3.71.27 \(\int \frac {-4 x^3-(2 x^2-2 x^4) \log (\frac {4}{3})+2 \log (\frac {4}{3}) \log (x)-2 \log (\frac {4}{3}) \log ^2(x)}{x^3} \, dx\)

Optimal. Leaf size=27 \[ 4 \left (1-x+\frac {1}{4} \log \left (\frac {4}{3}\right ) \left (-x+\frac {\log (x)}{x}\right )^2\right ) \]

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Rubi [B]  time = 0.06, antiderivative size = 80, normalized size of antiderivative = 2.96, number of steps used = 7, number of rules used = 3, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {14, 2304, 2305} \begin {gather*} \frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-4 x-2 \log \left (\frac {4}{3}\right ) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4*x^3 - (2*x^2 - 2*x^4)*Log[4/3] + 2*Log[4/3]*Log[x] - 2*Log[4/3]*Log[x]^2)/x^3,x]

[Out]

-4*x + Log[4/3]/(2*x^2) - Log[16/9]/(4*x^2) + (x^2*Log[16/9])/2 - 2*Log[4/3]*Log[x] + (Log[4/3]*Log[x])/x^2 -
(Log[16/9]*Log[x])/(2*x^2) + (Log[4/3]*Log[x]^2)/x^2

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
]

Rule 2305

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[((d*x)^(m + 1)*(a + b*Lo
g[c*x^n])^p)/(d*(m + 1)), x] - Dist[(b*n*p)/(m + 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] && GtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {-4 x-2 \log \left (\frac {4}{3}\right )+x^2 \log \left (\frac {16}{9}\right )}{x}+\frac {\log \left (\frac {16}{9}\right ) \log (x)}{x^3}-\frac {2 \log \left (\frac {4}{3}\right ) \log ^2(x)}{x^3}\right ) \, dx\\ &=-\left (\left (2 \log \left (\frac {4}{3}\right )\right ) \int \frac {\log ^2(x)}{x^3} \, dx\right )+\log \left (\frac {16}{9}\right ) \int \frac {\log (x)}{x^3} \, dx+\int \frac {-4 x-2 \log \left (\frac {4}{3}\right )+x^2 \log \left (\frac {16}{9}\right )}{x} \, dx\\ &=-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}-\left (2 \log \left (\frac {4}{3}\right )\right ) \int \frac {\log (x)}{x^3} \, dx+\int \left (-4-\frac {2 \log \left (\frac {4}{3}\right )}{x}+x \log \left (\frac {16}{9}\right )\right ) \, dx\\ &=-4 x+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-2 \log \left (\frac {4}{3}\right ) \log (x)+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 80, normalized size = 2.96 \begin {gather*} -4 x+\frac {\log \left (\frac {4}{3}\right )}{2 x^2}-\frac {\log \left (\frac {16}{9}\right )}{4 x^2}+\frac {1}{2} x^2 \log \left (\frac {16}{9}\right )-2 \log \left (\frac {4}{3}\right ) \log (x)+\frac {\log \left (\frac {4}{3}\right ) \log (x)}{x^2}-\frac {\log \left (\frac {16}{9}\right ) \log (x)}{2 x^2}+\frac {\log \left (\frac {4}{3}\right ) \log ^2(x)}{x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4*x^3 - (2*x^2 - 2*x^4)*Log[4/3] + 2*Log[4/3]*Log[x] - 2*Log[4/3]*Log[x]^2)/x^3,x]

[Out]

-4*x + Log[4/3]/(2*x^2) - Log[16/9]/(4*x^2) + (x^2*Log[16/9])/2 - 2*Log[4/3]*Log[x] + (Log[4/3]*Log[x])/x^2 -
(Log[16/9]*Log[x])/(2*x^2) + (Log[4/3]*Log[x]^2)/x^2

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fricas [A]  time = 0.73, size = 33, normalized size = 1.22 \begin {gather*} -\frac {x^{4} \log \left (\frac {3}{4}\right ) - 2 \, x^{2} \log \left (\frac {3}{4}\right ) \log \relax (x) + 4 \, x^{3} + \log \left (\frac {3}{4}\right ) \log \relax (x)^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(3/4)*log(x)^2-2*log(3/4)*log(x)+(-2*x^4+2*x^2)*log(3/4)-4*x^3)/x^3,x, algorithm="fricas")

[Out]

-(x^4*log(3/4) - 2*x^2*log(3/4)*log(x) + 4*x^3 + log(3/4)*log(x)^2)/x^2

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giac [B]  time = 0.24, size = 43, normalized size = 1.59 \begin {gather*} -x^{2} {\left (\log \relax (3) - 2 \, \log \relax (2)\right )} + 2 \, {\left (\log \relax (3) - 2 \, \log \relax (2)\right )} \log \relax (x) - 4 \, x - \frac {{\left (\log \relax (3) - 2 \, \log \relax (2)\right )} \log \relax (x)^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(3/4)*log(x)^2-2*log(3/4)*log(x)+(-2*x^4+2*x^2)*log(3/4)-4*x^3)/x^3,x, algorithm="giac")

