3.21.63 \(\int \frac {-6 x^3+2 x^7+(4 x^4+8 x^6+4 x^8) \log (5)+(6-8 x^4+2 x^8) \log (5) \log (3-x^4)}{-3 x^3+x^7} \, dx\) [2063]

Optimal. Leaf size=25 \[ 2 x+\frac {\left (1+x^2\right )^2 \log (5) \log \left (3-x^4\right )}{x^2} \]

[Out]

2*x+ln(-x^4+3)/x^2*ln(5)*(x^2+1)^2

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Rubi [B] Leaf count is larger than twice the leaf count of optimal. \(174\) vs. \(2(25)=50\).
time = 0.37, antiderivative size = 174, normalized size of antiderivative = 6.96, number of steps used = 22, number of rules used = 14, integrand size = 65, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.215, Rules used = {1607, 6857, 1899, 28, 1262, 716, 647, 31, 531, 2608, 2505, 281, 212, 327} \begin {gather*} \frac {\left (\sqrt {3} \left (4 \log ^2(5)+3 \log ^2(25)\right )+12 \log (5) \log (25)\right ) \log \left (\sqrt {3}-x^2\right )}{6 \log (25)}+\frac {\left (12 \log (5) \log (25)-\sqrt {3} \left (4 \log ^2(5)+3 \log ^2(25)\right )\right ) \log \left (x^2+\sqrt {3}\right )}{6 \log (25)}+x^2 \log (25)-2 x^2 \log (5)+2 \sqrt {3} \log (5) \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right )+\frac {2 \log (5) \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right )}{\sqrt {3}}+x^2 \log (5) \log \left (3-x^4\right )+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+2 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-6*x^3 + 2*x^7 + (4*x^4 + 8*x^6 + 4*x^8)*Log[5] + (6 - 8*x^4 + 2*x^8)*Log[5]*Log[3 - x^4])/(-3*x^3 + x^7)
,x]

[Out]

2*x - 2*x^2*Log[5] + (2*ArcTanh[x^2/Sqrt[3]]*Log[5])/Sqrt[3] + 2*Sqrt[3]*ArcTanh[x^2/Sqrt[3]]*Log[5] + x^2*Log
[25] + ((12*Log[5]*Log[25] + Sqrt[3]*(4*Log[5]^2 + 3*Log[25]^2))*Log[Sqrt[3] - x^2])/(6*Log[25]) + ((12*Log[5]
*Log[25] - Sqrt[3]*(4*Log[5]^2 + 3*Log[25]^2))*Log[Sqrt[3] + x^2])/(6*Log[25]) + (Log[5]*Log[3 - x^4])/x^2 + x
^2*Log[5]*Log[3 - x^4]

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 212

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

Rule 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 327

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[c^(n - 1)*(c*x)^(m - n + 1)*((a + b*x^n
)^(p + 1)/(b*(m + n*p + 1))), x] - Dist[a*c^n*((m - n + 1)/(b*(m + n*p + 1))), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 531

Int[(u_.)*((c_) + (d_.)*(x_)^(n_.))^(q_.)*((a1_) + (b1_.)*(x_)^(non2_.))^(p_.)*((a2_) + (b2_.)*(x_)^(non2_.))^
(p_.), x_Symbol] :> Int[u*(a1*a2 + b1*b2*x^n)^p*(c + d*x^n)^q, x] /; FreeQ[{a1, b1, a2, b2, c, d, n, p, q}, x]
&& EqQ[non2, n/2] && EqQ[a2*b1 + a1*b2, 0] && (IntegerQ[p] || (GtQ[a1, 0] && GtQ[a2, 0]))

Rule 647

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q)),
Int[1/(-q + c*x), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x), x], x]] /; FreeQ[{a, c, d, e}, x] && NiceS
qrtQ[(-a)*c]

Rule 716

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 1262

Int[(x_)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2, Subst[Int[(d + e*x)^q
*(a + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, c, d, e, p, q}, x]

Rule 1607

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

Rule 1899

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2*((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b
, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] &&  !PolyQ[Pq, x^(n/2)]

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rule 2608

Int[((a_.) + Log[(c_.)*(RFx_)^(p_.)]*(b_.))^(n_.)*(RGx_), x_Symbol] :> With[{u = ExpandIntegrand[(a + b*Log[c*
RFx^p])^n, RGx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, p}, x] && RationalFunctionQ[RFx, x] && RationalF
unctionQ[RGx, x] && IGtQ[n, 0]

