3.4.13 \(\int \frac {-14-4 x^2+4 \log (5)-\log (x)}{4 x^2} \, dx\)

Optimal. Leaf size=24 \[ \frac {1}{x}-x+\frac {3-4 (-2+\log (5))+\log (x)}{4 x} \]

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Rubi [A]  time = 0.03, antiderivative size = 33, normalized size of antiderivative = 1.38, number of steps used = 6, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {12, 14, 2304} \begin {gather*} -x+\frac {1}{4 x}+\frac {\log (x)}{4 x}+\frac {7-\log (25)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-14 - 4*x^2 + 4*Log[5] - Log[x])/(4*x^2),x]

[Out]

1/(4*x) - x + (7 - Log[25])/(2*x) + Log[x]/(4*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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2304

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Log[c*x^
n]))/(d*(m + 1)), x] - Simp[(b*n*(d*x)^(m + 1))/(d*(m + 1)^2), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{4} \int \frac {-14-4 x^2+4 \log (5)-\log (x)}{x^2} \, dx\\ &=\frac {1}{4} \int \left (-\frac {2 \left (7+2 x^2-2 \log (5)\right )}{x^2}-\frac {\log (x)}{x^2}\right ) \, dx\\ &=-\left (\frac {1}{4} \int \frac {\log (x)}{x^2} \, dx\right )-\frac {1}{2} \int \frac {7+2 x^2-2 \log (5)}{x^2} \, dx\\ &=\frac {1}{4 x}+\frac {\log (x)}{4 x}-\frac {1}{2} \int \left (2+\frac {7-2 \log (5)}{x^2}\right ) \, dx\\ &=\frac {1}{4 x}-x+\frac {7-\log (25)}{2 x}+\frac {\log (x)}{4 x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 24, normalized size = 1.00 \begin {gather*} \frac {15}{4 x}-x+\frac {\log \left (\frac {x}{625}\right )}{4 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-14 - 4*x^2 + 4*Log[5] - Log[x])/(4*x^2),x]

[Out]

15/(4*x) - x + Log[x/625]/(4*x)

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fricas [A]  time = 1.05, size = 20, normalized size = 0.83 \begin {gather*} -\frac {4 \, x^{2} + 4 \, \log \relax (5) - \log \relax (x) - 15}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-log(x)+4*log(5)-4*x^2-14)/x^2,x, algorithm="fricas")

[Out]

-1/4*(4*x^2 + 4*log(5) - log(x) - 15)/x

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giac [A]  time = 0.31, size = 22, normalized size = 0.92 \begin {gather*} -x - \frac {4 \, \log \relax (5) - 15}{4 \, x} + \frac {\log \relax (x)}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-log(x)+4*log(5)-4*x^2-14)/x^2,x, algorithm="giac")

[Out]

-x - 1/4*(4*log(5) - 15)/x + 1/4*log(x)/x

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maple [A]  time = 0.02, size = 24, normalized size = 1.00

method	result	size
default	\(-x +\frac {\ln \relax (x )}{4 x}+\frac {15}{4 x}-\frac {\ln \relax (5)}{x}\)	\(24\)
risch	\(\frac {\ln \relax (x )}{4 x}-\frac {4 x^{2}+4 \ln \relax (5)-15}{4 x}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/4*(-ln(x)+4*ln(5)-4*x^2-14)/x^2,x,method=_RETURNVERBOSE)

[Out]

-x+1/4*ln(x)/x+15/4/x-ln(5)/x

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maxima [A]  time = 0.40, size = 23, normalized size = 0.96 \begin {gather*} -x - \frac {\log \relax (5)}{x} + \frac {\log \relax (x)}{4 \, x} + \frac {15}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-log(x)+4*log(5)-4*x^2-14)/x^2,x, algorithm="maxima")

[Out]

-x - log(5)/x + 1/4*log(x)/x + 15/4/x

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mupad [B]  time = 0.38, size = 18, normalized size = 0.75 \begin {gather*} \frac {\frac {\ln \relax (x)}{4}-\ln \relax (5)+\frac {15}{4}}{x}-x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)/4 - log(5) + x^2 + 7/2)/x^2,x)

[Out]

(log(x)/4 - log(5) + 15/4)/x - x

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sympy [A]  time = 0.13, size = 17, normalized size = 0.71 \begin {gather*} - x + \frac {\log {\relax (x )}}{4 x} - \frac {-15 + 4 \log {\relax (5 )}}{4 x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/4*(-ln(x)+4*ln(5)-4*x**2-14)/x**2,x)

[Out]

-x + log(x)/(4*x) - (-15 + 4*log(5))/(4*x)

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