\(\int \frac {x^2}{\log ^2(c x)} \, dx\) [30]

Optimal result

Integrand size = 10, antiderivative size = 24 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \operatorname {ExpIntegralEi}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2343, 2346, 2209} \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \operatorname {ExpIntegralEi}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \]

[In]

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

[Out]

(3*ExpIntegralEi[3*Log[c*x]])/c^3 - x^3/Log[c*x]

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 2343

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

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 {x^3}{\log (c x)}+3 \int \frac {x^2}{\log (c x)} \, dx \\ & = -\frac {x^3}{\log (c x)}+\frac {3 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (c x)\right )}{c^3} \\ & = \frac {3 \text {Ei}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \operatorname {ExpIntegralEi}(3 \log (c x))}{c^3}-\frac {x^3}{\log (c x)} \]

[In]

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

[Out]

(3*ExpIntegralEi[3*Log[c*x]])/c^3 - x^3/Log[c*x]

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08

method	result	size
risch	\(-\frac {x^{3}}{\ln \left (x c \right )}-\frac {3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\)	\(26\)
derivativedivides	\(\frac {-\frac {x^{3} c^{3}}{\ln \left (x c \right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\)	\(30\)
default	\(\frac {-\frac {x^{3} c^{3}}{\ln \left (x c \right )}-3 \,\operatorname {Ei}_{1}\left (-3 \ln \left (x c \right )\right )}{c^{3}}\)	\(30\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.38 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=-\frac {c^{3} x^{3} - 3 \, \log \left (c x\right ) \operatorname {log\_integral}\left (c^{3} x^{3}\right )}{c^{3} \log \left (c x\right )} \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {x^2}{\log ^2(c x)} \, dx=- \frac {x^{3}}{\log {\left (c x \right )}} + 3 \int \frac {x^{2}}{\log {\left (c x \right )}}\, dx \]

[In]

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

[Out]

-x**3/log(c*x) + 3*Integral(x**2/log(c*x), x)

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.54 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=\frac {3 \, \Gamma \left (-1, -3 \, \log \left (c x\right )\right )}{c^{3}} \]

[In]

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

[Out]

3*gamma(-1, -3*log(c*x))/c^3

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {x^2}{\log ^2(c x)} \, dx=-\frac {x^{3}}{\log \left (c x\right )} + \frac {3 \, {\rm Ei}\left (3 \, \log \left (c x\right )\right )}{c^{3}} \]

[In]

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

[Out]

-x^3/log(c*x) + 3*Ei(3*log(c*x))/c^3

Mupad [F(-1)]

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

[In]

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

[Out]

int(x^2/log(c*x)^2, x)