \(31\) vs. \(2(15)=30\).
Time = 0.18 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.88
method | result | size |
risch | \(\frac {\ln \left (2\right )^{4}+112 \ln \left (2\right )^{3}+4704 \ln \left (2\right )^{2}+87808 \ln \left (2\right )+614656}{4096 x^{4} \ln \left (x \right )^{4}}\) | \(32\) |
parallelrisch | \(\frac {\ln \left (2\right )^{4}+112 \ln \left (2\right )^{3}+4704 \ln \left (2\right )^{2}+87808 \ln \left (2\right )+614656}{4096 x^{4} \ln \left (x \right )^{4}}\) | \(32\) |
parts | \(\left (-\frac {\ln \left (2\right )^{4}}{1024}-\frac {7 \ln \left (2\right )^{3}}{64}-\frac {147 \ln \left (2\right )^{2}}{32}-\frac {343 \ln \left (2\right )}{4}-\frac {2401}{4}\right ) \left (-\frac {1}{4 x^{4} \ln \left (x \right )^{4}}+\frac {1}{3 x^{4} \ln \left (x \right )^{3}}-\frac {2}{3 x^{4} \ln \left (x \right )^{2}}+\frac {8}{3 x^{4} \ln \left (x \right )}-\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )+\left (-\frac {\ln \left (2\right )^{4}}{1024}-\frac {7 \ln \left (2\right )^{3}}{64}-\frac {147 \ln \left (2\right )^{2}}{32}-\frac {343 \ln \left (2\right )}{4}-\frac {2401}{4}\right ) \left (-\frac {1}{3 x^{4} \ln \left (x \right )^{3}}+\frac {2}{3 x^{4} \ln \left (x \right )^{2}}-\frac {8}{3 x^{4} \ln \left (x \right )}+\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )\) | \(133\) |
default | \(-\frac {\ln \left (2\right )^{4} \left (-\frac {1}{3 x^{4} \ln \left (x \right )^{3}}+\frac {2}{3 x^{4} \ln \left (x \right )^{2}}-\frac {8}{3 x^{4} \ln \left (x \right )}+\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{1024}-\frac {\ln \left (2\right )^{4} \left (-\frac {1}{4 x^{4} \ln \left (x \right )^{4}}+\frac {1}{3 x^{4} \ln \left (x \right )^{3}}-\frac {2}{3 x^{4} \ln \left (x \right )^{2}}+\frac {8}{3 x^{4} \ln \left (x \right )}-\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{1024}-\frac {7 \ln \left (2\right )^{3} \left (-\frac {1}{3 x^{4} \ln \left (x \right )^{3}}+\frac {2}{3 x^{4} \ln \left (x \right )^{2}}-\frac {8}{3 x^{4} \ln \left (x \right )}+\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{64}-\frac {7 \ln \left (2\right )^{3} \left (-\frac {1}{4 x^{4} \ln \left (x \right )^{4}}+\frac {1}{3 x^{4} \ln \left (x \right )^{3}}-\frac {2}{3 x^{4} \ln \left (x \right )^{2}}+\frac {8}{3 x^{4} \ln \left (x \right )}-\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{64}-\frac {147 \ln \left (2\right )^{2} \left (-\frac {1}{3 x^{4} \ln \left (x \right )^{3}}+\frac {2}{3 x^{4} \ln \left (x \right )^{2}}-\frac {8}{3 x^{4} \ln \left (x \right )}+\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{32}-\frac {147 \ln \left (2\right )^{2} \left (-\frac {1}{4 x^{4} \ln \left (x \right )^{4}}+\frac {1}{3 x^{4} \ln \left (x \right )^{3}}-\frac {2}{3 x^{4} \ln \left (x \right )^{2}}+\frac {8}{3 x^{4} \ln \left (x \right )}-\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{32}-\frac {343 \ln \left (2\right ) \left (-\frac {1}{3 x^{4} \ln \left (x \right )^{3}}+\frac {2}{3 x^{4} \ln \left (x \right )^{2}}-\frac {8}{3 x^{4} \ln \left (x \right )}+\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{4}-\frac {343 \ln \left (2\right ) \left (-\frac {1}{4 x^{4} \ln \left (x \right )^{4}}+\frac {1}{3 x^{4} \ln \left (x \right )^{3}}-\frac {2}{3 x^{4} \ln \left (x \right )^{2}}+\frac {8}{3 x^{4} \ln \left (x \right )}-\frac {32 \,\operatorname {Ei}_{1}\left (4 \ln \left (x \right )\right )}{3}\right )}{4}+\frac {2401}{16 x^{4} \ln \left (x \right )^{4}}\) | \(379\) |
int(1/1024*((-ln(2)^4-112*ln(2)^3-4704*ln(2)^2-87808*ln(2)-614656)*ln(x)-l n(2)^4-112*ln(2)^3-4704*ln(2)^2-87808*ln(2)-614656)/x^5/ln(x)^5,x,method=_ RETURNVERBOSE)
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.24 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {\log \left (2\right )^{4} + 112 \, \log \left (2\right )^{3} + 4704 \, \log \left (2\right )^{2} + 87808 \, \log \left (2\right ) + 614656}{4096 \, x^{4} \log \left (x\right )^{4}} \]
integrate(1/1024*((-log(2)^4-112*log(2)^3-4704*log(2)^2-87808*log(2)-61465 6)*log(x)-log(2)^4-112*log(2)^3-4704*log(2)^2-87808*log(2)-614656)/x^5/log (x)^5,x, algorithm=\
Leaf count of result is larger than twice the leaf count of optimal. 34 vs. \(2 (15) = 30\).
