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

Optimal. Leaf size=29 \[ -2 x+\frac {\log \left (\frac {1}{3} (-2-x)-2 x-\frac {\log (4)}{x}\right )}{x} \]

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Rubi [B]  time = 0.77, antiderivative size = 107, normalized size of antiderivative = 3.69, number of steps used = 16, number of rules used = 8, integrand size = 82, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.098, Rules used = {1594, 6728, 1628, 634, 618, 204, 628, 2525} \begin {gather*} -2 x+\frac {\log \left (-\frac {7 x}{3}-\frac {\log (4)}{x}-\frac {2}{3}\right )}{x}+\frac {\sqrt {7 \log (64)-1} \log (4096) \tan ^{-1}\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 ) \tan ^{-1}\left (\frac {7 x+1}{\sqrt {7 \log (64)-1}}\right )}{\log ^2(64) \sqrt {7 \log (64)-1}} \end {gather*}

Antiderivative was successfully verified.

[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 204

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

Rule 618

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 628

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 634

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 1594

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 1628

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 2525

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 6728

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 {gather*} \begin {aligned} \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 ) \operatorname {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 ) \operatorname {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 {aligned} \end {gather*}

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Mathematica [A]  time = 0.23, size = 33, normalized size = 1.14 \begin {gather*} -\frac {\log (3)+\frac {x^2 \log (4096)}{\log (64)}-\log \left (-2-7 x-\frac {\log (64)}{x}\right )}{x} \end {gather*}

Antiderivative was successfully verified.

[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)

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fricas [A]  time = 0.73, size = 32, normalized size = 1.10 \begin {gather*} -\frac {2 \, x^{2} - \log \left (-\frac {7 \, x^{2} + 2 \, x + 6 \, \log \relax (2)}{3 \, x}\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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giac [A]  time = 0.21, size = 31, normalized size = 1.07 \begin {gather*} -2 \, x + \frac {\log \left (-7 \, x^{2} - 2 \, x - 6 \, \log \relax (2)\right )}{x} - \frac {\log \left (3 \, x\right )}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.14, size = 28, normalized size = 0.97

method	result	size
risch	\(\frac {\ln \left (\frac {-6 \ln \relax (2)-7 x^{2}-2 x}{3 x}\right )}{x}-2 x\)	\(28\)
norman	\(\frac {-2 x^{2}+\ln \left (\frac {-6 \ln \relax (2)-7 x^{2}-2 x}{3 x}\right )}{x}\)	\(30\)
default	\(-\frac {\ln \relax (3)}{x}+\frac {\ln \left (\frac {-6 \ln \relax (2)-7 x^{2}-2 x}{x}\right )}{x}-\frac {\sqrt {-1+42 \ln \relax (2)}\, \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \relax (2)}}\right )}{3 \ln \relax (2)}-2 x -\frac {\arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \relax (2)}}\right )}{3 \ln \relax (2) \sqrt {-1+42 \ln \relax (2)}}+\frac {14 \arctan \left (\frac {14 x +2}{2 \sqrt {-1+42 \ln \relax (2)}}\right )}{\sqrt {-1+42 \ln \relax (2)}}\)	\(120\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [B]  time = 1.11, size = 170, normalized size = 5.86 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, {\left (21 \, \log \relax (2) - 1\right )} \arctan \left (\frac {7 \, x + 1}{\sqrt {42 \, \log \relax (2) - 1}}\right )}{\sqrt {42 \, \log \relax (2) - 1} \log \relax (2)^{2}} - \frac {\log \left (7 \, x^{2} + 2 \, x + 6 \, \log \relax (2)\right )}{\log \relax (2)^{2}} + \frac {2 \, \log \relax (x)}{\log \relax (2)^{2}} + \frac {6}{x \log \relax (2)}\right )} \log \relax (2) - \frac {{\left (21 \, \log \relax (2) - 1\right )} \arctan \left (\frac {7 \, x + 1}{\sqrt {42 \, \log \relax (2) - 1}}\right )}{3 \, \sqrt {42 \, \log \relax (2) - 1} \log \relax (2)} - \frac {12 \, x^{2} \log \relax (2) + 6 \, {\left (\log \relax (3) + 1\right )} \log \relax (2) - {\left (x + 6 \, \log \relax (2)\right )} \log \left (-7 \, x^{2} - 2 \, x - 6 \, \log \relax (2)\right ) + 2 \, {\left (x + 3 \, \log \relax (2)\right )} \log \relax (x)}{6 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [B]  time = 4.51, size = 25, normalized size = 0.86 \begin {gather*} \frac {\ln \left (-\frac {\frac {7\,x^2}{3}+\frac {2\,x}{3}+\ln \relax (4)}{x}\right )}{x}-2\,x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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sympy [A]  time = 0.22, size = 26, normalized size = 0.90 \begin {gather*} - 2 x + \frac {\log {\left (\frac {- \frac {7 x^{2}}{3} - \frac {2 x}{3} - 2 \log {\relax (2 )}}{x} \right )}}{x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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