\(\int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+(16 x-4 x^2+2 x^3) \log (3)+(-16 x+4 x^2-2 x^3) \log (2 x^2)}{x^3} \, dx\) [8069]

Optimal result

Integrand size = 62, antiderivative size = 25 \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=\left (5-\frac {2 (4-x)}{x}+x+\log (3)-\log \left (2 x^2\right )\right )^2 \]

[Out]

(x+ln(3)-2*(-x+4)/x+5-ln(2*x^2))^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(69\) vs. \(2(25)=50\).

Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 2.76, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {14, 1642, 2404, 2332, 2341, 2338} \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=x^2+\frac {64}{x^2}+\log ^2\left (2 x^2\right )-2 x \log \left (2 x^2\right )+\frac {16 \log \left (2 x^2\right )}{x}+4 x+\frac {32}{x}+2 x (5+\log (3))-4 (7+\log (3)) \log (x)-\frac {16 (9+\log (3))}{x} \]

[In]

Int[(-128 + 144*x - 28*x^2 + 10*x^3 + 2*x^4 + (16*x - 4*x^2 + 2*x^3)*Log[3] + (-16*x + 4*x^2 - 2*x^3)*Log[2*x^
2])/x^3,x]

[Out]

64/x^2 + 32/x + 4*x + x^2 + 2*x*(5 + Log[3]) - (16*(9 + Log[3]))/x - 4*(7 + Log[3])*Log[x] + (16*Log[2*x^2])/x
- 2*x*Log[2*x^2] + Log[2*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 1642

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2332

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

Rule 2338

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 2341

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 2404

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{align*} \text {integral}& = \int \left (\frac {2 \left (8-2 x+x^2\right ) \left (-8+x^2+x (7+\log (3))\right )}{x^3}-\frac {2 \left (8-2 x+x^2\right ) \log \left (2 x^2\right )}{x^2}\right ) \, dx \\ & = 2 \int \frac {\left (8-2 x+x^2\right ) \left (-8+x^2+x (7+\log (3))\right )}{x^3} \, dx-2 \int \frac {\left (8-2 x+x^2\right ) \log \left (2 x^2\right )}{x^2} \, dx \\ & = 2 \int \left (-\frac {64}{x^3}+x+5 \left (1+\frac {\log (3)}{5}\right )-\frac {2 (7+\log (3))}{x}+\frac {8 (9+\log (3))}{x^2}\right ) \, dx-2 \int \left (\log \left (2 x^2\right )+\frac {8 \log \left (2 x^2\right )}{x^2}-\frac {2 \log \left (2 x^2\right )}{x}\right ) \, dx \\ & = \frac {64}{x^2}+x^2+2 x (5+\log (3))-\frac {16 (9+\log (3))}{x}-4 (7+\log (3)) \log (x)-2 \int \log \left (2 x^2\right ) \, dx+4 \int \frac {\log \left (2 x^2\right )}{x} \, dx-16 \int \frac {\log \left (2 x^2\right )}{x^2} \, dx \\ & = \frac {64}{x^2}+\frac {32}{x}+4 x+x^2+2 x (5+\log (3))-\frac {16 (9+\log (3))}{x}-4 (7+\log (3)) \log (x)+\frac {16 \log \left (2 x^2\right )}{x}-2 x \log \left (2 x^2\right )+\log ^2\left (2 x^2\right ) \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(58\) vs. \(2(25)=50\).

Time = 0.08 (sec) , antiderivative size = 58, normalized size of antiderivative = 2.32 \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=\frac {64}{x^2}-\frac {112}{x}+4 x+x^2+2 x (5+\log (3))+\left (-7+\log \left (\frac {2 x^2}{3}\right )\right )^2+\frac {16 \log \left (\frac {2 x^2}{3}\right )}{x}-2 x \log \left (2 x^2\right ) \]

[In]

Integrate[(-128 + 144*x - 28*x^2 + 10*x^3 + 2*x^4 + (16*x - 4*x^2 + 2*x^3)*Log[3] + (-16*x + 4*x^2 - 2*x^3)*Lo
g[2*x^2])/x^3,x]

[Out]

64/x^2 - 112/x + 4*x + x^2 + 2*x*(5 + Log[3]) + (-7 + Log[(2*x^2)/3])^2 + (16*Log[(2*x^2)/3])/x - 2*x*Log[2*x^
2]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(75\) vs. \(2(25)=50\).

