3.41.16 \(\int \frac {50}{625 x+50 x \log (x^2)+x \log ^2(x^2)} \, dx\)

Optimal. Leaf size=17 \[ 4+\frac {x}{x+\frac {25 x}{\log \left (x^2\right )}} \]

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Rubi [A]  time = 0.01, antiderivative size = 10, normalized size of antiderivative = 0.59, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 32} \begin {gather*} -\frac {25}{\log \left (x^2\right )+25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[50/(625*x + 50*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

-25/(25 + Log[x^2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=50 \int \frac {1}{625 x+50 x \log \left (x^2\right )+x \log ^2\left (x^2\right )} \, dx\\ &=25 \operatorname {Subst}\left (\int \frac {1}{(25+x)^2} \, dx,x,\log \left (x^2\right )\right )\\ &=-\frac {25}{25+\log \left (x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 10, normalized size = 0.59 \begin {gather*} -\frac {25}{25+\log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[50/(625*x + 50*x*Log[x^2] + x*Log[x^2]^2),x]

[Out]

-25/(25 + Log[x^2])

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fricas [A]  time = 0.58, size = 10, normalized size = 0.59 \begin {gather*} -\frac {25}{\log \left (x^{2}\right ) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50/(x*log(x^2)^2+50*x*log(x^2)+625*x),x, algorithm="fricas")

[Out]

-25/(log(x^2) + 25)

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giac [A]  time = 0.17, size = 10, normalized size = 0.59 \begin {gather*} -\frac {25}{\log \left (x^{2}\right ) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50/(x*log(x^2)^2+50*x*log(x^2)+625*x),x, algorithm="giac")

[Out]

-25/(log(x^2) + 25)

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maple [A]  time = 0.02, size = 11, normalized size = 0.65

method	result	size
default	\(-\frac {25}{\ln \left (x^{2}\right )+25}\)	\(11\)
norman	\(-\frac {25}{\ln \left (x^{2}\right )+25}\)	\(11\)
risch	\(-\frac {25}{\ln \left (x^{2}\right )+25}\)	\(11\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50/(x*ln(x^2)^2+50*x*ln(x^2)+625*x),x,method=_RETURNVERBOSE)

[Out]

-25/(ln(x^2)+25)

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maxima [A]  time = 0.36, size = 10, normalized size = 0.59 \begin {gather*} -\frac {25}{2 \, \log \relax (x) + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50/(x*log(x^2)^2+50*x*log(x^2)+625*x),x, algorithm="maxima")

[Out]

-25/(2*log(x) + 25)

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mupad [B]  time = 3.08, size = 10, normalized size = 0.59 \begin {gather*} -\frac {25}{\ln \left (x^2\right )+25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(50/(625*x + 50*x*log(x^2) + x*log(x^2)^2),x)

[Out]

-25/(log(x^2) + 25)

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sympy [A]  time = 0.08, size = 8, normalized size = 0.47 \begin {gather*} - \frac {25}{\log {\left (x^{2} \right )} + 25} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(50/(x*ln(x**2)**2+50*x*ln(x**2)+625*x),x)

[Out]

-25/(log(x**2) + 25)

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