\(\int \frac {x^2}{\log (c x)} \, dx\) [23]

Optimal result

Integrand size = 10, antiderivative size = 11 \[ \int \frac {x^2}{\log (c x)} \, dx=\frac {\operatorname {ExpIntegralEi}(3 \log (c x))}{c^3} \]

[Out]

Ei(3*ln(c*x))/c^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2346, 2209} \[ \int \frac {x^2}{\log (c x)} \, dx=\frac {\operatorname {ExpIntegralEi}(3 \log (c x))}{c^3} \]

[In]

Int[x^2/Log[c*x],x]

[Out]

ExpIntegralEi[3*Log[c*x]]/c^3

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2346

Int[((a_.) + Log[(c_.)*(x_)]*(b_.))^(p_)*(x_)^(m_.), x_Symbol] :> Dist[1/c^(m + 1), Subst[Int[E^((m + 1)*x)*(a
+ b*x)^p, x], x, Log[c*x]], x] /; FreeQ[{a, b, c, p}, x] && IntegerQ[m]

Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (c x)\right )}{c^3} \\ & = \frac {\text {Ei}(3 \log (c x))}{c^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log (c x)} \, dx=\frac {\operatorname {ExpIntegralEi}(3 \log (c x))}{c^3} \]

[In]

Integrate[x^2/Log[c*x],x]

[Out]

ExpIntegralEi[3*Log[c*x]]/c^3

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.27

method	result	size
derivativedivides	\(-\frac {\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\)	\(14\)
default	\(-\frac {\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\)	\(14\)
risch	\(-\frac {\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\)	\(14\)

[In]

int(x^2/ln(x*c),x,method=_RETURNVERBOSE)

[Out]

-1/c^3*Ei(1,-3*ln(x*c))

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.09 \[ \int \frac {x^2}{\log (c x)} \, dx=\frac {\operatorname {log\_integral}\left (c^{3} x^{3}\right )}{c^{3}} \]

[In]

integrate(x^2/log(c*x),x, algorithm="fricas")

[Out]

log_integral(c^3*x^3)/c^3

Sympy [F]

\[ \int \frac {x^2}{\log (c x)} \, dx=\int \frac {x^{2}}{\log {\left (c x \right )}}\, dx \]

[In]

integrate(x**2/ln(c*x),x)

[Out]

Integral(x**2/log(c*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log (c x)} \, dx=\frac {{\rm Ei}\left (3 \, \log \left (c x\right )\right )}{c^{3}} \]

[In]

integrate(x^2/log(c*x),x, algorithm="maxima")

[Out]

Ei(3*log(c*x))/c^3

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log (c x)} \, dx=\frac {{\rm Ei}\left (3 \, \log \left (c x\right )\right )}{c^{3}} \]

[In]

integrate(x^2/log(c*x),x, algorithm="giac")

[Out]

Ei(3*log(c*x))/c^3

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2}{\log (c x)} \, dx=\int \frac {x^2}{\ln \left (c\,x\right )} \,d x \]

[In]

int(x^2/log(c*x),x)

[Out]

int(x^2/log(c*x), x)