3.70.74 \(\int \frac {48 x-1168 x^2-1216 x^3+336 x^4+2000 x^5+(32-784 x-800 x^2-48 x^3+1600 x^4) \log (x)+(16-384 x-800 x^2-400 x \log (x)) \log (x^2)}{x} \, dx\)

Optimal. Leaf size=29 \[ 16 \left (x-25 x^2\right ) (x+\log (x)) \left (-x^2+\frac {x+\log \left (x^2\right )}{x}\right ) \]

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Rubi [B]  time = 0.42, antiderivative size = 86, normalized size of antiderivative = 2.97, number of steps used = 19, number of rules used = 7, integrand size = 71, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.099, Rules used = {14, 6742, 2357, 2295, 2301, 2304, 2361} \begin {gather*} 400 x^5-16 x^4+400 x^4 \log (x)-400 x^3-16 x^3 \log (x)+16 x^2+4 \log ^2\left (x^2\right )-400 x^2 \log (x)-400 x^2 \log \left (x^2\right )-400 x \log (x) \log \left (x^2\right )+16 x \log \left (x^2\right )+16 \log ^2(x)+16 x \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(48*x - 1168*x^2 - 1216*x^3 + 336*x^4 + 2000*x^5 + (32 - 784*x - 800*x^2 - 48*x^3 + 1600*x^4)*Log[x] + (16
- 384*x - 800*x^2 - 400*x*Log[x])*Log[x^2])/x,x]

[Out]

16*x^2 - 400*x^3 - 16*x^4 + 400*x^5 + 16*x*Log[x] - 400*x^2*Log[x] - 16*x^3*Log[x] + 400*x^4*Log[x] + 16*Log[x
]^2 + 16*x*Log[x^2] - 400*x^2*Log[x^2] - 400*x*Log[x]*Log[x^2] + 4*Log[x^2]^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 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]

Rule 2361

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.)), x_Symbol] :> With[{u =
IntHide[(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[SimplifyIntegrand[u/x, x], x],
x]] /; FreeQ[{a, b, c, d, e, f, n, p, r}, x]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {16 (-1+x) \left (-3 x+70 x^2+146 x^3+125 x^4-2 \log (x)+47 x \log (x)+97 x^2 \log (x)+100 x^3 \log (x)\right )}{x}-\frac {16 \left (-1+24 x+50 x^2+25 x \log (x)\right ) \log \left (x^2\right )}{x}\right ) \, dx\\ &=16 \int \frac {(-1+x) \left (-3 x+70 x^2+146 x^3+125 x^4-2 \log (x)+47 x \log (x)+97 x^2 \log (x)+100 x^3 \log (x)\right )}{x} \, dx-16 \int \frac {\left (-1+24 x+50 x^2+25 x \log (x)\right ) \log \left (x^2\right )}{x} \, dx\\ &=16 \int \left (3-73 x-76 x^2+21 x^3+125 x^4+\frac {\left (2-49 x-50 x^2-3 x^3+100 x^4\right ) \log (x)}{x}\right ) \, dx-16 \int \left (24 \log \left (x^2\right )-\frac {\log \left (x^2\right )}{x}+50 x \log \left (x^2\right )+25 \log (x) \log \left (x^2\right )\right ) \, dx\\ &=48 x-584 x^2-\frac {1216 x^3}{3}+84 x^4+400 x^5+16 \int \frac {\left (2-49 x-50 x^2-3 x^3+100 x^4\right ) \log (x)}{x} \, dx+16 \int \frac {\log \left (x^2\right )}{x} \, dx-384 \int \log \left (x^2\right ) \, dx-400 \int \log (x) \log \left (x^2\right ) \, dx-800 \int x \log \left (x^2\right ) \, dx\\ &=816 x-184 x^2-\frac {1216 x^3}{3}+84 x^4+400 x^5+16 x \log \left (x^2\right )-400 x^2 \log \left (x^2\right )-400 x \log (x) \log \left (x^2\right )+4 \log ^2\left (x^2\right )+16 \int \left (-49 \log (x)+\frac {2 \log (x)}{x}-50 x \log (x)-3 x^2 \log (x)+100 x^3 \log (x)\right ) \, dx+800 \int (-1+\log (x)) \, dx\\ &=16 x-184 x^2-\frac {1216 x^3}{3}+84 x^4+400 x^5+16 x \log \left (x^2\right )-400 x^2 \log \left (x^2\right )-400 x \log (x) \log \left (x^2\right )+4 \log ^2\left (x^2\right )+32 \int \frac {\log (x)}{x} \, dx-48 \int x^2 \log (x) \, dx-784 \int \log (x) \, dx+800 \int \log (x) \, dx-800 \int x \log (x) \, dx+1600 \int x^3 \log (x) \, dx\\ &=16 x^2-400 x^3-16 x^4+400 x^5+16 x \log (x)-400 x^2 \log (x)-16 x^3 \log (x)+400 x^4 \log (x)+16 \log ^2(x)+16 x \log \left (x^2\right )-400 x^2 \log \left (x^2\right )-400 x \log (x) \log \left (x^2\right )+4 \log ^2\left (x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 73, normalized size = 2.52 \begin {gather*} 4 \left (4 x^2 \left (1-25 x-x^2+25 x^3\right )+4 \log ^2(x)+4 x \log (x) \left (1-25 x-x^2+25 x^3-25 \log \left (x^2\right )\right )+4 (1-25 x) x \log \left (x^2\right )+\log ^2\left (x^2\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(48*x - 1168*x^2 - 1216*x^3 + 336*x^4 + 2000*x^5 + (32 - 784*x - 800*x^2 - 48*x^3 + 1600*x^4)*Log[x]
+ (16 - 384*x - 800*x^2 - 400*x*Log[x])*Log[x^2])/x,x]

