\(\int \frac {7 x^2-4 x^3-14 x^4+(-3-6 x^2) \log (4)+(-2 x-7 x^2-3 \log (4)) \log (\frac {-2 x-7 x^2-3 \log (4)}{3 x})}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx\) [6498]

Optimal result

Integrand size = 82, antiderivative size = 29 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-2 x+\frac {\log \left (\frac {1}{3} (-2-x)-2 x-\frac {\log (4)}{x}\right )}{x} \]

[Out]

ln(-7/3*x-2*ln(2)/x-2/3)/x-2*x

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(107\) vs. \(2(29)=58\).

Time = 0.47 (sec) , antiderivative size = 107, normalized size of antiderivative = 3.69, number of steps used = 16, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1608, 6860, 1642, 648, 632, 210, 642, 2605} \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=\frac {\sqrt {7 \log (64)-1} \log (4096) \arctan \left (\frac {7 x+1}{\sqrt {7 \log (64)-1}}\right )}{\log ^2(64)}+\frac {\left (\log (4) (6-21 \log (64))-7 \log ^2(64)\right ) \arctan \left (\frac {7 x+1}{\sqrt {7 \log (64)-1}}\right )}{\log ^2(64) \sqrt {7 \log (64)-1}}-2 x+\frac {\log \left (-\frac {7 x}{3}-\frac {\log (4)}{x}-\frac {2}{3}\right )}{x} \]

[In]

Int[(7*x^2 - 4*x^3 - 14*x^4 + (-3 - 6*x^2)*Log[4] + (-2*x - 7*x^2 - 3*Log[4])*Log[(-2*x - 7*x^2 - 3*Log[4])/(3
*x)])/(2*x^3 + 7*x^4 + 3*x^2*Log[4]),x]

[Out]

-2*x + (ArcTan[(1 + 7*x)/Sqrt[-1 + 7*Log[64]]]*(Log[4]*(6 - 21*Log[64]) - 7*Log[64]^2))/(Log[64]^2*Sqrt[-1 + 7
*Log[64]]) + (ArcTan[(1 + 7*x)/Sqrt[-1 + 7*Log[64]]]*Sqrt[-1 + 7*Log[64]]*Log[4096])/Log[64]^2 + Log[-2/3 - (7
*x)/3 - Log[4]/x]/x

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 1608

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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 2605

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x_Symbol] :> Simp[(d + e*x)^(m +
1)*((a + b*Log[c*RFx^p])^n/(e*(m + 1))), x] - Dist[b*n*(p/(e*(m + 1))), Int[SimplifyIntegrand[(d + e*x)^(m +
1)*(a + b*Log[c*RFx^p])^(n - 1)*(D[RFx, x]/RFx), x], x], x] /; FreeQ[{a, b, c, d, e, m, p}, x] && RationalFunc
tionQ[RFx, x] && IGtQ[n, 0] && (EqQ[n, 1] || IntegerQ[m]) && NeQ[m, -1]