Rule 6857

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-6 x^3+2 x^7+\left (4 x^4+8 x^6+4 x^8\right ) \log (5)+\left (6-8 x^4+2 x^8\right ) \log (5) \log \left (3-x^4\right )}{x^3 \left (-3+x^4\right )} \, dx\\ &=\int \left (\frac {2 \left (-3+x^4+4 x^3 \log (5)+x \log (25)+x^5 \log (25)\right )}{-3+x^4}+\frac {2 (-1+x) (1+x) \left (1+x^2\right ) \log (5) \log \left (3-x^4\right )}{x^3}\right ) \, dx\\ &=2 \int \frac {-3+x^4+4 x^3 \log (5)+x \log (25)+x^5 \log (25)}{-3+x^4} \, dx+(2 \log (5)) \int \frac {(-1+x) (1+x) \left (1+x^2\right ) \log \left (3-x^4\right )}{x^3} \, dx\\ &=2 \int \left (1+\frac {x \left (4 x^2 \log (5)+\log (25)+x^4 \log (25)\right )}{-3+x^4}\right ) \, dx+(2 \log (5)) \int \frac {\left (-1+x^2\right ) \left (1+x^2\right ) \log \left (3-x^4\right )}{x^3} \, dx\\ &=2 x+2 \int \frac {x \left (4 x^2 \log (5)+\log (25)+x^4 \log (25)\right )}{-3+x^4} \, dx+(2 \log (5)) \int \left (-\frac {\log \left (3-x^4\right )}{x^3}+x \log \left (3-x^4\right )\right ) \, dx\\ &=2 x-(2 \log (5)) \int \frac {\log \left (3-x^4\right )}{x^3} \, dx+(2 \log (5)) \int x \log \left (3-x^4\right ) \, dx+\frac {2 \int \frac {x \left (2 \log (5)+x^2 \log (25)\right )^2}{-3+x^4} \, dx}{\log (25)}\\ &=2 x+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+x^2 \log (5) \log \left (3-x^4\right )+(4 \log (5)) \int \frac {x}{3-x^4} \, dx+(4 \log (5)) \int \frac {x^5}{3-x^4} \, dx+\frac {\text {Subst}\left (\int \frac {(2 \log (5)+x \log (25))^2}{-3+x^2} \, dx,x,x^2\right )}{\log (25)}\\ &=2 x+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+x^2 \log (5) \log \left (3-x^4\right )+(2 \log (5)) \text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,x^2\right )+(2 \log (5)) \text {Subst}\left (\int \frac {x^2}{3-x^2} \, dx,x,x^2\right )+\frac {\text {Subst}\left (\int \left (\log ^2(25)+\frac {4 \log ^2(5)+4 x \log (5) \log (25)+3 \log ^2(25)}{-3+x^2}\right ) \, dx,x,x^2\right )}{\log (25)}\\ &=2 x-2 x^2 \log (5)+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)}{\sqrt {3}}+x^2 \log (25)+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+x^2 \log (5) \log \left (3-x^4\right )+(6 \log (5)) \text {Subst}\left (\int \frac {1}{3-x^2} \, dx,x,x^2\right )+\frac {\text {Subst}\left (\int \frac {4 \log ^2(5)+4 x \log (5) \log (25)+3 \log ^2(25)}{-3+x^2} \, dx,x,x^2\right )}{\log (25)}\\ &=2 x-2 x^2 \log (5)+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)}{\sqrt {3}}+2 \sqrt {3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)+x^2 \log (25)+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+x^2 \log (5) \log \left (3-x^4\right )+\frac {\left (2 \log (5) \log (25)-\frac {4 \log ^2(5)+3 \log ^2(25)}{2 \sqrt {3}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {3}+x} \, dx,x,x^2\right )}{\log (25)}+\frac {\left (2 \log (5) \log (25)+\frac {4 \log ^2(5)+3 \log ^2(25)}{2 \sqrt {3}}\right ) \text {Subst}\left (\int \frac {1}{-\sqrt {3}+x} \, dx,x,x^2\right )}{\log (25)}\\ &=2 x-2 x^2 \log (5)+\frac {2 \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)}{\sqrt {3}}+2 \sqrt {3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)+x^2 \log (25)+\frac {\left (12 \log (5) \log (25)+\sqrt {3} \left (4 \log ^2(5)+3 \log ^2(25)\right )\right ) \log \left (\sqrt {3}-x^2\right )}{6 \log (25)}+\frac {\left (12 \log (5) \log (25)-\sqrt {3} \left (4 \log ^2(5)+3 \log ^2(25)\right )\right ) \log \left (\sqrt {3}+x^2\right )}{6 \log (25)}+\frac {\log (5) \log \left (3-x^4\right )}{x^2}+x^2 \log (5) \log \left (3-x^4\right )\\ \end {aligned} \end {gather*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(111\) vs. \(2(25)=50\).
time = 0.13, size = 111, normalized size = 4.44 \begin {gather*} \frac {1}{3} \left (6 x+8 \sqrt {3} \tanh ^{-1}\left (\frac {x^2}{\sqrt {3}}\right ) \log (5)+\left (\sqrt {3} \log (625)+\log (15625)\right ) \log \left (\sqrt {3}-x^2\right )-\sqrt {3} \log (625) \log \left (\sqrt {3}+x^2\right )+\log (15625) \log \left (\sqrt {3}+x^2\right )+\frac {\log (125) \log \left (3-x^4\right )}{x^2}+x^2 \log (125) \log \left (3-x^4\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-6*x^3 + 2*x^7 + (4*x^4 + 8*x^6 + 4*x^8)*Log[5] + (6 - 8*x^4 + 2*x^8)*Log[5]*Log[3 - x^4])/(-3*x^3
+ x^7),x]