Time = 0.11 (sec) , antiderivative size = 76, normalized size of antiderivative = 3.04

method	result	size
norman	\(\frac {64+x^{4}+x^{2} \ln \left (2 x^{2}\right )^{2}+\left (-112-16 \ln \left (3\right )\right ) x +\left (14+2 \ln \left (3\right )\right ) x^{3}+\left (-14-2 \ln \left (3\right )\right ) x^{2} \ln \left (2 x^{2}\right )+16 x \ln \left (2 x^{2}\right )-2 x^{3} \ln \left (2 x^{2}\right )}{x^{2}}\)	\(76\)
default	\(14 x +x^{2}+\frac {64}{x^{2}}-28 \ln \left (x \right )-\frac {112}{x}-2 \ln \left (2\right ) \left (x -2 \ln \left (x \right )-\frac {8}{x}\right )+2 \ln \left (3\right ) \left (x -2 \ln \left (x \right )-\frac {8}{x}\right )-2 x \ln \left (x^{2}\right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}+\frac {16 \ln \left (x^{2}\right )}{x}\)	\(82\)
risch	\(2 x \ln \left (3\right )+x^{2}+14 x +\frac {64}{x^{2}}-4 \ln \left (3\right ) \ln \left (x \right )-28 \ln \left (x \right )-\frac {16 \ln \left (3\right )}{x}-\frac {112}{x}-2 x \ln \left (2\right )+4 \ln \left (2\right ) \ln \left (x \right )+\frac {16 \ln \left (2\right )}{x}-2 x \ln \left (x^{2}\right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}+\frac {16 \ln \left (x^{2}\right )}{x}\)	\(88\)
parts	\(2 x \ln \left (3\right )+x^{2}+14 x +\frac {64}{x^{2}}+2 \left (-14-2 \ln \left (3\right )\right ) \ln \left (x \right )-\frac {2 \left (8 \ln \left (3\right )+72\right )}{x}-2 \ln \left (2\right ) \left (x -2 \ln \left (x \right )-\frac {8}{x}\right )-2 x \ln \left (x^{2}\right )+4 \ln \left (x \right ) \ln \left (x^{2}\right )-4 \ln \left (x \right )^{2}+\frac {16 \ln \left (x^{2}\right )}{x}+\frac {32}{x}\)	\(89\)
parallelrisch	\(\frac {4 x^{3} \ln \left (3\right )-4 \ln \left (3\right ) x^{2} \ln \left (2 x^{2}\right )+128+2 x^{4}-4 x^{3} \ln \left (2 x^{2}\right )+2 x^{2} \ln \left (2 x^{2}\right )^{2}+28 x^{3}-28 x^{2} \ln \left (2 x^{2}\right )-32 x \ln \left (3\right )+32 x \ln \left (2 x^{2}\right )-224 x}{2 x^{2}}\)	\(90\)

[In]

int(((-2*x^3+4*x^2-16*x)*ln(2*x^2)+(2*x^3-4*x^2+16*x)*ln(3)+2*x^4+10*x^3-28*x^2+144*x-128)/x^3,x,method=_RETUR
NVERBOSE)

[Out]

(64+x^4+x^2*ln(2*x^2)^2+(-112-16*ln(3))*x+(14+2*ln(3))*x^3+(-14-2*ln(3))*x^2*ln(2*x^2)+16*x*ln(2*x^2)-2*x^3*ln
(2*x^2))/x^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 66 vs. \(2 (23) = 46\).

Time = 0.24 (sec) , antiderivative size = 66, normalized size of antiderivative = 2.64 \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=\frac {x^{4} + x^{2} \log \left (2 \, x^{2}\right )^{2} + 14 \, x^{3} + 2 \, {\left (x^{3} - 8 \, x\right )} \log \left (3\right ) - 2 \, {\left (x^{3} + x^{2} \log \left (3\right ) + 7 \, x^{2} - 8 \, x\right )} \log \left (2 \, x^{2}\right ) - 112 \, x + 64}{x^{2}} \]

[In]

integrate(((-2*x^3+4*x^2-16*x)*log(2*x^2)+(2*x^3-4*x^2+16*x)*log(3)+2*x^4+10*x^3-28*x^2+144*x-128)/x^3,x, algo
rithm="fricas")

[Out]

(x^4 + x^2*log(2*x^2)^2 + 14*x^3 + 2*(x^3 - 8*x)*log(3) - 2*(x^3 + x^2*log(3) + 7*x^2 - 8*x)*log(2*x^2) - 112*
x + 64)/x^2

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 60 vs. \(2 (20) = 40\).

