3.82.45 \(\int \frac {-20+8 x+19 x^2-9 x^3+(-8 x-38 x^2+27 x^3) \log (x)}{x \log ^2(x)} \, dx\) [8145]

Optimal. Leaf size=21 \[ \frac {5 \left (2-\frac {9 x}{5}\right ) \left (2+x-x^2\right )}{\log (x)} \]

[Out]

3*(2-9/5*x)/ln(x)*(-5/3*x^2+5/3*x+10/3)

________________________________________________________________________________________

Rubi [F]
time = 0.22, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {-20+8 x+19 x^2-9 x^3+\left (-8 x-38 x^2+27 x^3\right ) \log (x)}{x \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(-20 + 8*x + 19*x^2 - 9*x^3 + (-8*x - 38*x^2 + 27*x^3)*Log[x])/(x*Log[x]^2),x]

[Out]

-38*ExpIntegralEi[2*Log[x]] + 27*ExpIntegralEi[3*Log[x]] - 8*LogIntegral[x] + Defer[Int][(-20 + 8*x + 19*x^2 -
9*x^3)/(x*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-20+8 x+19 x^2-9 x^3}{x \log ^2(x)}+\frac {-8-38 x+27 x^2}{\log (x)}\right ) \, dx\\ &=\int \frac {-20+8 x+19 x^2-9 x^3}{x \log ^2(x)} \, dx+\int \frac {-8-38 x+27 x^2}{\log (x)} \, dx\\ &=\int \left (-\frac {8}{\log (x)}-\frac {38 x}{\log (x)}+\frac {27 x^2}{\log (x)}\right ) \, dx+\int \frac {-20+8 x+19 x^2-9 x^3}{x \log ^2(x)} \, dx\\ &=-\left (8 \int \frac {1}{\log (x)} \, dx\right )+27 \int \frac {x^2}{\log (x)} \, dx-38 \int \frac {x}{\log (x)} \, dx+\int \frac {-20+8 x+19 x^2-9 x^3}{x \log ^2(x)} \, dx\\ &=-8 \text {li}(x)+27 \text {Subst}\left (\int \frac {e^{3 x}}{x} \, dx,x,\log (x)\right )-38 \text {Subst}\left (\int \frac {e^{2 x}}{x} \, dx,x,\log (x)\right )+\int \frac {-20+8 x+19 x^2-9 x^3}{x \log ^2(x)} \, dx\\ &=-38 \text {Ei}(2 \log (x))+27 \text {Ei}(3 \log (x))-8 \text {li}(x)+\int \frac {-20+8 x+19 x^2-9 x^3}{x \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]
time = 0.05, size = 32, normalized size = 1.52 \begin {gather*} \frac {20}{\log (x)}-\frac {8 x}{\log (x)}-\frac {19 x^2}{\log (x)}+\frac {9 x^3}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-20 + 8*x + 19*x^2 - 9*x^3 + (-8*x - 38*x^2 + 27*x^3)*Log[x])/(x*Log[x]^2),x]

[Out]

20/Log[x] - (8*x)/Log[x] - (19*x^2)/Log[x] + (9*x^3)/Log[x]

________________________________________________________________________________________

Maple [A]
time = 0.16, size = 33, normalized size = 1.57

method	result	size
norman	\(\frac {9 x^{3}-19 x^{2}-8 x +20}{\ln \left (x \right )}\)	\(21\)
risch	\(\frac {9 x^{3}-19 x^{2}-8 x +20}{\ln \left (x \right )}\)	\(21\)
default	\(\frac {9 x^{3}}{\ln \left (x \right )}-\frac {19 x^{2}}{\ln \left (x \right )}-\frac {8 x}{\ln \left (x \right )}+\frac {20}{\ln \left (x \right )}\)	\(33\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((27*x^3-38*x^2-8*x)*ln(x)-9*x^3+19*x^2+8*x-20)/x/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

9*x^3/ln(x)-19*x^2/ln(x)-8*x/ln(x)+20/ln(x)

________________________________________________________________________________________

Maxima [C] Result contains higher order function than in optimal. Order 4 vs. order 3.
time = 0.34, size = 50, normalized size = 2.38 \begin {gather*} \frac {20}{\log \left (x\right )} + 27 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - 38 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) - 8 \, {\rm Ei}\left (\log \left (x\right )\right ) + 8 \, \Gamma \left (-1, -\log \left (x\right )\right ) + 38 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - 27 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^3-38*x^2-8*x)*log(x)-9*x^3+19*x^2+8*x-20)/x/log(x)^2,x, algorithm="maxima")

[Out]

20/log(x) + 27*Ei(3*log(x)) - 38*Ei(2*log(x)) - 8*Ei(log(x)) + 8*gamma(-1, -log(x)) + 38*gamma(-1, -2*log(x))
- 27*gamma(-1, -3*log(x))

________________________________________________________________________________________

Fricas [A]
time = 0.36, size = 20, normalized size = 0.95 \begin {gather*} \frac {9 \, x^{3} - 19 \, x^{2} - 8 \, x + 20}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^3-38*x^2-8*x)*log(x)-9*x^3+19*x^2+8*x-20)/x/log(x)^2,x, algorithm="fricas")

[Out]

(9*x^3 - 19*x^2 - 8*x + 20)/log(x)

________________________________________________________________________________________

Sympy [A]
time = 0.03, size = 17, normalized size = 0.81 \begin {gather*} \frac {9 x^{3} - 19 x^{2} - 8 x + 20}{\log {\left (x \right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x**3-38*x**2-8*x)*ln(x)-9*x**3+19*x**2+8*x-20)/x/ln(x)**2,x)

[Out]

(9*x**3 - 19*x**2 - 8*x + 20)/log(x)

________________________________________________________________________________________

Giac [A]
time = 0.41, size = 20, normalized size = 0.95 \begin {gather*} \frac {9 \, x^{3} - 19 \, x^{2} - 8 \, x + 20}{\log \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((27*x^3-38*x^2-8*x)*log(x)-9*x^3+19*x^2+8*x-20)/x/log(x)^2,x, algorithm="giac")

[Out]

(9*x^3 - 19*x^2 - 8*x + 20)/log(x)

________________________________________________________________________________________

Mupad [B]
time = 5.96, size = 16, normalized size = 0.76 \begin {gather*} \frac {\left (9\,x-10\right )\,\left (x+1\right )\,\left (x-2\right )}{\ln \left (x\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(9*x^3 - 19*x^2 - 8*x + log(x)*(8*x + 38*x^2 - 27*x^3) + 20)/(x*log(x)^2),x)

[Out]

((9*x - 10)*(x + 1)*(x - 2))/log(x)

________________________________________________________________________________________