\(\int \frac {4 x-6 x^3-3 \log (4)+(x-x^3-\log (4)) \log (\frac {-x^2+x^4+x \log (4)}{\log (4)})}{-x+x^3+\log (4)} \, dx\) [3903]

Optimal result

Integrand size = 56, antiderivative size = 23 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=x-x \left (3+\log \left (x+\frac {x \left (-x+x^3\right )}{\log (4)}\right )\right ) \]

[Out]

x-x*(ln(1/2*x*(x^3-x)/ln(2)+x)+3)

Rubi [A] (verified)

Time = 6.41 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17, number of steps used = 21, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {6874, 2104, 814, 648, 632, 210, 642, 2603} \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=x \left (-\log \left (-x \left (-x^3+x-\log (4)\right )\right )\right )-2 x+x \log (\log (4)) \]

[In]

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

[Out]

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

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 648

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 2104

Int[((e_.) + (f_.)*(x_))^(m_.)*((a_) + (b_.)*(x_) + (d_.)*(x_)^3)^(p_), x_Symbol] :> With[{r = Rt[-9*a*d^2 + S
qrt[3]*d*Sqrt[4*b^3*d + 27*a^2*d^2], 3]}, Dist[1/d^(2*p), Int[(e + f*x)^m*Simp[18^(1/3)*b*(d/(3*r)) - r/18^(1/
3) + d*x, x]^p*Simp[b*(d/3) + 12^(1/3)*b^2*(d^2/(3*r^2)) + r^2/(3*12^(1/3)) - d*(2^(1/3)*b*(d/(3^(1/3)*r)) - r
/18^(1/3))*x + d^2*x^2, x]^p, x], x]] /; FreeQ[{a, b, d, e, f, m}, x] && NeQ[4*b^3 + 27*a^2*d, 0] && ILtQ[p, 0
]

Rule 2603

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.), x_Symbol] :> Simp[x*(a + b*Log[c*RFx^p])^n, x] - Dist[b*n*p
, Int[SimplifyIntegrand[x*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, p}, x] &
& RationalFunctionQ[RFx, x] && IGtQ[n, 0]

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {4 x-6 x^3-3 \log (4)}{-x+x^3+\log (4)}+\log (\log (4))-\log \left (x \left (-x+x^3+\log (4)\right )\right )\right ) \, dx \\ & = x \log (\log (4))+\int \frac {4 x-6 x^3-3 \log (4)}{-x+x^3+\log (4)} \, dx-\int \log \left (x \left (-x+x^3+\log (4)\right )\right ) \, dx \\ & = -x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {2 x-4 x^3-\log (4)}{x-x^3-\log (4)} \, dx+\int \left (-6+\frac {-2 x+\log (64)}{-x+x^3+\log (4)}\right ) \, dx \\ & = -6 x-x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {-2 x+\log (64)}{-x+x^3+\log (4)} \, dx+\int \left (4+\frac {-2 x+\log (64)}{x-x^3-\log (4)}\right ) \, dx \\ & = -2 x-x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {-2 x+\log (64)}{x-x^3-\log (4)} \, dx+\int \frac {-2 x+\log (64)}{\left (x+\frac {2 \sqrt [3]{\frac {3}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{2 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}}{6^{2/3}}\right ) \left (x^2-\frac {1}{3} x \left (3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{\frac {3}{2} \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}\right )+\frac {1}{18} \left (-6+6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}\right )\right )} \, dx \\ & = -2 x-x \log \left (-x \left (x-x^3-\log (4)\right )\right )+x \log (\log (4))+\int \frac {-2 x+\log (64)}{\left (-x-\frac {2 \sqrt [3]{\frac {3}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{2 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}}{6^{2/3}}\right ) \left (x^2-\frac {1}{3} x \left (3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{\frac {3}{2} \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}\right )+\frac {1}{18} \left (-6+6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}\right )\right )} \, dx+\int \left (\frac {12 \left (-6+6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}-6\ 3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}} \log (64)-3\ 2^{2/3} \sqrt [3]{3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )} \log (64)+3 x \left (2\ 3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+2^{2/3} \sqrt [3]{3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}+3 \log (64)\right )\right )}{\left (6+6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}\right ) \left (6-18 x^2-6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}-\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}+3 \sqrt [3]{6} x \left (2 \sqrt [3]{\frac {3}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+\sqrt [3]{2 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}\right )\right )}+\frac {12 \left (2\ 3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}}+2^{2/3} \sqrt [3]{3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )}+3 \log (64)\right )}{\left (6+6 \sqrt [3]{3} \left (\frac {2}{9 \log (4)-\sqrt {-12+81 \log ^2(4)}}\right )^{2/3}+\sqrt [3]{2} \left (3 \left (9 \log (4)-\sqrt {-12+81 \log ^2(4)}\right )\right )^{2/3}\right ) \left (6 x+2\ 3^{2/3} \sqrt [3]{\frac {2}{9 \log (4)-\sqrt {3 \left (-4+27 \log ^2(4)\right )}}}+2^{2/3} \sqrt [3]{3 \left (-\sqrt {3 \left (-4+27 \log ^2(4)\right )}+2 \log (512)\right )}\right )}\right ) \, dx \\ & = \text {Too large to display} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=-2 x-x \log \left (\frac {x \left (-x+x^3+\log (4)\right )}{\log (4)}\right ) \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.06 (sec) , antiderivative size = 27, normalized size of antiderivative = 1.17