Time = 0.18 (sec) , antiderivative size = 60, normalized size of antiderivative = 2.40 \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=x^{2} + x \left (2 \log {\left (3 \right )} + 14\right ) - 4 \left (\log {\left (3 \right )} + 7\right ) \log {\left (x \right )} + \log {\left (2 x^{2} \right )}^{2} + \frac {\left (16 - 2 x^{2}\right ) \log {\left (2 x^{2} \right )}}{x} + \frac {x \left (-112 - 16 \log {\left (3 \right )}\right ) + 64}{x^{2}} \]

[In]

integrate(((-2*x**3+4*x**2-16*x)*ln(2*x**2)+(2*x**3-4*x**2+16*x)*ln(3)+2*x**4+10*x**3-28*x**2+144*x-128)/x**3,
x)

[Out]

x**2 + x*(2*log(3) + 14) - 4*(log(3) + 7)*log(x) + log(2*x**2)**2 + (16 - 2*x**2)*log(2*x**2)/x + (x*(-112 - 1
6*log(3)) + 64)/x**2

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 67 vs. \(2 (23) = 46\).

Time = 0.19 (sec) , antiderivative size = 67, normalized size of antiderivative = 2.68 \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=x^{2} + 2 \, x \log \left (3\right ) - 2 \, x \log \left (2 \, x^{2}\right ) + \log \left (2 \, x^{2}\right )^{2} - 4 \, \log \left (3\right ) \log \left (x\right ) + 14 \, x - \frac {16 \, \log \left (3\right )}{x} + \frac {16 \, \log \left (2 \, x^{2}\right )}{x} - \frac {112}{x} + \frac {64}{x^{2}} - 28 \, \log \left (x\right ) \]

[In]

integrate(((-2*x^3+4*x^2-16*x)*log(2*x^2)+(2*x^3-4*x^2+16*x)*log(3)+2*x^4+10*x^3-28*x^2+144*x-128)/x^3,x, algo
rithm="maxima")

[Out]

x^2 + 2*x*log(3) - 2*x*log(2*x^2) + log(2*x^2)^2 - 4*log(3)*log(x) + 14*x - 16*log(3)/x + 16*log(2*x^2)/x - 11
2/x + 64/x^2 - 28*log(x)

Giac [F]

\[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx=\int { \frac {2 \, {\left (x^{4} + 5 \, x^{3} - 14 \, x^{2} + {\left (x^{3} - 2 \, x^{2} + 8 \, x\right )} \log \left (3\right ) - {\left (x^{3} - 2 \, x^{2} + 8 \, x\right )} \log \left (2 \, x^{2}\right ) + 72 \, x - 64\right )}}{x^{3}} \,d x } \]

[In]

integrate(((-2*x^3+4*x^2-16*x)*log(2*x^2)+(2*x^3-4*x^2+16*x)*log(3)+2*x^4+10*x^3-28*x^2+144*x-128)/x^3,x, algo
rithm="giac")

[Out]

integrate(2*(x^4 + 5*x^3 - 14*x^2 + (x^3 - 2*x^2 + 8*x)*log(3) - (x^3 - 2*x^2 + 8*x)*log(2*x^2) + 72*x - 64)/x
^3, x)

Mupad [B] (verification not implemented)

Time = 12.49 (sec) , antiderivative size = 63, normalized size of antiderivative = 2.52 \[ \int \frac {-128+144 x-28 x^2+10 x^3+2 x^4+\left (16 x-4 x^2+2 x^3\right ) \log (3)+\left (-16 x+4 x^2-2 x^3\right ) \log \left (2 x^2\right )}{x^3} \, dx={\ln \left (2\,x^2\right )}^2-\ln \left (x^2\right )\,\left (\ln \left (9\right )+14\right )+x\,\left (\ln \left (9\right )-2\,\ln \left (2\,x^2\right )+14\right )+\frac {64\,x-x^2\,\left (16\,\ln \left (3\right )-16\,\ln \left (2\,x^2\right )+112\right )}{x^3}+x^2 \]

[In]

int((144*x + log(3)*(16*x - 4*x^2 + 2*x^3) - log(2*x^2)*(16*x - 4*x^2 + 2*x^3) - 28*x^2 + 10*x^3 + 2*x^4 - 128
)/x^3,x)

[Out]

log(2*x^2)^2 - log(x^2)*(log(9) + 14) + x*(log(9) - 2*log(2*x^2) + 14) + (64*x - x^2*(16*log(3) - 16*log(2*x^2
) + 112))/x^3 + x^2