\(\int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+(-16-31 x^2-16 x^3+48 x^4) \log (x)}{x^2 \log ^2(x)} \, dx\) [6883]

Optimal result

Integrand size = 48, antiderivative size = 33 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {-x+\left (-x+\frac {-1+\left (1-2 x^2\right )^2}{x^2}\right )^2}{x \log (x)} \]

[Out]

((((-2*x^2+1)^2-1)/x^2-x)^2-x)/x/ln(x)

Rubi [F]

\[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx \]

[In]

Int[(-16 - 7*x + 31*x^2 + 8*x^3 - 16*x^4 + (-16 - 31*x^2 - 16*x^3 + 48*x^4)*Log[x])/(x^2*Log[x]^2),x]

[Out]

Defer[Int][(-16 - 7*x + 31*x^2 + 8*x^3 - 16*x^4)/(x^2*Log[x]^2), x] + Defer[Int][(-16 - 31*x^2 - 16*x^3 + 48*x
^4)/(x^2*Log[x]), x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {-16-7 x+31 x^2+8 x^3-16 x^4}{x^2 \log ^2(x)}+\frac {-16-31 x^2-16 x^3+48 x^4}{x^2 \log (x)}\right ) \, dx \\ & = \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4}{x^2 \log ^2(x)} \, dx+\int \frac {-16-31 x^2-16 x^3+48 x^4}{x^2 \log (x)} \, dx \\ \end{align*}

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.24 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {7}{\log (x)}+\frac {16}{x \log (x)}-\frac {31 x}{\log (x)}-\frac {8 x^2}{\log (x)}+\frac {16 x^3}{\log (x)} \]

[In]

Integrate[(-16 - 7*x + 31*x^2 + 8*x^3 - 16*x^4 + (-16 - 31*x^2 - 16*x^3 + 48*x^4)*Log[x])/(x^2*Log[x]^2),x]

[Out]

7/Log[x] + 16/(x*Log[x]) - (31*x)/Log[x] - (8*x^2)/Log[x] + (16*x^3)/Log[x]

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.88

method	result	size
norman	\(\frac {16 x^{4}-8 x^{3}-31 x^{2}+7 x +16}{x \ln \left (x \right )}\)	\(29\)
risch	\(\frac {16 x^{4}-8 x^{3}-31 x^{2}+7 x +16}{x \ln \left (x \right )}\)	\(29\)
parallelrisch	\(\frac {16 x^{4}-8 x^{3}-31 x^{2}+7 x +16}{x \ln \left (x \right )}\)	\(29\)
default	\(\frac {16 x^{3}}{\ln \left (x \right )}-\frac {8 x^{2}}{\ln \left (x \right )}-\frac {31 x}{\ln \left (x \right )}+\frac {7}{\ln \left (x \right )}+\frac {16}{x \ln \left (x \right )}\)	\(42\)
parts	\(\frac {16 x^{3}}{\ln \left (x \right )}-\frac {8 x^{2}}{\ln \left (x \right )}-\frac {31 x}{\ln \left (x \right )}+\frac {7}{\ln \left (x \right )}+\frac {16}{x \ln \left (x \right )}\)	\(42\)

[In]

int(((48*x^4-16*x^3-31*x^2-16)*ln(x)-16*x^4+8*x^3+31*x^2-7*x-16)/x^2/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

(16*x^4-8*x^3-31*x^2+7*x+16)/x/ln(x)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {16 \, x^{4} - 8 \, x^{3} - 31 \, x^{2} + 7 \, x + 16}{x \log \left (x\right )} \]

[In]

integrate(((48*x^4-16*x^3-31*x^2-16)*log(x)-16*x^4+8*x^3+31*x^2-7*x-16)/x^2/log(x)^2,x, algorithm="fricas")

[Out]

(16*x^4 - 8*x^3 - 31*x^2 + 7*x + 16)/(x*log(x))

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.73 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {16 x^{4} - 8 x^{3} - 31 x^{2} + 7 x + 16}{x \log {\left (x \right )}} \]

[In]

integrate(((48*x**4-16*x**3-31*x**2-16)*ln(x)-16*x**4+8*x**3+31*x**2-7*x-16)/x**2/ln(x)**2,x)

[Out]

(16*x**4 - 8*x**3 - 31*x**2 + 7*x + 16)/(x*log(x))

Maxima [C] (verification not implemented)

Result contains higher order function than in optimal. Order 4 vs. order 3.

Time = 0.24 (sec) , antiderivative size = 63, normalized size of antiderivative = 1.91 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {7}{\log \left (x\right )} + 48 \, {\rm Ei}\left (3 \, \log \left (x\right )\right ) - 16 \, {\rm Ei}\left (2 \, \log \left (x\right )\right ) - 16 \, {\rm Ei}\left (-\log \left (x\right )\right ) - 31 \, {\rm Ei}\left (\log \left (x\right )\right ) + 31 \, \Gamma \left (-1, -\log \left (x\right )\right ) + 16 \, \Gamma \left (-1, -2 \, \log \left (x\right )\right ) - 48 \, \Gamma \left (-1, -3 \, \log \left (x\right )\right ) + 16 \, \Gamma \left (-1, \log \left (x\right )\right ) \]

[In]

integrate(((48*x^4-16*x^3-31*x^2-16)*log(x)-16*x^4+8*x^3+31*x^2-7*x-16)/x^2/log(x)^2,x, algorithm="maxima")

[Out]

7/log(x) + 48*Ei(3*log(x)) - 16*Ei(2*log(x)) - 16*Ei(-log(x)) - 31*Ei(log(x)) + 31*gamma(-1, -log(x)) + 16*gam
ma(-1, -2*log(x)) - 48*gamma(-1, -3*log(x)) + 16*gamma(-1, log(x))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {16 \, x^{4} - 8 \, x^{3} - 31 \, x^{2} + 7 \, x + 16}{x \log \left (x\right )} \]

[In]

integrate(((48*x^4-16*x^3-31*x^2-16)*log(x)-16*x^4+8*x^3+31*x^2-7*x-16)/x^2/log(x)^2,x, algorithm="giac")

[Out]

(16*x^4 - 8*x^3 - 31*x^2 + 7*x + 16)/(x*log(x))

Mupad [B] (verification not implemented)

Time = 11.41 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.85 \[ \int \frac {-16-7 x+31 x^2+8 x^3-16 x^4+\left (-16-31 x^2-16 x^3+48 x^4\right ) \log (x)}{x^2 \log ^2(x)} \, dx=\frac {16\,x^4-8\,x^3-31\,x^2+7\,x+16}{x\,\ln \left (x\right )} \]

[In]

int(-(7*x + log(x)*(31*x^2 + 16*x^3 - 48*x^4 + 16) - 31*x^2 - 8*x^3 + 16*x^4 + 16)/(x^2*log(x)^2),x)

[Out]

(7*x - 31*x^2 - 8*x^3 + 16*x^4 + 16)/(x*log(x))