3.74.33 \(\int \frac {-135-27 \log ^2(5)}{52 x^2+12 x^2 \log ^2(5)} \, dx\)

Optimal. Leaf size=25 \[ 5-\frac {9}{4 \left (-x+\frac {2 x}{3 \left (5+\log ^2(5)\right )}\right )} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 0.92, number of steps used = 3, number of rules used = 3, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.115, Rules used = {6, 12, 30} \begin {gather*} \frac {27 \left (5+\log ^2(5)\right )}{4 x \left (13+3 \log ^2(5)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-135 - 27*Log[5]^2)/(52*x^2 + 12*x^2*Log[5]^2),x]

[Out]

(27*(5 + Log[5]^2))/(4*x*(13 + 3*Log[5]^2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

Rule 12

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-135-27 \log ^2(5)}{x^2 \left (52+12 \log ^2(5)\right )} \, dx\\ &=-\frac {\left (27 \left (5+\log ^2(5)\right )\right ) \int \frac {1}{x^2} \, dx}{4 \left (13+3 \log ^2(5)\right )}\\ &=\frac {27 \left (5+\log ^2(5)\right )}{4 x \left (13+3 \log ^2(5)\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 23, normalized size = 0.92 \begin {gather*} \frac {27 \left (5+\log ^2(5)\right )}{4 x \left (13+3 \log ^2(5)\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-135 - 27*Log[5]^2)/(52*x^2 + 12*x^2*Log[5]^2),x]

[Out]

(27*(5 + Log[5]^2))/(4*x*(13 + 3*Log[5]^2))

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fricas [A]  time = 0.64, size = 21, normalized size = 0.84 \begin {gather*} \frac {27 \, {\left (\log \relax (5)^{2} + 5\right )}}{4 \, {\left (3 \, x \log \relax (5)^{2} + 13 \, x\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*log(5)^2-135)/(12*x^2*log(5)^2+52*x^2),x, algorithm="fricas")

[Out]

27/4*(log(5)^2 + 5)/(3*x*log(5)^2 + 13*x)

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giac [A]  time = 0.13, size = 21, normalized size = 0.84 \begin {gather*} \frac {27 \, {\left (\log \relax (5)^{2} + 5\right )}}{4 \, {\left (3 \, \log \relax (5)^{2} + 13\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*log(5)^2-135)/(12*x^2*log(5)^2+52*x^2),x, algorithm="giac")

[Out]

27/4*(log(5)^2 + 5)/((3*log(5)^2 + 13)*x)

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maple [A]  time = 0.15, size = 22, normalized size = 0.88

method	result	size
gosper	\(\frac {\frac {27 \ln \relax (5)^{2}}{4}+\frac {135}{4}}{x \left (3 \ln \relax (5)^{2}+13\right )}\)	\(22\)
norman	\(\frac {\frac {27 \ln \relax (5)^{2}}{4}+\frac {135}{4}}{x \left (3 \ln \relax (5)^{2}+13\right )}\)	\(22\)
default	\(-\frac {-\frac {27 \ln \relax (5)^{2}}{4}-\frac {135}{4}}{\left (3 \ln \relax (5)^{2}+13\right ) x}\)	\(24\)
risch	\(\frac {27 \ln \relax (5)^{2}}{4 x \left (3 \ln \relax (5)^{2}+13\right )}+\frac {135}{4 \left (3 \ln \relax (5)^{2}+13\right ) x}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-27*ln(5)^2-135)/(12*x^2*ln(5)^2+52*x^2),x,method=_RETURNVERBOSE)

[Out]

27/4/x*(ln(5)^2+5)/(3*ln(5)^2+13)

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maxima [A]  time = 0.36, size = 21, normalized size = 0.84 \begin {gather*} \frac {27 \, {\left (\log \relax (5)^{2} + 5\right )}}{4 \, {\left (3 \, \log \relax (5)^{2} + 13\right )} x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*log(5)^2-135)/(12*x^2*log(5)^2+52*x^2),x, algorithm="maxima")

[Out]

27/4*(log(5)^2 + 5)/((3*log(5)^2 + 13)*x)

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mupad [B]  time = 0.08, size = 21, normalized size = 0.84 \begin {gather*} \frac {27\,\left ({\ln \relax (5)}^2+5\right )}{4\,x\,\left (3\,{\ln \relax (5)}^2+13\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(27*log(5)^2 + 135)/(12*x^2*log(5)^2 + 52*x^2),x)

[Out]

(27*(log(5)^2 + 5))/(4*x*(3*log(5)^2 + 13))

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sympy [A]  time = 0.07, size = 20, normalized size = 0.80 \begin {gather*} - \frac {-135 - 27 \log {\relax (5 )}^{2}}{x \left (12 \log {\relax (5 )}^{2} + 52\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-27*ln(5)**2-135)/(12*x**2*ln(5)**2+52*x**2),x)

[Out]

-(-135 - 27*log(5)**2)/(x*(12*log(5)**2 + 52))

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