[Out]

4*(4*x^2*(1 - 25*x - x^2 + 25*x^3) + 4*Log[x]^2 + 4*x*Log[x]*(1 - 25*x - x^2 + 25*x^3 - 25*Log[x^2]) + 4*(1 -
25*x)*x*Log[x^2] + Log[x^2]^2)

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fricas [A]  time = 0.63, size = 55, normalized size = 1.90 \begin {gather*} 400 \, x^{5} - 16 \, x^{4} - 400 \, x^{3} - 32 \, {\left (25 \, x - 1\right )} \log \relax (x)^{2} + 16 \, x^{2} + 16 \, {\left (25 \, x^{4} - x^{3} - 75 \, x^{2} + 3 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800*x^2-784*x+32)*log(x)+2000*x^5+336*x^
4-1216*x^3-1168*x^2+48*x)/x,x, algorithm="fricas")

[Out]

400*x^5 - 16*x^4 - 400*x^3 - 32*(25*x - 1)*log(x)^2 + 16*x^2 + 16*(25*x^4 - x^3 - 75*x^2 + 3*x)*log(x)

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giac [A]  time = 0.16, size = 55, normalized size = 1.90 \begin {gather*} 400 \, x^{5} - 16 \, x^{4} - 400 \, x^{3} - 32 \, {\left (25 \, x - 1\right )} \log \relax (x)^{2} + 16 \, x^{2} + 16 \, {\left (25 \, x^{4} - x^{3} - 75 \, x^{2} + 3 \, x\right )} \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800*x^2-784*x+32)*log(x)+2000*x^5+336*x^
4-1216*x^3-1168*x^2+48*x)/x,x, algorithm="giac")

[Out]

400*x^5 - 16*x^4 - 400*x^3 - 32*(25*x - 1)*log(x)^2 + 16*x^2 + 16*(25*x^4 - x^3 - 75*x^2 + 3*x)*log(x)

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maple [C]  time = 0.08, size = 271, normalized size = 9.34

method	result	size
risch	\(\left (-800 x +32\right ) \ln \relax (x )^{2}+\left (200 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-400 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+200 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+400 x^{4}-16 x^{3}-1200 x^{2}+48 x \right ) \ln \relax (x )+200 i \pi \,x^{2} \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )-400 i \pi \,x^{2} \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}+200 i \pi \,x^{2} \mathrm {csgn}\left (i x^{2}\right )^{3}-8 i \pi x \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+16 i \pi x \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-8 i \pi x \mathrm {csgn}\left (i x^{2}\right )^{3}+400 x^{5}-16 x^{4}-400 x^{3}+16 x^{2}-8 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right )^{2} \mathrm {csgn}\left (i x^{2}\right )+16 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i x^{2}\right )^{2}-8 i \pi \ln \relax (x ) \mathrm {csgn}\left (i x^{2}\right )^{3}\)	\(271\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-400*x*ln(x)-800*x^2-384*x+16)*ln(x^2)+(1600*x^4-48*x^3-800*x^2-784*x+32)*ln(x)+2000*x^5+336*x^4-1216*x^
3-1168*x^2+48*x)/x,x,method=_RETURNVERBOSE)

