3.79.53 \(\int \frac {-400+9 x^4+(40 x^2+6 x^4) \log (\frac {x}{4})+6 x^4 \log ^2(\frac {x}{4})+2 x^4 \log ^3(\frac {x}{4})+x^4 \log ^4(\frac {x}{4})}{200 x^3} \, dx\)

Optimal. Leaf size=25 \[ \frac {\left (1+\frac {1}{20} x^2 \left (3+\log ^2\left (\frac {x}{4}\right )\right )\right )^2}{x^2} \]

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Rubi [B]  time = 0.14, antiderivative size = 53, normalized size of antiderivative = 2.12, number of steps used = 18, number of rules used = 6, integrand size = 70, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {12, 14, 2351, 2301, 2304, 2305} \begin {gather*} \frac {9 x^2}{400}+\frac {1}{x^2}+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{10} \log ^2\left (\frac {x}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-400 + 9*x^4 + (40*x^2 + 6*x^4)*Log[x/4] + 6*x^4*Log[x/4]^2 + 2*x^4*Log[x/4]^3 + x^4*Log[x/4]^4)/(200*x^3
),x]

[Out]

x^(-2) + (9*x^2)/400 + Log[x/4]^2/10 + (3*x^2*Log[x/4]^2)/200 + (x^2*Log[x/4]^4)/400

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 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 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]

Rule 2351

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Wit
h[{u = ExpandIntegrand[a + b*Log[c*x^n], (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c,
d, e, f, m, n, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[m] && IntegerQ[r]))

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{200} \int \frac {-400+9 x^4+\left (40 x^2+6 x^4\right ) \log \left (\frac {x}{4}\right )+6 x^4 \log ^2\left (\frac {x}{4}\right )+2 x^4 \log ^3\left (\frac {x}{4}\right )+x^4 \log ^4\left (\frac {x}{4}\right )}{x^3} \, dx\\ &=\frac {1}{200} \int \left (\frac {-400+9 x^4}{x^3}+\frac {2 \left (20+3 x^2\right ) \log \left (\frac {x}{4}\right )}{x}+6 x \log ^2\left (\frac {x}{4}\right )+2 x \log ^3\left (\frac {x}{4}\right )+x \log ^4\left (\frac {x}{4}\right )\right ) \, dx\\ &=\frac {1}{200} \int \frac {-400+9 x^4}{x^3} \, dx+\frac {1}{200} \int x \log ^4\left (\frac {x}{4}\right ) \, dx+\frac {1}{100} \int \frac {\left (20+3 x^2\right ) \log \left (\frac {x}{4}\right )}{x} \, dx+\frac {1}{100} \int x \log ^3\left (\frac {x}{4}\right ) \, dx+\frac {3}{100} \int x \log ^2\left (\frac {x}{4}\right ) \, dx\\ &=\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{200} x^2 \log ^3\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )+\frac {1}{200} \int \left (-\frac {400}{x^3}+9 x\right ) \, dx-\frac {1}{100} \int x \log ^3\left (\frac {x}{4}\right ) \, dx+\frac {1}{100} \int \left (\frac {20 \log \left (\frac {x}{4}\right )}{x}+3 x \log \left (\frac {x}{4}\right )\right ) \, dx-\frac {3}{200} \int x \log ^2\left (\frac {x}{4}\right ) \, dx-\frac {3}{100} \int x \log \left (\frac {x}{4}\right ) \, dx\\ &=\frac {1}{x^2}+\frac {3 x^2}{100}-\frac {3}{200} x^2 \log \left (\frac {x}{4}\right )+\frac {3}{400} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )+\frac {3}{200} \int x \log \left (\frac {x}{4}\right ) \, dx+\frac {3}{200} \int x \log ^2\left (\frac {x}{4}\right ) \, dx+\frac {3}{100} \int x \log \left (\frac {x}{4}\right ) \, dx+\frac {1}{5} \int \frac {\log \left (\frac {x}{4}\right )}{x} \, dx\\ &=\frac {1}{x^2}+\frac {3 x^2}{160}+\frac {3}{400} x^2 \log \left (\frac {x}{4}\right )+\frac {1}{10} \log ^2\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )-\frac {3}{200} \int x \log \left (\frac {x}{4}\right ) \, dx\\ &=\frac {1}{x^2}+\frac {9 x^2}{400}+\frac {1}{10} \log ^2\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.02, size = 53, normalized size = 2.12 \begin {gather*} \frac {1}{x^2}+\frac {9 x^2}{400}+\frac {1}{10} \log ^2\left (\frac {x}{4}\right )+\frac {3}{200} x^2 \log ^2\left (\frac {x}{4}\right )+\frac {1}{400} x^2 \log ^4\left (\frac {x}{4}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-400 + 9*x^4 + (40*x^2 + 6*x^4)*Log[x/4] + 6*x^4*Log[x/4]^2 + 2*x^4*Log[x/4]^3 + x^4*Log[x/4]^4)/(2
00*x^3),x]