[Out]

-x^2*(log(3) - 2*log(2)) + 2*(log(3) - 2*log(2))*log(x) - 4*x - (log(3) - 2*log(2))*log(x)^2/x^2

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maple [A]  time = 0.03, size = 48, normalized size = 1.78

method	result	size
risch	\(\frac {\left (-\ln \relax (3)+2 \ln \relax (2)\right ) \ln \relax (x )^{2}}{x^{2}}+2 x^{2} \ln \relax (2)-x^{2} \ln \relax (3)-4 \ln \relax (2) \ln \relax (x )+2 \ln \relax (3) \ln \relax (x )-4 x\)	\(48\)
norman	\(\frac {\left (-\ln \relax (3)+2 \ln \relax (2)\right ) x^{4}+\left (-\ln \relax (3)+2 \ln \relax (2)\right ) \ln \relax (x )^{2}+\left (2 \ln \relax (3)-4 \ln \relax (2)\right ) x^{2} \ln \relax (x )-4 x^{3}}{x^{2}}\)	\(53\)
default	\(-x^{2} \ln \relax (3)+2 x^{2} \ln \relax (2)+2 \ln \relax (3) \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )-4 \ln \relax (2) \left (-\frac {\ln \relax (x )^{2}}{2 x^{2}}-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+2 \ln \relax (3) \ln \relax (x )-4 \ln \relax (2) \ln \relax (x )-4 x -2 \ln \relax (3) \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )+4 \ln \relax (2) \left (-\frac {\ln \relax (x )}{2 x^{2}}-\frac {1}{4 x^{2}}\right )\)	\(117\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*ln(3/4)*ln(x)^2-2*ln(3/4)*ln(x)+(-2*x^4+2*x^2)*ln(3/4)-4*x^3)/x^3,x,method=_RETURNVERBOSE)

[Out]

(-ln(3)+2*ln(2))/x^2*ln(x)^2+2*x^2*ln(2)-x^2*ln(3)-4*ln(2)*ln(x)+2*ln(3)*ln(x)-4*x

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maxima [B]  time = 0.37, size = 51, normalized size = 1.89 \begin {gather*} -x^{2} \log \left (\frac {3}{4}\right ) + \frac {1}{2} \, {\left (\frac {2 \, \log \relax (x)}{x^{2}} + \frac {1}{x^{2}}\right )} \log \left (\frac {3}{4}\right ) + 2 \, \log \left (\frac {3}{4}\right ) \log \relax (x) - 4 \, x - \frac {{\left (2 \, \log \relax (x)^{2} + 2 \, \log \relax (x) + 1\right )} \log \left (\frac {3}{4}\right )}{2 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*log(3/4)*log(x)^2-2*log(3/4)*log(x)+(-2*x^4+2*x^2)*log(3/4)-4*x^3)/x^3,x, algorithm="maxima")

[Out]

-x^2*log(3/4) + 1/2*(2*log(x)/x^2 + 1/x^2)*log(3/4) + 2*log(3/4)*log(x) - 4*x - 1/2*(2*log(x)^2 + 2*log(x) + 1
)*log(3/4)/x^2

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mupad [B]  time = 4.15, size = 25, normalized size = 0.93 \begin {gather*} x^2\,\ln \left (\frac {4}{3}\right )-4\,x+\ln \left (\frac {9}{16}\right )\,\ln \relax (x)+\frac {\ln \left (\frac {4}{3}\right )\,{\ln \relax (x)}^2}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*log(3/4)*log(x)^2 + log(3/4)*(2*x^2 - 2*x^4) - 2*log(3/4)*log(x) - 4*x^3)/x^3,x)

[Out]

x^2*log(4/3) - 4*x + log(9/16)*log(x) + (log(4/3)*log(x)^2)/x^2

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sympy [A]  time = 0.23, size = 44, normalized size = 1.63 \begin {gather*} x^{2} \left (- \log {\relax (3 )} + 2 \log {\relax (2 )}\right ) - 4 x - 2 \left (- \log {\relax (3 )} + 2 \log {\relax (2 )}\right ) \log {\relax (x )} + \frac {\left (- \log {\relax (3 )} + 2 \log {\relax (2 )}\right ) \log {\relax (x )}^{2}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*ln(3/4)*ln(x)**2-2*ln(3/4)*ln(x)+(-2*x**4+2*x**2)*ln(3/4)-4*x**3)/x**3,x)

[Out]

x**2*(-log(3) + 2*log(2)) - 4*x - 2*(-log(3) + 2*log(2))*log(x) + (-log(3) + 2*log(2))*log(x)**2/x**2

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