3.73.89 \(\int \frac {-1+\log (x)+(-1+e^{\frac {1}{4} (1+8 x)} (-1-2 x)) \log ^2(x)}{\log ^2(x)} \, dx\)

Optimal. Leaf size=30 \[ x \left (-e^{\frac {1}{4}+2 x}+\frac {-x+\log (4)+\frac {x}{\log (x)}}{x}\right ) \]

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 41, normalized size of antiderivative = 1.37, number of steps used = 9, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {6742, 2176, 2194, 2297, 2298} \begin {gather*} -x+\frac {1}{2} e^{2 x+\frac {1}{4}}-\frac {1}{2} e^{2 x+\frac {1}{4}} (2 x+1)+\frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-1 + Log[x] + (-1 + E^((1 + 8*x)/4)*(-1 - 2*x))*Log[x]^2)/Log[x]^2,x]

[Out]

E^(1/4 + 2*x)/2 - x - (E^(1/4 + 2*x)*(1 + 2*x))/2 + x/Log[x]

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2297

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[(x*(a + b*Log[c*x^n])^(p + 1))/(b*n*(p + 1))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2298

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-e^{\frac {1}{4}+2 x} (1+2 x)+\frac {-1+\log (x)-\log ^2(x)}{\log ^2(x)}\right ) \, dx\\ &=-\int e^{\frac {1}{4}+2 x} (1+2 x) \, dx+\int \frac {-1+\log (x)-\log ^2(x)}{\log ^2(x)} \, dx\\ &=-\frac {1}{2} e^{\frac {1}{4}+2 x} (1+2 x)+\int e^{\frac {1}{4}+2 x} \, dx+\int \left (-1-\frac {1}{\log ^2(x)}+\frac {1}{\log (x)}\right ) \, dx\\ &=\frac {1}{2} e^{\frac {1}{4}+2 x}-x-\frac {1}{2} e^{\frac {1}{4}+2 x} (1+2 x)-\int \frac {1}{\log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=\frac {1}{2} e^{\frac {1}{4}+2 x}-x-\frac {1}{2} e^{\frac {1}{4}+2 x} (1+2 x)+\frac {x}{\log (x)}+\text {li}(x)-\int \frac {1}{\log (x)} \, dx\\ &=\frac {1}{2} e^{\frac {1}{4}+2 x}-x-\frac {1}{2} e^{\frac {1}{4}+2 x} (1+2 x)+\frac {x}{\log (x)}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.04, size = 22, normalized size = 0.73 \begin {gather*} -x-e^{\frac {1}{4}+2 x} x+\frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-1 + Log[x] + (-1 + E^((1 + 8*x)/4)*(-1 - 2*x))*Log[x]^2)/Log[x]^2,x]

[Out]

-x - E^(1/4 + 2*x)*x + x/Log[x]

________________________________________________________________________________________

fricas [A]  time = 0.57, size = 23, normalized size = 0.77 \begin {gather*} -\frac {{\left (x e^{\left (2 \, x + \frac {1}{4}\right )} + x\right )} \log \relax (x) - x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x+1/4)-1)*log(x)^2+log(x)-1)/log(x)^2,x, algorithm="fricas")

[Out]

-((x*e^(2*x + 1/4) + x)*log(x) - x)/log(x)

________________________________________________________________________________________

giac [A]  time = 0.17, size = 24, normalized size = 0.80 \begin {gather*} -\frac {x e^{\left (2 \, x + \frac {1}{4}\right )} \log \relax (x) + x \log \relax (x) - x}{\log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x+1/4)-1)*log(x)^2+log(x)-1)/log(x)^2,x, algorithm="giac")

[Out]

-(x*e^(2*x + 1/4)*log(x) + x*log(x) - x)/log(x)

________________________________________________________________________________________

maple [A]  time = 0.02, size = 20, normalized size = 0.67

method	result	size
risch	\(-x \,{\mathrm e}^{2 x +\frac {1}{4}}-x +\frac {x}{\ln \relax (x )}\)	\(20\)
norman	\(\frac {x -x \ln \relax (x )-x \,{\mathrm e}^{2 x +\frac {1}{4}} \ln \relax (x )}{\ln \relax (x )}\)	\(24\)
default	\(-x -\frac {{\mathrm e}^{2 x +\frac {1}{4}} \left (2 x +\frac {1}{4}\right )}{2}+\frac {{\mathrm e}^{2 x +\frac {1}{4}}}{8}+\frac {x}{\ln \relax (x )}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-2*x-1)*exp(2*x+1/4)-1)*ln(x)^2+ln(x)-1)/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

-x*exp(2*x+1/4)-x+x/ln(x)

________________________________________________________________________________________

maxima [C]  time = 0.38, size = 39, normalized size = 1.30 \begin {gather*} -\frac {1}{2} \, {\left (2 \, x e^{\frac {1}{4}} - e^{\frac {1}{4}}\right )} e^{\left (2 \, x\right )} - x + {\rm Ei}\left (\log \relax (x)\right ) - \frac {1}{2} \, e^{\left (2 \, x + \frac {1}{4}\right )} - \Gamma \left (-1, -\log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x+1/4)-1)*log(x)^2+log(x)-1)/log(x)^2,x, algorithm="maxima")

[Out]

-1/2*(2*x*e^(1/4) - e^(1/4))*e^(2*x) - x + Ei(log(x)) - 1/2*e^(2*x + 1/4) - gamma(-1, -log(x))

________________________________________________________________________________________

mupad [B]  time = 4.44, size = 18, normalized size = 0.60 \begin {gather*} \frac {x}{\ln \relax (x)}-x\,\left ({\mathrm {e}}^{2\,x+\frac {1}{4}}+1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)^2*(exp(2*x + 1/4)*(2*x + 1) + 1) - log(x) + 1)/log(x)^2,x)

[Out]

x/log(x) - x*(exp(2*x + 1/4) + 1)

________________________________________________________________________________________

sympy [A]  time = 0.29, size = 15, normalized size = 0.50 \begin {gather*} - x e^{2 x + \frac {1}{4}} - x + \frac {x}{\log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-2*x-1)*exp(2*x+1/4)-1)*ln(x)**2+ln(x)-1)/ln(x)**2,x)

[Out]

-x*exp(2*x + 1/4) - x + x/log(x)

________________________________________________________________________________________