[Out]

(6*x + 8*Sqrt[3]*ArcTanh[x^2/Sqrt[3]]*Log[5] + (Sqrt[3]*Log[625] + Log[15625])*Log[Sqrt[3] - x^2] - Sqrt[3]*Lo
g[625]*Log[Sqrt[3] + x^2] + Log[15625]*Log[Sqrt[3] + x^2] + (Log[125]*Log[3 - x^4])/x^2 + x^2*Log[125]*Log[3 -
x^4])/3

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Maple [A]
time = 0.25, size = 43, normalized size = 1.72

method	result	size
risch	\(\frac {\ln \left (5\right ) \left (x^{4}+1\right ) \ln \left (-x^{4}+3\right )}{x^{2}}+2 x +2 \ln \left (5\right ) \ln \left (x^{4}-3\right )\)	\(34\)
default	\(2 x +\ln \left (5\right ) \ln \left (-x^{4}+3\right ) x^{2}+\frac {\ln \left (5\right ) \ln \left (-x^{4}+3\right )}{x^{2}}+2 \ln \left (5\right ) \ln \left (x^{4}-3\right )\)	\(43\)
norman	\(\frac {\ln \left (5\right ) \ln \left (-x^{4}+3\right )+2 \ln \left (5\right ) \ln \left (-x^{4}+3\right ) x^{2}+\ln \left (5\right ) \ln \left (-x^{4}+3\right ) x^{4}+2 x^{3}}{x^{2}}\)	\(51\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((2*x^8-8*x^4+6)*ln(5)*ln(-x^4+3)+(4*x^8+8*x^6+4*x^4)*ln(5)+2*x^7-6*x^3)/(x^7-3*x^3),x,method=_RETURNVERBO
SE)

[Out]

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

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 113 vs. \(2 (25) = 50\).
time = 0.52, size = 113, normalized size = 4.52 \begin {gather*} {\left (2 \, x^{2} - \sqrt {3} \log \left (x^{2} + \sqrt {3}\right ) + \sqrt {3} \log \left (x^{2} - \sqrt {3}\right )\right )} \log \left (5\right ) + {\left (\sqrt {3} \log \left (x^{2} + \sqrt {3}\right ) - \sqrt {3} \log \left (x^{2} - \sqrt {3}\right )\right )} \log \left (5\right ) + 2 \, \log \left (5\right ) \log \left (x^{4} - 3\right ) + 2 \, x - \frac {2 \, x^{4} \log \left (5\right ) - {\left (x^{4} \log \left (5\right ) + \log \left (5\right )\right )} \log \left (-x^{4} + 3\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2*x^7-6*x^3)/(x^7-3*x^3),x, algorithm
="maxima")

[Out]

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

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Fricas [A]
time = 0.36, size = 31, normalized size = 1.24 \begin {gather*} \frac {2 \, x^{3} + {\left (x^{4} + 2 \, x^{2} + 1\right )} \log \left (5\right ) \log \left (-x^{4} + 3\right )}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2*x^7-6*x^3)/(x^7-3*x^3),x, algorithm
="fricas")

[Out]

(2*x^3 + (x^4 + 2*x^2 + 1)*log(5)*log(-x^4 + 3))/x^2

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Sympy [A]
time = 0.14, size = 34, normalized size = 1.36 \begin {gather*} 2 x + 2 \log {\left (5 \right )} \log {\left (x^{4} - 3 \right )} + \frac {\left (x^{4} \log {\left (5 \right )} + \log {\left (5 \right )}\right ) \log {\left (3 - x^{4} \right )}}{x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x**8-8*x**4+6)*ln(5)*ln(-x**4+3)+(4*x**8+8*x**6+4*x**4)*ln(5)+2*x**7-6*x**3)/(x**7-3*x**3),x)

[Out]

2*x + 2*log(5)*log(x**4 - 3) + (x**4*log(5) + log(5))*log(3 - x**4)/x**2

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Giac [A]
time = 0.43, size = 36, normalized size = 1.44 \begin {gather*} 2 \, \log \left (5\right ) \log \left (x^{4} - 3\right ) + {\left (x^{2} \log \left (5\right ) + \frac {\log \left (5\right )}{x^{2}}\right )} \log \left (-x^{4} + 3\right ) + 2 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((2*x^8-8*x^4+6)*log(5)*log(-x^4+3)+(4*x^8+8*x^6+4*x^4)*log(5)+2*x^7-6*x^3)/(x^7-3*x^3),x, algorithm
="giac")

[Out]

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

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Mupad [B]
time = 1.39, size = 35, normalized size = 1.40 \begin {gather*} 2\,x+2\,\ln \left (5\right )\,\ln \left (x^4-3\right )+\frac {\ln \left (3-x^4\right )\,\left (\ln \left (5\right )\,x^4+\ln \left (5\right )\right )}{x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(log(5)*(4*x^4 + 8*x^6 + 4*x^8) - 6*x^3 + 2*x^7 + log(3 - x^4)*log(5)*(2*x^8 - 8*x^4 + 6))/(3*x^3 - x^7),
x)

[Out]

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

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