3.50.7 \(\int \frac {7 x+2 x^2+(-6-2 x) \log (2)+(-2 x \log (2)+2 \log ^2(2)) \log (x)}{x} \, dx\)

Optimal. Leaf size=19 \[ -1+x-3 \log (25)+(-3-x+\log (2) \log (x))^2 \]

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Rubi [B]  time = 0.06, antiderivative size = 39, normalized size of antiderivative = 2.05, number of steps used = 7, number of rules used = 4, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {14, 2346, 2301, 2295} \begin {gather*} x^2+\log ^2(2) \log ^2(x)-2 x \log (2) \log (x)+x (7-\log (4))+2 x \log (2)-6 \log (2) \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(7*x + 2*x^2 + (-6 - 2*x)*Log[2] + (-2*x*Log[2] + 2*Log[2]^2)*Log[x])/x,x]

[Out]

x^2 + 2*x*Log[2] + x*(7 - Log[4]) - 6*Log[2]*Log[x] - 2*x*Log[2]*Log[x] + Log[2]^2*Log[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 2295

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

Rule 2301

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))/(x_), x_Symbol] :> Simp[(a + b*Log[c*x^n])^2/(2*b*n), x] /; FreeQ[{a
, b, c, n}, x]

Rule 2346

Int[(((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.))/(x_), x_Symbol] :> Dist[d, Int[((d
+ e*x)^(q - 1)*(a + b*Log[c*x^n])^p)/x, x], x] + Dist[e, Int[(d + e*x)^(q - 1)*(a + b*Log[c*x^n])^p, x], x] /
; FreeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && GtQ[q, 0] && IntegerQ[2*q]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {2 x^2-6 \log (2)+x (7-\log (4))}{x}-\frac {2 (x-\log (2)) \log (2) \log (x)}{x}\right ) \, dx\\ &=-\left ((2 \log (2)) \int \frac {(x-\log (2)) \log (x)}{x} \, dx\right )+\int \frac {2 x^2-6 \log (2)+x (7-\log (4))}{x} \, dx\\ &=-((2 \log (2)) \int \log (x) \, dx)+\left (2 \log ^2(2)\right ) \int \frac {\log (x)}{x} \, dx+\int \left (7+2 x-\frac {6 \log (2)}{x}-\log (4)\right ) \, dx\\ &=x^2+2 x \log (2)+x (7-\log (4))-6 \log (2) \log (x)-2 x \log (2) \log (x)+\log ^2(2) \log ^2(x)\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.01, size = 39, normalized size = 2.05 \begin {gather*} 7 x+x^2+2 x \log (2)-x \log (4)-6 \log (2) \log (x)-2 x \log (2) \log (x)+\log ^2(2) \log ^2(x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(7*x + 2*x^2 + (-6 - 2*x)*Log[2] + (-2*x*Log[2] + 2*Log[2]^2)*Log[x])/x,x]

[Out]

7*x + x^2 + 2*x*Log[2] - x*Log[4] - 6*Log[2]*Log[x] - 2*x*Log[2]*Log[x] + Log[2]^2*Log[x]^2

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fricas [A]  time = 0.56, size = 25, normalized size = 1.32 \begin {gather*} \log \relax (2)^{2} \log \relax (x)^{2} - 2 \, {\left (x + 3\right )} \log \relax (2) \log \relax (x) + x^{2} + 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)^2-2*x*log(2))*log(x)+(-2*x-6)*log(2)+2*x^2+7*x)/x,x, algorithm="fricas")

[Out]

log(2)^2*log(x)^2 - 2*(x + 3)*log(2)*log(x) + x^2 + 7*x

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giac [A]  time = 0.15, size = 29, normalized size = 1.53 \begin {gather*} \log \relax (2)^{2} \log \relax (x)^{2} - 2 \, x \log \relax (2) \log \relax (x) + x^{2} - 6 \, \log \relax (2) \log \relax (x) + 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)^2-2*x*log(2))*log(x)+(-2*x-6)*log(2)+2*x^2+7*x)/x,x, algorithm="giac")

[Out]

log(2)^2*log(x)^2 - 2*x*log(2)*log(x) + x^2 - 6*log(2)*log(x) + 7*x

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maple [A]  time = 0.04, size = 30, normalized size = 1.58

method	result	size
norman	\(x^{2}-6 \ln \relax (2) \ln \relax (x )+\ln \relax (2)^{2} \ln \relax (x )^{2}+7 x -2 x \ln \relax (2) \ln \relax (x )\)	\(30\)
risch	\(x^{2}-6 \ln \relax (2) \ln \relax (x )+\ln \relax (2)^{2} \ln \relax (x )^{2}+7 x -2 x \ln \relax (2) \ln \relax (x )\)	\(30\)
default	\(\ln \relax (2)^{2} \ln \relax (x )^{2}-2 \ln \relax (2) \left (x \ln \relax (x )-x \right )-2 x \ln \relax (2)+x^{2}-6 \ln \relax (2) \ln \relax (x )+7 x\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*ln(2)^2-2*x*ln(2))*ln(x)+(-2*x-6)*ln(2)+2*x^2+7*x)/x,x,method=_RETURNVERBOSE)

[Out]

x^2-6*ln(2)*ln(x)+ln(2)^2*ln(x)^2+7*x-2*x*ln(2)*ln(x)

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maxima [B]  time = 0.36, size = 39, normalized size = 2.05 \begin {gather*} \log \relax (2)^{2} \log \relax (x)^{2} + x^{2} - 2 \, {\left (x \log \relax (x) - x\right )} \log \relax (2) - 2 \, x \log \relax (2) - 6 \, \log \relax (2) \log \relax (x) + 7 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*log(2)^2-2*x*log(2))*log(x)+(-2*x-6)*log(2)+2*x^2+7*x)/x,x, algorithm="maxima")

[Out]

log(2)^2*log(x)^2 + x^2 - 2*(x*log(x) - x)*log(2) - 2*x*log(2) - 6*log(2)*log(x) + 7*x

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mupad [B]  time = 4.11, size = 29, normalized size = 1.53 \begin {gather*} x^2-2\,\ln \relax (2)\,x\,\ln \relax (x)+7\,x+{\ln \relax (2)}^2\,{\ln \relax (x)}^2-6\,\ln \relax (2)\,\ln \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x - log(2)*(2*x + 6) - log(x)*(2*x*log(2) - 2*log(2)^2) + 2*x^2)/x,x)

[Out]

7*x + log(2)^2*log(x)^2 - 6*log(2)*log(x) + x^2 - 2*x*log(2)*log(x)

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sympy [A]  time = 0.19, size = 34, normalized size = 1.79 \begin {gather*} x^{2} - 2 x \log {\relax (2 )} \log {\relax (x )} + 7 x + \log {\relax (2 )}^{2} \log {\relax (x )}^{2} - 6 \log {\relax (2 )} \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*ln(2)**2-2*x*ln(2))*ln(x)+(-2*x-6)*ln(2)+2*x**2+7*x)/x,x)

[Out]

x**2 - 2*x*log(2)*log(x) + 7*x + log(2)**2*log(x)**2 - 6*log(2)*log(x)

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