\(\int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx\) [4203]

Optimal result

Integrand size = 29, antiderivative size = 18 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {3}{5} \left (-\frac {5}{3}+e^{225 \log ^2(x)}\right )^2 \]

[Out]

3/5*(-5/3+exp(225*ln(x)^2))^2

Rubi [A] (verified)

Time = 0.08 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.28, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {14, 2308, 2235, 2240} \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {3}{5} e^{450 \log ^2(x)}-2 e^{225 \log ^2(x)} \]

[In]

Int[(-900*E^(225*Log[x]^2)*Log[x] + 540*E^(450*Log[x]^2)*Log[x])/x,x]

[Out]

-2*E^(225*Log[x]^2) + (3*E^(450*Log[x]^2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2240

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^(n_))*((e_.) + (f_.)*(x_))^(m_.), x_Symbol] :> Simp[(e + f*x)^n*(
F^(a + b*(c + d*x)^n)/(b*f*n*(c + d*x)^n*Log[F])), x] /; FreeQ[{F, a, b, c, d, e, f, n}, x] && EqQ[m, n - 1] &
& EqQ[d*e - c*f, 0]

Rule 2308

Int[(F_)^(((a_.) + Log[(c_.)*((d_.) + (e_.)*(x_))^(n_.)]^2*(b_.))*(f_.))*((g_.) + (h_.)*(x_))^(m_.), x_Symbol]
:> Dist[(g + h*x)^(m + 1)/(h*n*(c*(d + e*x)^n)^((m + 1)/n)), Subst[Int[E^(a*f*Log[F] + ((m + 1)*x)/n + b*f*Lo
g[F]*x^2), x], x, Log[c*(d + e*x)^n]], x] /; FreeQ[{F, a, b, c, d, e, f, g, h, m, n}, x] && EqQ[e*g - d*h, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (-\frac {900 e^{225 \log ^2(x)} \log (x)}{x}+\frac {540 e^{450 \log ^2(x)} \log (x)}{x}\right ) \, dx \\ & = 540 \int \frac {e^{450 \log ^2(x)} \log (x)}{x} \, dx-900 \int \frac {e^{225 \log ^2(x)} \log (x)}{x} \, dx \\ & = 540 \text {Subst}\left (\int e^{450 x^2} x \, dx,x,\log (x)\right )-900 \text {Subst}\left (\int e^{225 x^2} x \, dx,x,\log (x)\right ) \\ & = -2 e^{225 \log ^2(x)}+\frac {3}{5} e^{450 \log ^2(x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {1}{15} \left (-5+3 e^{225 \log ^2(x)}\right )^2 \]

[In]

Integrate[(-900*E^(225*Log[x]^2)*Log[x] + 540*E^(450*Log[x]^2)*Log[x])/x,x]

[Out]

(-5 + 3*E^(225*Log[x]^2))^2/15

Maple [A] (verified)

Time = 1.43 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11

method	result	size
risch	\(-2 \,{\mathrm e}^{225 \ln \left (x \right )^{2}}+\frac {3 \,{\mathrm e}^{450 \ln \left (x \right )^{2}}}{5}\)	\(20\)
derivativedivides	\(-2 \,{\mathrm e}^{225 \ln \left (x \right )^{2}}+\frac {3 \,{\mathrm e}^{450 \ln \left (x \right )^{2}}}{5}\)	\(22\)
default	\(-2 \,{\mathrm e}^{225 \ln \left (x \right )^{2}}+\frac {3 \,{\mathrm e}^{450 \ln \left (x \right )^{2}}}{5}\)	\(22\)
norman	\(-2 \,{\mathrm e}^{225 \ln \left (x \right )^{2}}+\frac {3 \,{\mathrm e}^{450 \ln \left (x \right )^{2}}}{5}\)	\(22\)
parallelrisch	\(-2 \,{\mathrm e}^{225 \ln \left (x \right )^{2}}+\frac {3 \,{\mathrm e}^{450 \ln \left (x \right )^{2}}}{5}\)	\(22\)
parts	\(-2 \,{\mathrm e}^{225 \ln \left (x \right )^{2}}+\frac {3 \,{\mathrm e}^{450 \ln \left (x \right )^{2}}}{5}\)	\(22\)

[In]

int((540*ln(x)*exp(225*ln(x)^2)^2-900*ln(x)*exp(225*ln(x)^2))/x,x,method=_RETURNVERBOSE)

[Out]

-2*exp(225*ln(x)^2)+3/5*exp(450*ln(x)^2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {3}{5} \, e^{\left (450 \, \log \left (x\right )^{2}\right )} - 2 \, e^{\left (225 \, \log \left (x\right )^{2}\right )} \]

[In]

integrate((540*log(x)*exp(225*log(x)^2)^2-900*log(x)*exp(225*log(x)^2))/x,x, algorithm="fricas")

[Out]

3/5*e^(450*log(x)^2) - 2*e^(225*log(x)^2)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {3 e^{450 \log {\left (x \right )}^{2}}}{5} - 2 e^{225 \log {\left (x \right )}^{2}} \]

[In]

integrate((540*ln(x)*exp(225*ln(x)**2)**2-900*ln(x)*exp(225*ln(x)**2))/x,x)

[Out]

3*exp(450*log(x)**2)/5 - 2*exp(225*log(x)**2)

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {3}{5} \, e^{\left (450 \, \log \left (x\right )^{2}\right )} - 2 \, e^{\left (225 \, \log \left (x\right )^{2}\right )} \]

[In]

integrate((540*log(x)*exp(225*log(x)^2)^2-900*log(x)*exp(225*log(x)^2))/x,x, algorithm="maxima")

[Out]

3/5*e^(450*log(x)^2) - 2*e^(225*log(x)^2)

Giac [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.06 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {3}{5} \, e^{\left (450 \, \log \left (x\right )^{2}\right )} - 2 \, e^{\left (225 \, \log \left (x\right )^{2}\right )} \]

[In]

integrate((540*log(x)*exp(225*log(x)^2)^2-900*log(x)*exp(225*log(x)^2))/x,x, algorithm="giac")

[Out]

3/5*e^(450*log(x)^2) - 2*e^(225*log(x)^2)

Mupad [B] (verification not implemented)

Time = 10.08 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int \frac {-900 e^{225 \log ^2(x)} \log (x)+540 e^{450 \log ^2(x)} \log (x)}{x} \, dx=\frac {{\mathrm {e}}^{225\,{\ln \left (x\right )}^2}\,\left (3\,{\mathrm {e}}^{225\,{\ln \left (x\right )}^2}-10\right )}{5} \]

[In]

int(-(900*exp(225*log(x)^2)*log(x) - 540*exp(450*log(x)^2)*log(x))/x,x)

[Out]

(exp(225*log(x)^2)*(3*exp(225*log(x)^2) - 10))/5