method	result	size
parallelrisch	\(-\ln \left (\frac {x \left (2 \ln \left (2\right )+x^{3}-x \right )}{2 \ln \left (2\right )}\right ) x -2 x\)	\(27\)
norman	\(-2 x -x \ln \left (\frac {2 x \ln \left (2\right )+x^{4}-x^{2}}{2 \ln \left (2\right )}\right )\)	\(29\)
risch	\(-2 x -x \ln \left (\frac {2 x \ln \left (2\right )+x^{4}-x^{2}}{2 \ln \left (2\right )}\right )\)	\(29\)
parts	\(-2 x +x \ln \left (2\right )+x \ln \left (\ln \left (2\right )\right )-x \ln \left (x \left (2 \ln \left (2\right )+x^{3}-x \right )\right )\)	\(31\)
default	\(x \ln \left (2\right )-2 x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \ln \left (2\right )+\textit {\_Z}^{3}-\textit {\_Z} \right )}{\sum }\frac {\left (-\textit {\_R} +3 \ln \left (2\right )\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )-x \ln \left (x \left (2 \ln \left (2\right )+x^{3}-x \right )\right )+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (2 \ln \left (2\right )+\textit {\_Z}^{3}-\textit {\_Z} \right )}{\sum }\frac {\left (\textit {\_R} -3 \ln \left (2\right )\right ) \ln \left (x -\textit {\_R} \right )}{3 \textit {\_R}^{2}-1}\right )+x \ln \left (\ln \left (2\right )\right )\)	\(111\)

[In]

int(((-2*ln(2)-x^3+x)*ln(1/2*(2*x*ln(2)+x^4-x^2)/ln(2))-6*ln(2)-6*x^3+4*x)/(2*ln(2)+x^3-x),x,method=_RETURNVER
BOSE)

[Out]

-ln(1/2*x*(2*ln(2)+x^3-x)/ln(2))*x-2*x

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=-x \log \left (\frac {x^{4} - x^{2} + 2 \, x \log \left (2\right )}{2 \, \log \left (2\right )}\right ) - 2 \, x \]

[In]

integrate(((-2*log(2)-x^3+x)*log(1/2*(2*x*log(2)+x^4-x^2)/log(2))-6*log(2)-6*x^3+4*x)/(2*log(2)+x^3-x),x, algo
rithm="fricas")

[Out]

-x*log(1/2*(x^4 - x^2 + 2*x*log(2))/log(2)) - 2*x

Sympy [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.13 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=- x \log {\left (\frac {\frac {x^{4}}{2} - \frac {x^{2}}{2} + x \log {\left (2 \right )}}{\log {\left (2 \right )}} \right )} - 2 x \]

[In]

integrate(((-2*ln(2)-x**3+x)*ln(1/2*(2*x*ln(2)+x**4-x**2)/ln(2))-6*ln(2)-6*x**3+4*x)/(2*ln(2)+x**3-x),x)

[Out]

-x*log((x**4/2 - x**2/2 + x*log(2))/log(2)) - 2*x

Maxima [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.30 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right ) - 2\right )} - x \log \left (x^{3} - x + 2 \, \log \left (2\right )\right ) - x \log \left (x\right ) \]

[In]

integrate(((-2*log(2)-x^3+x)*log(1/2*(2*x*log(2)+x^4-x^2)/log(2))-6*log(2)-6*x^3+4*x)/(2*log(2)+x^3-x),x, algo
rithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.22 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=x {\left (\log \left (2\right ) + \log \left (\log \left (2\right )\right ) - 2\right )} - x \log \left (x^{4} - x^{2} + 2 \, x \log \left (2\right )\right ) \]

[In]

integrate(((-2*log(2)-x^3+x)*log(1/2*(2*x*log(2)+x^4-x^2)/log(2))-6*log(2)-6*x^3+4*x)/(2*log(2)+x^3-x),x, algo
rithm="giac")

[Out]

x*(log(2) + log(log(2)) - 2) - x*log(x^4 - x^2 + 2*x*log(2))

Mupad [B] (verification not implemented)

Time = 9.78 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.35 \[ \int \frac {4 x-6 x^3-3 \log (4)+\left (x-x^3-\log (4)\right ) \log \left (\frac {-x^2+x^4+x \log (4)}{\log (4)}\right )}{-x+x^3+\log (4)} \, dx=x\,\ln \left (2\right )-2\,x+x\,\ln \left (\ln \left (2\right )\right )-x\,\ln \left (x^4-x^2+2\,\ln \left (2\right )\,x\right ) \]

[In]

int(-(6*log(2) - 4*x + 6*x^3 + log((x*log(2) - x^2/2 + x^4/2)/log(2))*(2*log(2) - x + x^3))/(2*log(2) - x + x^
3),x)

[Out]

x*log(2) - 2*x + x*log(log(2)) - x*log(2*x*log(2) - x^2 + x^4)