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

Optimal. Leaf size=33 \[ \frac {-x+\left (-x+\frac {-1+\left (1-2 x^2\right )^2}{x^2}\right )^2}{x \log (x)} \]

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Rubi [F]  time = 0.39, 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 {-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 \end {gather*}

Verification is not applicable to the result.

[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 {gather*} \begin {aligned} \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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.16, size = 28, normalized size = 0.85 \begin {gather*} \frac {16+7 x-31 x^2-8 x^3+16 x^4}{x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[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]

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

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fricas [A]  time = 0.59, size = 28, normalized size = 0.85 \begin {gather*} \frac {16 \, x^{4} - 8 \, x^{3} - 31 \, x^{2} + 7 \, x + 16}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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giac [A]  time = 0.15, size = 28, normalized size = 0.85 \begin {gather*} \frac {16 \, x^{4} - 8 \, x^{3} - 31 \, x^{2} + 7 \, x + 16}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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maple [A]  time = 0.02, size = 29, normalized size = 0.88

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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maxima [C]  time = 0.42, size = 63, normalized size = 1.91 \begin {gather*} \frac {7}{\log \relax (x)} + 48 \, {\rm Ei}\left (3 \, \log \relax (x)\right ) - 16 \, {\rm Ei}\left (2 \, \log \relax (x)\right ) - 16 \, {\rm Ei}\left (-\log \relax (x)\right ) - 31 \, {\rm Ei}\left (\log \relax (x)\right ) + 31 \, \Gamma \left (-1, -\log \relax (x)\right ) + 16 \, \Gamma \left (-1, -2 \, \log \relax (x)\right ) - 48 \, \Gamma \left (-1, -3 \, \log \relax (x)\right ) + 16 \, \Gamma \left (-1, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 4.17, size = 28, normalized size = 0.85 \begin {gather*} \frac {16\,x^4-8\,x^3-31\,x^2+7\,x+16}{x\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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sympy [A]  time = 0.12, size = 24, normalized size = 0.73 \begin {gather*} \frac {16 x^{4} - 8 x^{3} - 31 x^{2} + 7 x + 16}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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