Rule 6860

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{x^2 \left (2 x+7 x^2+3 \log (4)\right )} \, dx \\ & = \int \left (\frac {-4 x^3-14 x^4+x^2 (7-6 \log (4))-3 \log (4)}{x^2 \left (2 x+7 x^2+\log (64)\right )}-\frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x^2}\right ) \, dx \\ & = \int \frac {-4 x^3-14 x^4+x^2 (7-6 \log (4))-3 \log (4)}{x^2 \left (2 x+7 x^2+\log (64)\right )} \, dx-\int \frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x^2} \, dx \\ & = \frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x}-\int \frac {7 x^2-3 \log (4)}{x^2 \left (2 x+7 x^2+\log (64)\right )} \, dx+\int \left (-2-\frac {1}{x^2}+\frac {6 \log (4)}{x \log ^2(64)}+\frac {-12 \log (4)-42 x \log (4)+21 \log (4) \log (64)+7 \log ^2(64)-6 \log (4) \log ^2(64)+\log ^2(64) \log (4096)}{\log ^2(64) \left (2 x+7 x^2+\log (64)\right )}\right ) \, dx \\ & = \frac {1}{x}-2 x+\frac {6 \log (4) \log (x)}{\log ^2(64)}+\frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x}+\frac {\int \frac {-12 \log (4)-42 x \log (4)+21 \log (4) \log (64)+7 \log ^2(64)-6 \log (4) \log ^2(64)+\log ^2(64) \log (4096)}{2 x+7 x^2+\log (64)} \, dx}{\log ^2(64)}-\int \left (-\frac {1}{x^2}+\frac {6 \log (4)}{x \log ^2(64)}+\frac {-12 \log (4)-42 x \log (4)+21 \log (4) \log (64)+7 \log ^2(64)}{\log ^2(64) \left (2 x+7 x^2+\log (64)\right )}\right ) \, dx \\ & = -2 x+\frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x}-\frac {\int \frac {-12 \log (4)-42 x \log (4)+21 \log (4) \log (64)+7 \log ^2(64)}{2 x+7 x^2+\log (64)} \, dx}{\log ^2(64)}-\frac {(3 \log (4)) \int \frac {2+14 x}{2 x+7 x^2+\log (64)} \, dx}{\log ^2(64)}+\frac {\left (7 \log ^2(64)-\log (4096)+\log (64) \log (4398046511104)\right ) \int \frac {1}{2 x+7 x^2+\log (64)} \, dx}{\log ^2(64)} \\ & = -2 x+\frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x}-\frac {3 \log (4) \log \left (2 x+7 x^2+\log (64)\right )}{\log ^2(64)}-\left (7-\frac {3 \log (4) (2-7 \log (64))}{\log ^2(64)}\right ) \int \frac {1}{2 x+7 x^2+\log (64)} \, dx+\frac {(3 \log (4)) \int \frac {2+14 x}{2 x+7 x^2+\log (64)} \, dx}{\log ^2(64)}-\frac {\left (2 \left (7 \log ^2(64)-\log (4096)+\log (64) \log (4398046511104)\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2+4 (1-7 \log (64))} \, dx,x,2+14 x\right )}{\log ^2(64)} \\ & = -2 x+\frac {\tan ^{-1}\left (\frac {1+7 x}{\sqrt {-1+7 \log (64)}}\right ) \sqrt {-1+7 \log (64)} \log (4096)}{\log ^2(64)}+\frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x}+\left (2 \left (7-\frac {3 \log (4) (2-7 \log (64))}{\log ^2(64)}\right )\right ) \text {Subst}\left (\int \frac {1}{-x^2+4 (1-7 \log (64))} \, dx,x,2+14 x\right ) \\ & = -2 x-\frac {\tan ^{-1}\left (\frac {1+7 x}{\sqrt {-1+7 \log (64)}}\right ) \left (7-\frac {3 \log (4) (2-7 \log (64))}{\log ^2(64)}\right )}{\sqrt {-1+7 \log (64)}}+\frac {\tan ^{-1}\left (\frac {1+7 x}{\sqrt {-1+7 \log (64)}}\right ) \sqrt {-1+7 \log (64)} \log (4096)}{\log ^2(64)}+\frac {\log \left (-\frac {2}{3}-\frac {7 x}{3}-\frac {\log (4)}{x}\right )}{x} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.14 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-\frac {\log (3)+\frac {x^2 \log (4096)}{\log (64)}-\log \left (-2-7 x-\frac {\log (64)}{x}\right )}{x} \]

[In]

Integrate[(7*x^2 - 4*x^3 - 14*x^4 + (-3 - 6*x^2)*Log[4] + (-2*x - 7*x^2 - 3*Log[4])*Log[(-2*x - 7*x^2 - 3*Log[
4])/(3*x)])/(2*x^3 + 7*x^4 + 3*x^2*Log[4]),x]

[Out]

-((Log[3] + (x^2*Log[4096])/Log[64] - Log[-2 - 7*x - Log[64]/x])/x)

Maple [A] (verified)

Time = 0.20 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.97

method	result	size
risch	\(\frac {\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{3 x}\right )}{x}-2 x\)	\(28\)
norman	\(\frac {-2 x^{2}+\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{3 x}\right )}{x}\)	\(30\)
parallelrisch	\(-\frac {98 x^{2}-56 x -49 \ln \left (-\frac {7 x^{2}+6 \ln \left (2\right )+2 x}{3 x}\right )}{49 x}\)	\(36\)
default	\(-\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{x}\right )}{x}-\frac {-\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\sqrt {-1+42 \ln \left (2\right )}\, \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{3 \ln \left (2\right )}-2 x -\frac {\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\frac {\left (1-42 \ln \left (2\right )\right ) \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{\sqrt {-1+42 \ln \left (2\right )}}}{3 \ln \left (2\right )}\)	\(136\)
parts	\(-\frac {\ln \left (3\right )}{x}+\frac {\ln \left (\frac {-6 \ln \left (2\right )-7 x^{2}-2 x}{x}\right )}{x}-\frac {-\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\sqrt {-1+42 \ln \left (2\right )}\, \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{3 \ln \left (2\right )}-2 x -\frac {\frac {\ln \left (7 x^{2}+6 \ln \left (2\right )+2 x \right )}{2}+\frac {\left (1-42 \ln \left (2\right )\right ) \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \left (2\right )}}\right )}{\sqrt {-1+42 \ln \left (2\right )}}}{3 \ln \left (2\right )}\)	\(136\)

[In]

int(((-6*ln(2)-7*x^2-2*x)*ln(1/3*(-6*ln(2)-7*x^2-2*x)/x)+2*(-6*x^2-3)*ln(2)-14*x^4-4*x^3+7*x^2)/(6*x^2*ln(2)+7
*x^4+2*x^3),x,method=_RETURNVERBOSE)

[Out]

1/x*ln(1/3*(-6*ln(2)-7*x^2-2*x)/x)-2*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.10 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-\frac {2 \, x^{2} - \log \left (-\frac {7 \, x^{2} + 2 \, x + 6 \, \log \left (2\right )}{3 \, x}\right )}{x} \]

