3.30.61 \(\int \frac {-10+2 \log (5)+(5-\log (5)) \log (4 x^2)}{4 x^2} \, dx\)

Optimal. Leaf size=23 \[ \frac {x^2+\frac {1}{4} x (-5+\log (5)) \log \left (4 x^2\right )}{x^2} \]

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 19, normalized size of antiderivative = 0.83, number of steps used = 2, number of rules used = 2, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {12, 2303} \begin {gather*} -\frac {(5-\log (5)) \log \left (4 x^2\right )}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-10 + 2*Log[5] + (5 - Log[5])*Log[4*x^2])/(4*x^2),x]

[Out]

-1/4*((5 - Log[5])*Log[4*x^2])/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 2303

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(b*(d*x)^(m + 1)*Log[c*x^n])/(
d*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1] && EqQ[a*(m + 1) - b*n, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-10+2 \log (5)+(5-\log (5)) \log \left (4 x^2\right )}{x^2} \, dx\\ &=-\frac {(5-\log (5)) \log \left (4 x^2\right )}{4 x}\\ \end {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.00, size = 17, normalized size = 0.74 \begin {gather*} \frac {(-5+\log (5)) \log \left (4 x^2\right )}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-10 + 2*Log[5] + (5 - Log[5])*Log[4*x^2])/(4*x^2),x]

[Out]

((-5 + Log[5])*Log[4*x^2])/(4*x)

________________________________________________________________________________________

fricas [A]  time = 0.64, size = 15, normalized size = 0.65 \begin {gather*} \frac {{\left (\log \relax (5) - 5\right )} \log \left (4 \, x^{2}\right )}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-log(5)+5)*log(4*x^2)+2*log(5)-10)/x^2,x, algorithm="fricas")

[Out]

1/4*(log(5) - 5)*log(4*x^2)/x

________________________________________________________________________________________

giac [A]  time = 0.24, size = 15, normalized size = 0.65 \begin {gather*} \frac {{\left (\log \relax (5) - 5\right )} \log \left (4 \, x^{2}\right )}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-log(5)+5)*log(4*x^2)+2*log(5)-10)/x^2,x, algorithm="giac")

[Out]

1/4*(log(5) - 5)*log(4*x^2)/x

________________________________________________________________________________________

maple [A]  time = 0.03, size = 16, normalized size = 0.70

method	result	size
risch	\(\frac {\left (\ln \relax (5)-5\right ) \ln \left (4 x^{2}\right )}{4 x}\)	\(16\)
norman	\(\frac {\left (\frac {\ln \relax (5)}{4}-\frac {5}{4}\right ) \ln \left (4 x^{2}\right )}{x}\)	\(17\)
default	\(-\frac {5 \ln \relax (2)}{2 x}+\frac {\ln \relax (2) \ln \relax (5)}{2 x}+\frac {\ln \relax (5) \ln \left (x^{2}\right )}{4 x}-\frac {5 \ln \left (x^{2}\right )}{4 x}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*((-ln(5)+5)*ln(4*x^2)+2*ln(5)-10)/x^2,x,method=_RETURNVERBOSE)

[Out]

1/4*(ln(5)-5)/x*ln(4*x^2)

________________________________________________________________________________________

maxima [A]  time = 0.39, size = 39, normalized size = 1.70 \begin {gather*} \frac {1}{4} \, {\left (\frac {\log \left (4 \, x^{2}\right )}{x} + \frac {2}{x}\right )} \log \relax (5) - \frac {\log \relax (5)}{2 \, x} - \frac {5 \, \log \left (4 \, x^{2}\right )}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-log(5)+5)*log(4*x^2)+2*log(5)-10)/x^2,x, algorithm="maxima")

[Out]

1/4*(log(4*x^2)/x + 2/x)*log(5) - 1/2*log(5)/x - 5/4*log(4*x^2)/x

________________________________________________________________________________________

mupad [B]  time = 1.73, size = 15, normalized size = 0.65 \begin {gather*} \frac {\ln \left (4\,x^2\right )\,\left (\ln \relax (5)-5\right )}{4\,x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-((log(4*x^2)*(log(5) - 5))/4 - log(5)/2 + 5/2)/x^2,x)

[Out]

(log(4*x^2)*(log(5) - 5))/(4*x)

________________________________________________________________________________________

sympy [A]  time = 0.14, size = 14, normalized size = 0.61 \begin {gather*} \frac {\left (-5 + \log {\relax (5 )}\right ) \log {\left (4 x^{2} \right )}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*((-ln(5)+5)*ln(4*x**2)+2*ln(5)-10)/x**2,x)

[Out]

(-5 + log(5))*log(4*x**2)/(4*x)

________________________________________________________________________________________