3.27.17 \(\int \frac {26-x-x^2+(26+x^2) \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=16 \[ x+\frac {\frac {-26+x}{x}+x}{\log (x)} \]

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Rubi [F]  time = 0.23, 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 {26-x-x^2+\left (26+x^2\right ) \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(26 - x - x^2 + (26 + x^2)*Log[x] + x^2*Log[x]^2)/(x^2*Log[x]^2),x]

[Out]

x + 26*ExpIntegralEi[-Log[x]] + LogIntegral[x] + Defer[Int][(26 - x - x^2)/(x^2*Log[x]^2), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {26-x-x^2}{x^2 \log ^2(x)}+\frac {26+x^2}{x^2 \log (x)}\right ) \, dx\\ &=x+\int \frac {26-x-x^2}{x^2 \log ^2(x)} \, dx+\int \frac {26+x^2}{x^2 \log (x)} \, dx\\ &=x+\int \left (\frac {1}{\log (x)}+\frac {26}{x^2 \log (x)}\right ) \, dx+\int \frac {26-x-x^2}{x^2 \log ^2(x)} \, dx\\ &=x+26 \int \frac {1}{x^2 \log (x)} \, dx+\int \frac {26-x-x^2}{x^2 \log ^2(x)} \, dx+\int \frac {1}{\log (x)} \, dx\\ &=x+\text {li}(x)+26 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )+\int \frac {26-x-x^2}{x^2 \log ^2(x)} \, dx\\ &=x+26 \text {Ei}(-\log (x))+\text {li}(x)+\int \frac {26-x-x^2}{x^2 \log ^2(x)} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 21, normalized size = 1.31 \begin {gather*} x+\frac {1}{\log (x)}-\frac {26}{x \log (x)}+\frac {x}{\log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(26 - x - x^2 + (26 + x^2)*Log[x] + x^2*Log[x]^2)/(x^2*Log[x]^2),x]

[Out]

x + Log[x]^(-1) - 26/(x*Log[x]) + x/Log[x]

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fricas [A]  time = 0.54, size = 20, normalized size = 1.25 \begin {gather*} \frac {x^{2} \log \relax (x) + x^{2} + x - 26}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(x)^2+(x^2+26)*log(x)-x^2-x+26)/x^2/log(x)^2,x, algorithm="fricas")

[Out]

(x^2*log(x) + x^2 + x - 26)/(x*log(x))

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giac [A]  time = 0.20, size = 16, normalized size = 1.00 \begin {gather*} x + \frac {x^{2} + x - 26}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(x)^2+(x^2+26)*log(x)-x^2-x+26)/x^2/log(x)^2,x, algorithm="giac")

[Out]

x + (x^2 + x - 26)/(x*log(x))

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maple [A]  time = 0.02, size = 17, normalized size = 1.06

method	result	size
risch	\(x +\frac {x^{2}+x -26}{x \ln \relax (x )}\)	\(17\)
norman	\(\frac {-26+x +x^{2}+x^{2} \ln \relax (x )}{x \ln \relax (x )}\)	\(21\)
default	\(x +\frac {x}{\ln \relax (x )}+\frac {1}{\ln \relax (x )}-\frac {26}{x \ln \relax (x )}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*ln(x)^2+(x^2+26)*ln(x)-x^2-x+26)/x^2/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

x+1/x*(x^2+x-26)/ln(x)

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maxima [C]  time = 0.55, size = 30, normalized size = 1.88 \begin {gather*} x + \frac {1}{\log \relax (x)} + 26 \, {\rm Ei}\left (-\log \relax (x)\right ) + {\rm Ei}\left (\log \relax (x)\right ) - \Gamma \left (-1, -\log \relax (x)\right ) - 26 \, \Gamma \left (-1, \log \relax (x)\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(x)^2+(x^2+26)*log(x)-x^2-x+26)/x^2/log(x)^2,x, algorithm="maxima")

[Out]

x + 1/log(x) + 26*Ei(-log(x)) + Ei(log(x)) - gamma(-1, -log(x)) - 26*gamma(-1, log(x))

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mupad [B]  time = 1.49, size = 16, normalized size = 1.00 \begin {gather*} x+\frac {x^2+x-26}{x\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(x)*(x^2 + 26) - x + x^2*log(x)^2 - x^2 + 26)/(x^2*log(x)^2),x)

[Out]

x + (x + x^2 - 26)/(x*log(x))

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sympy [A]  time = 0.09, size = 12, normalized size = 0.75 \begin {gather*} x + \frac {x^{2} + x - 26}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*ln(x)**2+(x**2+26)*ln(x)-x**2-x+26)/x**2/ln(x)**2,x)

[Out]

x + (x**2 + x - 26)/(x*log(x))

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