[In]

integrate(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^2-3)*log(2)-14*x^4-4*x^3+7*x^2)/(6*x
^2*log(2)+7*x^4+2*x^3),x, algorithm="fricas")

[Out]

-(2*x^2 - log(-1/3*(7*x^2 + 2*x + 6*log(2))/x))/x

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.90 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=- 2 x + \frac {\log {\left (\frac {- \frac {7 x^{2}}{3} - \frac {2 x}{3} - 2 \log {\left (2 \right )}}{x} \right )}}{x} \]

[In]

integrate(((-6*ln(2)-7*x**2-2*x)*ln(1/3*(-6*ln(2)-7*x**2-2*x)/x)+2*(-6*x**2-3)*ln(2)-14*x**4-4*x**3+7*x**2)/(6
*x**2*ln(2)+7*x**4+2*x**3),x)

[Out]

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 170 vs. \(2 (21) = 42\).

Time = 0.74 (sec) , antiderivative size = 170, normalized size of antiderivative = 5.86 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=\frac {1}{6} \, {\left (\frac {2 \, {\left (21 \, \log \left (2\right ) - 1\right )} \arctan \left (\frac {7 \, x + 1}{\sqrt {42 \, \log \left (2\right ) - 1}}\right )}{\sqrt {42 \, \log \left (2\right ) - 1} \log \left (2\right )^{2}} - \frac {\log \left (7 \, x^{2} + 2 \, x + 6 \, \log \left (2\right )\right )}{\log \left (2\right )^{2}} + \frac {2 \, \log \left (x\right )}{\log \left (2\right )^{2}} + \frac {6}{x \log \left (2\right )}\right )} \log \left (2\right ) - \frac {{\left (21 \, \log \left (2\right ) - 1\right )} \arctan \left (\frac {7 \, x + 1}{\sqrt {42 \, \log \left (2\right ) - 1}}\right )}{3 \, \sqrt {42 \, \log \left (2\right ) - 1} \log \left (2\right )} - \frac {12 \, x^{2} \log \left (2\right ) + 6 \, {\left (\log \left (3\right ) + 1\right )} \log \left (2\right ) - {\left (x + 6 \, \log \left (2\right )\right )} \log \left (-7 \, x^{2} - 2 \, x - 6 \, \log \left (2\right )\right ) + 2 \, {\left (x + 3 \, \log \left (2\right )\right )} \log \left (x\right )}{6 \, x \log \left (2\right )} \]

[In]

integrate(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^2-3)*log(2)-14*x^4-4*x^3+7*x^2)/(6*x
^2*log(2)+7*x^4+2*x^3),x, algorithm="maxima")

[Out]

1/6*(2*(21*log(2) - 1)*arctan((7*x + 1)/sqrt(42*log(2) - 1))/(sqrt(42*log(2) - 1)*log(2)^2) - log(7*x^2 + 2*x
+ 6*log(2))/log(2)^2 + 2*log(x)/log(2)^2 + 6/(x*log(2)))*log(2) - 1/3*(21*log(2) - 1)*arctan((7*x + 1)/sqrt(42
*log(2) - 1))/(sqrt(42*log(2) - 1)*log(2)) - 1/6*(12*x^2*log(2) + 6*(log(3) + 1)*log(2) - (x + 6*log(2))*log(-
7*x^2 - 2*x - 6*log(2)) + 2*(x + 3*log(2))*log(x))/(x*log(2))

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 31, normalized size of antiderivative = 1.07 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=-2 \, x + \frac {\log \left (-7 \, x^{2} - 2 \, x - 6 \, \log \left (2\right )\right )}{x} - \frac {\log \left (3 \, x\right )}{x} \]

[In]

integrate(((-6*log(2)-7*x^2-2*x)*log(1/3*(-6*log(2)-7*x^2-2*x)/x)+2*(-6*x^2-3)*log(2)-14*x^4-4*x^3+7*x^2)/(6*x
^2*log(2)+7*x^4+2*x^3),x, algorithm="giac")

[Out]

-2*x + log(-7*x^2 - 2*x - 6*log(2))/x - log(3*x)/x

Mupad [B] (verification not implemented)

Time = 12.96 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.86 \[ \int \frac {7 x^2-4 x^3-14 x^4+\left (-3-6 x^2\right ) \log (4)+\left (-2 x-7 x^2-3 \log (4)\right ) \log \left (\frac {-2 x-7 x^2-3 \log (4)}{3 x}\right )}{2 x^3+7 x^4+3 x^2 \log (4)} \, dx=\frac {\ln \left (-\frac {\frac {7\,x^2}{3}+\frac {2\,x}{3}+\ln \left (4\right )}{x}\right )}{x}-2\,x \]

[In]

int(-(2*log(2)*(6*x^2 + 3) + log(-((2*x)/3 + 2*log(2) + (7*x^2)/3)/x)*(2*x + 6*log(2) + 7*x^2) - 7*x^2 + 4*x^3
+ 14*x^4)/(6*x^2*log(2) + 2*x^3 + 7*x^4),x)

[Out]

log(-((2*x)/3 + log(4) + (7*x^2)/3)/x)/x - 2*x