3.78.87 \(\int \frac {121+22 x+4 x^2+(-22-2 x) \log (2)+\log ^2(2)+(-121+x^2+22 \log (2)-\log ^2(2)) \log (x)}{x^2} \, dx\)

Optimal. Leaf size=22 \[ 5+x+x \left (2+\frac {(11+x-\log (2))^2 \log (x)}{x^2}\right ) \]

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Rubi [A]  time = 0.10, antiderivative size = 31, normalized size of antiderivative = 1.41, number of steps used = 8, number of rules used = 4, integrand size = 44, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {14, 2357, 2295, 2304} \begin {gather*} 3 x+x \log (x)+(22-\log (4)) \log (x)+\frac {(11-\log (2))^2 \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(121 + 22*x + 4*x^2 + (-22 - 2*x)*Log[2] + Log[2]^2 + (-121 + x^2 + 22*Log[2] - Log[2]^2)*Log[x])/x^2,x]

[Out]

3*x + x*Log[x] + ((11 - Log[2])^2*Log[x])/x + (22 - Log[4])*Log[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 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

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 2357

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*x^
n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, n}, x] && RationalFunctionQ[RFx, x] && IGtQ[p, 0]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {4 x^2+(-11+\log (2))^2+x (22-\log (4))}{x^2}+\frac {(11+x-\log (2)) (-11+x+\log (2)) \log (x)}{x^2}\right ) \, dx\\ &=\int \frac {4 x^2+(-11+\log (2))^2+x (22-\log (4))}{x^2} \, dx+\int \frac {(11+x-\log (2)) (-11+x+\log (2)) \log (x)}{x^2} \, dx\\ &=\int \left (4+\frac {(-11+\log (2))^2}{x^2}+\frac {22-\log (4)}{x}\right ) \, dx+\int \left (\log (x)-\frac {(-11+\log (2))^2 \log (x)}{x^2}\right ) \, dx\\ &=4 x-\frac {(11-\log (2))^2}{x}+(22-\log (4)) \log (x)-(-11+\log (2))^2 \int \frac {\log (x)}{x^2} \, dx+\int \log (x) \, dx\\ &=3 x+x \log (x)+\frac {(11-\log (2))^2 \log (x)}{x}+(22-\log (4)) \log (x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 44, normalized size = 2.00 \begin {gather*} 3 x+22 \log (x)+\frac {121 \log (x)}{x}+x \log (x)-2 \log (2) \log (x)-\frac {22 \log (2) \log (x)}{x}+\frac {\log ^2(2) \log (x)}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(121 + 22*x + 4*x^2 + (-22 - 2*x)*Log[2] + Log[2]^2 + (-121 + x^2 + 22*Log[2] - Log[2]^2)*Log[x])/x^
2,x]

[Out]

3*x + 22*Log[x] + (121*Log[x])/x + x*Log[x] - 2*Log[2]*Log[x] - (22*Log[2]*Log[x])/x + (Log[2]^2*Log[x])/x

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fricas [A]  time = 0.69, size = 32, normalized size = 1.45 \begin {gather*} \frac {3 \, x^{2} + {\left (x^{2} - 2 \, {\left (x + 11\right )} \log \relax (2) + \log \relax (2)^{2} + 22 \, x + 121\right )} \log \relax (x)}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(2)^2+22*log(2)+x^2-121)*log(x)+log(2)^2+(-2*x-22)*log(2)+4*x^2+22*x+121)/x^2,x, algorithm="fr
icas")

[Out]

(3*x^2 + (x^2 - 2*(x + 11)*log(2) + log(2)^2 + 22*x + 121)*log(x))/x

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giac [A]  time = 0.14, size = 31, normalized size = 1.41 \begin {gather*} {\left (x + \frac {\log \relax (2)^{2} - 22 \, \log \relax (2) + 121}{x}\right )} \log \relax (x) - 2 \, {\left (\log \relax (2) - 11\right )} \log \relax (x) + 3 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(2)^2+22*log(2)+x^2-121)*log(x)+log(2)^2+(-2*x-22)*log(2)+4*x^2+22*x+121)/x^2,x, algorithm="gi
ac")

