3.22 \(\int \frac {x^3}{\log (c x)} \, dx\)

Optimal. Leaf size=11 \[ \frac {\text {Ei}(4 \log (c x))}{c^4} \]

[Out]

Ei(4*ln(c*x))/c^4

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Rubi [A]  time = 0.02, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2309, 2178} \[ \frac {\text {Ei}(4 \log (c x))}{c^4} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Log[c*x],x]

[Out]

ExpIntegralEi[4*Log[c*x]]/c^4

Rule 2178

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

Rule 2309

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*} \int \frac {x^3}{\log (c x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {e^{4 x}}{x} \, dx,x,\log (c x)\right )}{c^4}\\ &=\frac {\text {Ei}(4 \log (c x))}{c^4}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 11, normalized size = 1.00 \[ \frac {\text {Ei}(4 \log (c x))}{c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Log[c*x],x]

[Out]

ExpIntegralEi[4*Log[c*x]]/c^4

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fricas [A]  time = 0.45, size = 12, normalized size = 1.09 \[ \frac {\operatorname {log\_integral}\left (c^{4} x^{4}\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/log(c*x),x, algorithm="fricas")

[Out]

log_integral(c^4*x^4)/c^4

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giac [A]  time = 0.20, size = 11, normalized size = 1.00 \[ \frac {{\rm Ei}\left (4 \, \log \left (c x\right )\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/log(c*x),x, algorithm="giac")

[Out]

Ei(4*log(c*x))/c^4

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maple [A]  time = 0.04, size = 14, normalized size = 1.27 \[ -\frac {\Ei \left (1, -4 \ln \left (c x \right )\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/ln(c*x),x)

[Out]

-1/c^4*Ei(1,-4*ln(c*x))

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maxima [A]  time = 0.96, size = 11, normalized size = 1.00 \[ \frac {{\rm Ei}\left (4 \, \log \left (c x\right )\right )}{c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/log(c*x),x, algorithm="maxima")

[Out]

Ei(4*log(c*x))/c^4

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mupad [F]  time = 0.00, size = -1, normalized size = -0.09 \[ \int \frac {x^3}{\ln \left (c\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/log(c*x),x)

[Out]

int(x^3/log(c*x), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{3}}{\log {\left (c x \right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/ln(c*x),x)

[Out]

Integral(x**3/log(c*x), x)

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