Time = 0.06 (sec) , antiderivative size = 34, normalized size of antiderivative = 2.00 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {\log {\left (2 \right )}^{4} + 112 \log {\left (2 \right )}^{3} + 4704 \log {\left (2 \right )}^{2} + 87808 \log {\left (2 \right )} + 614656}{4096 x^{4} \log {\left (x \right )}^{4}} \]
integrate(1/1024*((-ln(2)**4-112*ln(2)**3-4704*ln(2)**2-87808*ln(2)-614656 )*ln(x)-ln(2)**4-112*ln(2)**3-4704*ln(2)**2-87808*ln(2)-614656)/x**5/ln(x) **5,x)
Result contains higher order function than in optimal. Order 4 vs. order 3.
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 6.41 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {1}{16} \, \Gamma \left (-3, 4 \, \log \left (x\right )\right ) \log \left (2\right )^{4} + \frac {1}{4} \, \Gamma \left (-4, 4 \, \log \left (x\right )\right ) \log \left (2\right )^{4} + 7 \, \Gamma \left (-3, 4 \, \log \left (x\right )\right ) \log \left (2\right )^{3} + 28 \, \Gamma \left (-4, 4 \, \log \left (x\right )\right ) \log \left (2\right )^{3} + 294 \, \Gamma \left (-3, 4 \, \log \left (x\right )\right ) \log \left (2\right )^{2} + 1176 \, \Gamma \left (-4, 4 \, \log \left (x\right )\right ) \log \left (2\right )^{2} + 5488 \, \Gamma \left (-3, 4 \, \log \left (x\right )\right ) \log \left (2\right ) + 21952 \, \Gamma \left (-4, 4 \, \log \left (x\right )\right ) \log \left (2\right ) + 38416 \, \Gamma \left (-3, 4 \, \log \left (x\right )\right ) + 153664 \, \Gamma \left (-4, 4 \, \log \left (x\right )\right ) \]
integrate(1/1024*((-log(2)^4-112*log(2)^3-4704*log(2)^2-87808*log(2)-61465 6)*log(x)-log(2)^4-112*log(2)^3-4704*log(2)^2-87808*log(2)-614656)/x^5/log (x)^5,x, algorithm=\
1/16*gamma(-3, 4*log(x))*log(2)^4 + 1/4*gamma(-4, 4*log(x))*log(2)^4 + 7*g amma(-3, 4*log(x))*log(2)^3 + 28*gamma(-4, 4*log(x))*log(2)^3 + 294*gamma( -3, 4*log(x))*log(2)^2 + 1176*gamma(-4, 4*log(x))*log(2)^2 + 5488*gamma(-3 , 4*log(x))*log(2) + 21952*gamma(-4, 4*log(x))*log(2) + 38416*gamma(-3, 4* log(x)) + 153664*gamma(-4, 4*log(x))
Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (15) = 30\).
Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.82 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {\log \left (2\right )^{4} + 112 \, \log \left (2\right )^{3} + 4704 \, \log \left (2\right )^{2} + 87808 \, \log \left (2\right ) + 614656}{4096 \, x^{4} \log \left (x\right )^{4}} \]
integrate(1/1024*((-log(2)^4-112*log(2)^3-4704*log(2)^2-87808*log(2)-61465 6)*log(x)-log(2)^4-112*log(2)^3-4704*log(2)^2-87808*log(2)-614656)/x^5/log (x)^5,x, algorithm=\
Time = 10.80 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.88 \[ \int \frac {-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)+\left (-614656-87808 \log (2)-4704 \log ^2(2)-112 \log ^3(2)-\log ^4(2)\right ) \log (x)}{1024 x^5 \log ^5(x)} \, dx=\frac {{\left (\ln \left (2\right )+28\right )}^4}{4096\,x^4\,{\ln \left (x\right )}^4} \]