[Out]

(x + (log(2)^2 - 22*log(2) + 121)/x)*log(x) - 2*(log(2) - 11)*log(x) + 3*x

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maple [A]  time = 0.03, size = 34, normalized size = 1.55

method	result	size
risch	\(\frac {\left (\ln \relax (2)^{2}+x^{2}-22 \ln \relax (2)+121\right ) \ln \relax (x )}{x}-2 \ln \relax (2) \ln \relax (x )+22 \ln \relax (x )+3 x\)	\(34\)
norman	\(\frac {x^{2} \ln \relax (x )+\left (\ln \relax (2)^{2}-22 \ln \relax (2)+121\right ) \ln \relax (x )+\left (-2 \ln \relax (2)+22\right ) x \ln \relax (x )+3 x^{2}}{x}\)	\(40\)
default	\(-\ln \relax (2)^{2} \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )+x \ln \relax (x )+3 x +22 \ln \relax (2) \left (-\frac {\ln \relax (x )}{x}-\frac {1}{x}\right )-\frac {\ln \relax (2)^{2}}{x}-2 \ln \relax (2) \ln \relax (x )+\frac {121 \ln \relax (x )}{x}+\frac {22 \ln \relax (2)}{x}+22 \ln \relax (x )\)	\(78\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-ln(2)^2+22*ln(2)+x^2-121)*ln(x)+ln(2)^2+(-2*x-22)*ln(2)+4*x^2+22*x+121)/x^2,x,method=_RETURNVERBOSE)

[Out]

(ln(2)^2+x^2-22*ln(2)+121)/x*ln(x)-2*ln(2)*ln(x)+22*ln(x)+3*x

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maxima [B]  time = 0.37, size = 70, normalized size = 3.18 \begin {gather*} {\left (\frac {\log \relax (x)}{x} + \frac {1}{x}\right )} \log \relax (2)^{2} - 22 \, {\left (\frac {\log \relax (x)}{x} + \frac {1}{x}\right )} \log \relax (2) + x \log \relax (x) - 2 \, \log \relax (2) \log \relax (x) + 3 \, x - \frac {\log \relax (2)^{2}}{x} + \frac {22 \, \log \relax (2)}{x} + \frac {121 \, \log \relax (x)}{x} + 22 \, \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-log(2)^2+22*log(2)+x^2-121)*log(x)+log(2)^2+(-2*x-22)*log(2)+4*x^2+22*x+121)/x^2,x, algorithm="ma
xima")

[Out]

(log(x)/x + 1/x)*log(2)^2 - 22*(log(x)/x + 1/x)*log(2) + x*log(x) - 2*log(2)*log(x) + 3*x - log(2)^2/x + 22*lo
g(2)/x + 121*log(x)/x + 22*log(x)

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mupad [B]  time = 5.20, size = 27, normalized size = 1.23 \begin {gather*} x\,\left (\ln \relax (x)+3\right )-\ln \relax (x)\,\left (\ln \relax (4)-22\right )+\frac {\ln \relax (x)\,{\left (\ln \relax (2)-11\right )}^2}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((22*x - log(2)*(2*x + 22) + log(x)*(22*log(2) - log(2)^2 + x^2 - 121) + log(2)^2 + 4*x^2 + 121)/x^2,x)

[Out]

x*(log(x) + 3) - log(x)*(log(4) - 22) + (log(x)*(log(2) - 11)^2)/x

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sympy [A]  time = 0.19, size = 32, normalized size = 1.45 \begin {gather*} 3 x - 2 \left (-11 + \log {\relax (2 )}\right ) \log {\relax (x )} + \frac {\left (x^{2} - 22 \log {\relax (2 )} + \log {\relax (2 )}^{2} + 121\right ) \log {\relax (x )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-ln(2)**2+22*ln(2)+x**2-121)*ln(x)+ln(2)**2+(-2*x-22)*ln(2)+4*x**2+22*x+121)/x**2,x)

[Out]

3*x - 2*(-11 + log(2))*log(x) + (x**2 - 22*log(2) + log(2)**2 + 121)*log(x)/x

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