[Out]

x^(-2) + (9*x^2)/400 + Log[x/4]^2/10 + (3*x^2*Log[x/4]^2)/200 + (x^2*Log[x/4]^4)/400

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fricas [A]  time = 0.61, size = 41, normalized size = 1.64 \begin {gather*} \frac {x^{4} \log \left (\frac {1}{4} \, x\right )^{4} + 9 \, x^{4} + 2 \, {\left (3 \, x^{4} + 20 \, x^{2}\right )} \log \left (\frac {1}{4} \, x\right )^{2} + 400}{400 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/200*(x^4*log(1/4*x)^4+2*x^4*log(1/4*x)^3+6*x^4*log(1/4*x)^2+(6*x^4+40*x^2)*log(1/4*x)+9*x^4-400)/x
^3,x, algorithm="fricas")

[Out]

1/400*(x^4*log(1/4*x)^4 + 9*x^4 + 2*(3*x^4 + 20*x^2)*log(1/4*x)^2 + 400)/x^2

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giac [A]  time = 0.18, size = 35, normalized size = 1.40 \begin {gather*} \frac {1}{400} \, x^{2} \log \left (\frac {1}{4} \, x\right )^{4} + \frac {1}{200} \, {\left (3 \, x^{2} + 20\right )} \log \left (\frac {1}{4} \, x\right )^{2} + \frac {9}{400} \, x^{2} + \frac {1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/200*(x^4*log(1/4*x)^4+2*x^4*log(1/4*x)^3+6*x^4*log(1/4*x)^2+(6*x^4+40*x^2)*log(1/4*x)+9*x^4-400)/x
^3,x, algorithm="giac")

[Out]

1/400*x^2*log(1/4*x)^4 + 1/200*(3*x^2 + 20)*log(1/4*x)^2 + 9/400*x^2 + 1/x^2

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maple [A]  time = 0.03, size = 40, normalized size = 1.60

method	result	size
derivativedivides	\(\frac {x^{2} \ln \left (\frac {x}{4}\right )^{4}}{400}+\frac {3 x^{2} \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {9 x^{2}}{400}+\frac {\ln \left (\frac {x}{4}\right )^{2}}{10}+\frac {1}{x^{2}}\)	\(40\)
default	\(\frac {x^{2} \ln \left (\frac {x}{4}\right )^{4}}{400}+\frac {3 x^{2} \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {9 x^{2}}{400}+\frac {\ln \left (\frac {x}{4}\right )^{2}}{10}+\frac {1}{x^{2}}\)	\(40\)
risch	\(\frac {x^{2} \ln \left (\frac {x}{4}\right )^{4}}{400}+\frac {\left (3 x^{2}+20\right ) \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {9 x^{4}+400}{400 x^{2}}\)	\(40\)
norman	\(\frac {1+\frac {9 x^{4}}{400}+\frac {x^{2} \ln \left (\frac {x}{4}\right )^{2}}{10}+\frac {3 x^{4} \ln \left (\frac {x}{4}\right )^{2}}{200}+\frac {x^{4} \ln \left (\frac {x}{4}\right )^{4}}{400}}{x^{2}}\)	\(45\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/200*(x^4*ln(1/4*x)^4+2*x^4*ln(1/4*x)^3+6*x^4*ln(1/4*x)^2+(6*x^4+40*x^2)*ln(1/4*x)+9*x^4-400)/x^3,x,metho
d=_RETURNVERBOSE)

