3.17.2 \(\int \frac {200+200 \log (x)+x^2 \log ^2(x)}{x^2 \log ^2(x)} \, dx\)

Optimal. Leaf size=12 \[ 2+x-\frac {200}{x \log (x)} \]

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Rubi [A]  time = 0.17, antiderivative size = 11, normalized size of antiderivative = 0.92, number of steps used = 7, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {6742, 2306, 2309, 2178} \begin {gather*} x-\frac {200}{x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(200 + 200*Log[x] + x^2*Log[x]^2)/(x^2*Log[x]^2),x]

[Out]

x - 200/(x*Log[x])

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 2306

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log
[c*x^n])^(p + 1))/(b*d*n*(p + 1)), x] - Dist[(m + 1)/(b*n*(p + 1)), Int[(d*x)^m*(a + b*Log[c*x^n])^(p + 1), x]
, x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && LtQ[p, -1]

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (1+\frac {200}{x^2 \log ^2(x)}+\frac {200}{x^2 \log (x)}\right ) \, dx\\ &=x+200 \int \frac {1}{x^2 \log ^2(x)} \, dx+200 \int \frac {1}{x^2 \log (x)} \, dx\\ &=x-\frac {200}{x \log (x)}-200 \int \frac {1}{x^2 \log (x)} \, dx+200 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=x+200 \text {Ei}(-\log (x))-\frac {200}{x \log (x)}-200 \operatorname {Subst}\left (\int \frac {e^{-x}}{x} \, dx,x,\log (x)\right )\\ &=x-\frac {200}{x \log (x)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 11, normalized size = 0.92 \begin {gather*} x-\frac {200}{x \log (x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(200 + 200*Log[x] + x^2*Log[x]^2)/(x^2*Log[x]^2),x]

[Out]

x - 200/(x*Log[x])

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fricas [A]  time = 0.61, size = 16, normalized size = 1.33 \begin {gather*} \frac {x^{2} \log \relax (x) - 200}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(x)^2+200*log(x)+200)/x^2/log(x)^2,x, algorithm="fricas")

[Out]

(x^2*log(x) - 200)/(x*log(x))

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giac [A]  time = 0.18, size = 11, normalized size = 0.92 \begin {gather*} x - \frac {200}{x \log \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x^2*log(x)^2+200*log(x)+200)/x^2/log(x)^2,x, algorithm="giac")

[Out]

x - 200/(x*log(x))

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maple [A]  time = 0.02, size = 12, normalized size = 1.00

method	result	size
default	\(x -\frac {200}{x \ln \relax (x )}\)	\(12\)
risch	\(x -\frac {200}{x \ln \relax (x )}\)	\(12\)
norman	\(\frac {-200+x^{2} \ln \relax (x )}{x \ln \relax (x )}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((x^2*ln(x)^2+200*ln(x)+200)/x^2/ln(x)^2,x,method=_RETURNVERBOSE)

[Out]

x-200/x/ln(x)

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maxima [C]  time = 0.45, size = 15, normalized size = 1.25 \begin {gather*} x + 200 \, {\rm Ei}\left (-\log \relax (x)\right ) - 200 \, \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+200*log(x)+200)/x^2/log(x)^2,x, algorithm="maxima")

[Out]

x + 200*Ei(-log(x)) - 200*gamma(-1, log(x))

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mupad [B]  time = 1.05, size = 11, normalized size = 0.92 \begin {gather*} x-\frac {200}{x\,\ln \relax (x)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((200*log(x) + x^2*log(x)^2 + 200)/(x^2*log(x)^2),x)

[Out]

x - 200/(x*log(x))

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sympy [A]  time = 0.08, size = 7, normalized size = 0.58 \begin {gather*} x - \frac {200}{x \log {\relax (x )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((x**2*ln(x)**2+200*ln(x)+200)/x**2/ln(x)**2,x)

[Out]

x - 200/(x*log(x))

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