\(\int \frac {10 e^{36 x^4}+x^2 \log (4 x^2)+e^{36 x^4} (-5+720 x^4) \log (4 x^2) \log (\log (4 x^2))}{x^2 \log (4 x^2)} \, dx\) [4856]

Optimal result

Integrand size = 60, antiderivative size = 21 \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=x+\frac {5 e^{36 x^4} \log \left (\log \left (4 x^2\right )\right )}{x} \]

[Out]

5/x*ln(ln(4*x^2))*exp(36*x^4)+x

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.033, Rules used = {6874, 2326} \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=\frac {5 e^{36 x^4} \log \left (\log \left (4 x^2\right )\right )}{x}+x \]

[In]

Int[(10*E^(36*x^4) + x^2*Log[4*x^2] + E^(36*x^4)*(-5 + 720*x^4)*Log[4*x^2]*Log[Log[4*x^2]])/(x^2*Log[4*x^2]),x
]

[Out]

x + (5*E^(36*x^4)*Log[Log[4*x^2]])/x

Rule 2326

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = v*(y/(Log[F]*D[u, x]))}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (1+\frac {5 e^{36 x^4} \left (2-\log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )+144 x^4 \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )\right )}{x^2 \log \left (4 x^2\right )}\right ) \, dx \\ & = x+5 \int \frac {e^{36 x^4} \left (2-\log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )+144 x^4 \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx \\ & = x+\frac {5 e^{36 x^4} \log \left (\log \left (4 x^2\right )\right )}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=x+\frac {5 e^{36 x^4} \log \left (\log \left (4 x^2\right )\right )}{x} \]

[In]

Integrate[(10*E^(36*x^4) + x^2*Log[4*x^2] + E^(36*x^4)*(-5 + 720*x^4)*Log[4*x^2]*Log[Log[4*x^2]])/(x^2*Log[4*x
^2]),x]

[Out]

x + (5*E^(36*x^4)*Log[Log[4*x^2]])/x

Maple [A] (verified)

Time = 0.88 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.29

method	result	size
parallelrisch	\(-\frac {-x^{2}-5 \,{\mathrm e}^{36 x^{4}} \ln \left (\ln \left (4 x^{2}\right )\right )}{x}\)	\(27\)
risch	\(\frac {5 \,{\mathrm e}^{36 x^{4}} \ln \left (2 \ln \left (2\right )+2 \ln \left (x \right )-\frac {i \pi \,\operatorname {csgn}\left (i x^{2}\right ) {\left (-\operatorname {csgn}\left (i x^{2}\right )+\operatorname {csgn}\left (i x \right )\right )}^{2}}{2}\right )}{x}+x\)	\(52\)

[In]

int(((720*x^4-5)*exp(36*x^4)*ln(4*x^2)*ln(ln(4*x^2))+x^2*ln(4*x^2)+10*exp(36*x^4))/x^2/ln(4*x^2),x,method=_RET
URNVERBOSE)

[Out]

-1/x*(-x^2-5*exp(36*x^4)*ln(ln(4*x^2)))

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=\frac {x^{2} + 5 \, e^{\left (36 \, x^{4}\right )} \log \left (\log \left (4 \, x^{2}\right )\right )}{x} \]

[In]

integrate(((720*x^4-5)*exp(36*x^4)*log(4*x^2)*log(log(4*x^2))+x^2*log(4*x^2)+10*exp(36*x^4))/x^2/log(4*x^2),x,
algorithm="fricas")

[Out]

(x^2 + 5*e^(36*x^4)*log(log(4*x^2)))/x

Sympy [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90 \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=x + \frac {5 e^{36 x^{4}} \log {\left (\log {\left (4 x^{2} \right )} \right )}}{x} \]

[In]

integrate(((720*x**4-5)*exp(36*x**4)*ln(4*x**2)*ln(ln(4*x**2))+x**2*ln(4*x**2)+10*exp(36*x**4))/x**2/ln(4*x**2
),x)

[Out]

x + 5*exp(36*x**4)*log(log(4*x**2))/x

Maxima [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.43 \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=x + \frac {5 \, {\left (e^{\left (36 \, x^{4}\right )} \log \left (2\right ) + e^{\left (36 \, x^{4}\right )} \log \left (\log \left (2\right ) + \log \left (x\right )\right )\right )}}{x} \]

[In]

integrate(((720*x^4-5)*exp(36*x^4)*log(4*x^2)*log(log(4*x^2))+x^2*log(4*x^2)+10*exp(36*x^4))/x^2/log(4*x^2),x,
algorithm="maxima")

[Out]

x + 5*(e^(36*x^4)*log(2) + e^(36*x^4)*log(log(2) + log(x)))/x

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=\frac {x^{2} + 5 \, e^{\left (36 \, x^{4}\right )} \log \left (\log \left (4 \, x^{2}\right )\right )}{x} \]

[In]

integrate(((720*x^4-5)*exp(36*x^4)*log(4*x^2)*log(log(4*x^2))+x^2*log(4*x^2)+10*exp(36*x^4))/x^2/log(4*x^2),x,
algorithm="giac")

[Out]

(x^2 + 5*e^(36*x^4)*log(log(4*x^2)))/x

Mupad [F(-1)]

Timed out. \[ \int \frac {10 e^{36 x^4}+x^2 \log \left (4 x^2\right )+e^{36 x^4} \left (-5+720 x^4\right ) \log \left (4 x^2\right ) \log \left (\log \left (4 x^2\right )\right )}{x^2 \log \left (4 x^2\right )} \, dx=\int \frac {10\,{\mathrm {e}}^{36\,x^4}+x^2\,\ln \left (4\,x^2\right )+\ln \left (\ln \left (4\,x^2\right )\right )\,{\mathrm {e}}^{36\,x^4}\,\ln \left (4\,x^2\right )\,\left (720\,x^4-5\right )}{x^2\,\ln \left (4\,x^2\right )} \,d x \]

[In]

int((10*exp(36*x^4) + x^2*log(4*x^2) + log(log(4*x^2))*exp(36*x^4)*log(4*x^2)*(720*x^4 - 5))/(x^2*log(4*x^2)),
x)

[Out]

int((10*exp(36*x^4) + x^2*log(4*x^2) + log(log(4*x^2))*exp(36*x^4)*log(4*x^2)*(720*x^4 - 5))/(x^2*log(4*x^2)),
x)