[Out]

1/400*x^2*ln(1/4*x)^4+3/200*x^2*ln(1/4*x)^2+9/400*x^2+1/10*ln(1/4*x)^2+1/x^2

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maxima [B]  time = 0.37, size = 113, normalized size = 4.52 \begin {gather*} \frac {1}{800} \, {\left (2 \, \log \left (\frac {1}{4} \, x\right )^{4} - 4 \, \log \left (\frac {1}{4} \, x\right )^{3} + 6 \, \log \left (\frac {1}{4} \, x\right )^{2} - 6 \, \log \left (\frac {1}{4} \, x\right ) + 3\right )} x^{2} + \frac {1}{800} \, {\left (4 \, \log \left (\frac {1}{4} \, x\right )^{3} - 6 \, \log \left (\frac {1}{4} \, x\right )^{2} + 6 \, \log \left (\frac {1}{4} \, x\right ) - 3\right )} x^{2} + \frac {3}{400} \, {\left (2 \, \log \left (\frac {1}{4} \, x\right )^{2} - 2 \, \log \left (\frac {1}{4} \, x\right ) + 1\right )} x^{2} + \frac {3}{200} \, x^{2} \log \left (\frac {1}{4} \, x\right ) + \frac {3}{200} \, x^{2} + \frac {1}{10} \, \log \left (\frac {1}{4} \, x\right )^{2} + \frac {1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/200*(x^4*log(1/4*x)^4+2*x^4*log(1/4*x)^3+6*x^4*log(1/4*x)^2+(6*x^4+40*x^2)*log(1/4*x)+9*x^4-400)/x
^3,x, algorithm="maxima")

[Out]

1/800*(2*log(1/4*x)^4 - 4*log(1/4*x)^3 + 6*log(1/4*x)^2 - 6*log(1/4*x) + 3)*x^2 + 1/800*(4*log(1/4*x)^3 - 6*lo
g(1/4*x)^2 + 6*log(1/4*x) - 3)*x^2 + 3/400*(2*log(1/4*x)^2 - 2*log(1/4*x) + 1)*x^2 + 3/200*x^2*log(1/4*x) + 3/
200*x^2 + 1/10*log(1/4*x)^2 + 1/x^2

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mupad [B]  time = 5.04, size = 34, normalized size = 1.36 \begin {gather*} x^2\,\left (\frac {{\ln \left (\frac {x}{4}\right )}^4}{400}+\frac {3\,{\ln \left (\frac {x}{4}\right )}^2}{200}+\frac {9}{400}\right )+\frac {{\ln \left (\frac {x}{4}\right )}^2}{10}+\frac {1}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((log(x/4)*(40*x^2 + 6*x^4))/200 + (9*x^4)/200 + (3*x^4*log(x/4)^2)/100 + (x^4*log(x/4)^3)/100 + (x^4*log(
x/4)^4)/200 - 2)/x^3,x)

[Out]

x^2*((3*log(x/4)^2)/200 + log(x/4)^4/400 + 9/400) + log(x/4)^2/10 + 1/x^2

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sympy [B]  time = 0.21, size = 39, normalized size = 1.56 \begin {gather*} \frac {x^{2} \log {\left (\frac {x}{4} \right )}^{4}}{400} + \frac {9 x^{2}}{400} + \left (\frac {3 x^{2}}{200} + \frac {1}{10}\right ) \log {\left (\frac {x}{4} \right )}^{2} + \frac {1}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/200*(x**4*ln(1/4*x)**4+2*x**4*ln(1/4*x)**3+6*x**4*ln(1/4*x)**2+(6*x**4+40*x**2)*ln(1/4*x)+9*x**4-4
00)/x**3,x)

[Out]

x**2*log(x/4)**4/400 + 9*x**2/400 + (3*x**2/200 + 1/10)*log(x/4)**2 + x**(-2)

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