[Out]

(-800*x+32)*ln(x)^2+(200*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)-400*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2+200*I*Pi*x*csgn(I*x
^2)^3+400*x^4-16*x^3-1200*x^2+48*x)*ln(x)+200*I*Pi*x^2*csgn(I*x)^2*csgn(I*x^2)-400*I*Pi*x^2*csgn(I*x)*csgn(I*x
^2)^2+200*I*Pi*x^2*csgn(I*x^2)^3-8*I*Pi*x*csgn(I*x)^2*csgn(I*x^2)+16*I*Pi*x*csgn(I*x)*csgn(I*x^2)^2-8*I*Pi*x*c
sgn(I*x^2)^3+400*x^5-16*x^4-400*x^3+16*x^2-8*I*Pi*ln(x)*csgn(I*x)^2*csgn(I*x^2)+16*I*Pi*ln(x)*csgn(I*x)*csgn(I
*x^2)^2-8*I*Pi*ln(x)*csgn(I*x^2)^3

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maxima [B]  time = 0.37, size = 91, normalized size = 3.14 \begin {gather*} 400 \, x^{5} + 400 \, x^{4} \log \relax (x) - 16 \, x^{4} - 16 \, x^{3} \log \relax (x) - 400 \, x^{3} - 400 \, x^{2} \log \left (x^{2}\right ) - 400 \, x^{2} \log \relax (x) + 16 \, x^{2} - 384 \, x \log \left (x^{2}\right ) + 4 \, \log \left (x^{2}\right )^{2} - 400 \, {\left (x \log \left (x^{2}\right ) - 2 \, x\right )} \log \relax (x) + 16 \, x \log \relax (x) + 16 \, \log \relax (x)^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-400*x*log(x)-800*x^2-384*x+16)*log(x^2)+(1600*x^4-48*x^3-800*x^2-784*x+32)*log(x)+2000*x^5+336*x^
4-1216*x^3-1168*x^2+48*x)/x,x, algorithm="maxima")

[Out]

400*x^5 + 400*x^4*log(x) - 16*x^4 - 16*x^3*log(x) - 400*x^3 - 400*x^2*log(x^2) - 400*x^2*log(x) + 16*x^2 - 384
*x*log(x^2) + 4*log(x^2)^2 - 400*(x*log(x^2) - 2*x)*log(x) + 16*x*log(x) + 16*log(x)^2

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mupad [B]  time = 4.06, size = 22, normalized size = 0.76 \begin {gather*} -16\,\left (25\,x-1\right )\,\left (x+\ln \relax (x)\right )\,\left (x+\ln \left (x^2\right )-x^3\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(x)*(784*x + 800*x^2 + 48*x^3 - 1600*x^4 - 32) - 48*x + log(x^2)*(384*x + 400*x*log(x) + 800*x^2 - 16
) + 1168*x^2 + 1216*x^3 - 336*x^4 - 2000*x^5)/x,x)

[Out]

-16*(25*x - 1)*(x + log(x))*(x + log(x^2) - x^3)

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sympy [B]  time = 0.27, size = 51, normalized size = 1.76 \begin {gather*} 400 x^{5} - 16 x^{4} - 400 x^{3} + 16 x^{2} + \left (32 - 800 x\right ) \log {\relax (x )}^{2} + \left (400 x^{4} - 16 x^{3} - 1200 x^{2} + 48 x\right ) \log {\relax (x )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-400*x*ln(x)-800*x**2-384*x+16)*ln(x**2)+(1600*x**4-48*x**3-800*x**2-784*x+32)*ln(x)+2000*x**5+336
*x**4-1216*x**3-1168*x**2+48*x)/x,x)

[Out]

400*x**5 - 16*x**4 - 400*x**3 + 16*x**2 + (32 - 800*x)*log(x)**2 + (400*x**4 - 16*x**3 - 1200*x**2 + 48*x)*log
(x)

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