\(\int \frac {2-2 x \log (x^2)+e^{\log (5) \log (x)} \log (5) \log (x^2)}{x \log (x^2)} \, dx\) [668]

Optimal result

Integrand size = 33, antiderivative size = 25 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=-x+\log \left (4 e^{20+e^{\log (5) \log (x)}-x} \log \left (x^2\right )\right ) \]

[Out]

ln(4*exp(20)/exp(x-exp(ln(5)*ln(x)))*ln(x^2))-x

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {6820, 2306, 30, 2339, 29} \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=x^{\log (5)}+\log \left (\log \left (x^2\right )\right )-2 x \]

[In]

Int[(2 - 2*x*Log[x^2] + E^(Log[5]*Log[x])*Log[5]*Log[x^2])/(x*Log[x^2]),x]

[Out]

-2*x + x^Log[5] + Log[Log[x^2]]

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2306

Int[(u_.)*(F_)^((a_.)*(Log[z_]*(b_.) + (v_.))), x_Symbol] :> Int[u*F^(a*v)*z^(a*b*Log[F]), x] /; FreeQ[{F, a,
b}, x]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 6820

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

Rubi steps \begin{align*} \text {integral}& = \int \left (-2+\frac {5^{\log (x)} \log (5)}{x}+\frac {2}{x \log \left (x^2\right )}\right ) \, dx \\ & = -2 x+2 \int \frac {1}{x \log \left (x^2\right )} \, dx+\log (5) \int \frac {5^{\log (x)}}{x} \, dx \\ & = -2 x+\log (5) \int x^{-1+\log (5)} \, dx+\text {Subst}\left (\int \frac {1}{x} \, dx,x,\log \left (x^2\right )\right ) \\ & = -2 x+x^{\log (5)}+\log \left (\log \left (x^2\right )\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=5^{\log (x)}-2 x+\log \left (\log \left (x^2\right )\right ) \]

[In]

Integrate[(2 - 2*x*Log[x^2] + E^(Log[5]*Log[x])*Log[5]*Log[x^2])/(x*Log[x^2]),x]

[Out]

5^Log[x] - 2*x + Log[Log[x^2]]

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.64

method	result	size
default	\(-2 x +{\mathrm e}^{\ln \left (5\right ) \ln \left (x \right )}+\ln \left (\ln \left (x^{2}\right )\right )\)	\(16\)
norman	\(-2 x +{\mathrm e}^{\ln \left (5\right ) \ln \left (x \right )}+\ln \left (\ln \left (x^{2}\right )\right )\)	\(16\)
parallelrisch	\(-2 x +{\mathrm e}^{\ln \left (5\right ) \ln \left (x \right )}+\ln \left (\ln \left (x^{2}\right )\right )\)	\(16\)
parts	\(-2 x +{\mathrm e}^{\ln \left (5\right ) \ln \left (x \right )}+\ln \left (\ln \left (x^{2}\right )\right )\)	\(16\)
risch	\(-2 x +\ln \left (\ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) \left (\operatorname {csgn}\left (i x \right )^{2}-2 \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )+\operatorname {csgn}\left (i x^{2}\right )^{2}\right )}{4}\right )+x^{\ln \left (5\right )}\)	\(55\)

[In]

int((ln(5)*ln(x^2)*exp(ln(5)*ln(x))-2*x*ln(x^2)+2)/x/ln(x^2),x,method=_RETURNVERBOSE)

[Out]

-2*x+exp(ln(5)*ln(x))+ln(ln(x^2))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=-2 \, x + e^{\left (\log \left (5\right ) \log \left (x\right )\right )} + \log \left (\log \left (x\right )\right ) \]

[In]

integrate((log(5)*log(x^2)*exp(log(5)*log(x))-2*x*log(x^2)+2)/x/log(x^2),x, algorithm="fricas")

[Out]

-2*x + e^(log(5)*log(x)) + log(log(x))

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.68 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=- 2 x + e^{\log {\left (5 \right )} \log {\left (x \right )}} + \log {\left (\log {\left (x^{2} \right )} \right )} \]

[In]

integrate((ln(5)*ln(x**2)*exp(ln(5)*ln(x))-2*x*ln(x**2)+2)/x/ln(x**2),x)

[Out]

-2*x + exp(log(5)*log(x)) + log(log(x**2))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=-2 \, x + e^{\left (\log \left (5\right ) \log \left (x\right )\right )} + \log \left (\log \left (x^{2}\right )\right ) \]

[In]

integrate((log(5)*log(x^2)*exp(log(5)*log(x))-2*x*log(x^2)+2)/x/log(x^2),x, algorithm="maxima")

[Out]

-2*x + e^(log(5)*log(x)) + log(log(x^2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.60 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=-2 \, x + e^{\left (\log \left (5\right ) \log \left (x\right )\right )} + \log \left (\log \left (x^{2}\right )\right ) \]

[In]

integrate((log(5)*log(x^2)*exp(log(5)*log(x))-2*x*log(x^2)+2)/x/log(x^2),x, algorithm="giac")

[Out]

-2*x + e^(log(5)*log(x)) + log(log(x^2))

Mupad [B] (verification not implemented)

Time = 8.10 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.52 \[ \int \frac {2-2 x \log \left (x^2\right )+e^{\log (5) \log (x)} \log (5) \log \left (x^2\right )}{x \log \left (x^2\right )} \, dx=\ln \left (\ln \left (x^2\right )\right )-2\,x+x^{\ln \left (5\right )} \]

[In]

int((log(x^2)*exp(log(5)*log(x))*log(5) - 2*x*log(x^2) + 2)/(x*log(x^2)),x)

[Out]

log(log(x^2)) - 2*x + x^log(5)