3.50.67 \(\int \frac {(-405+2 x+x^2) \log (\frac {1}{2} (405-2 x-x^2))+\log (x) (2 x+2 x^2+(405-2 x-x^2) \log (\frac {1}{2} (405-2 x-x^2)))}{(-810 x^2+4 x^3+2 x^4) \log (2)} \, dx\)

Optimal. Leaf size=29 \[ \frac {\log (x) \log \left (-x+\frac {1}{2} \left (405-x^2\right )\right )}{2 x \log (2)} \]

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Rubi [B]  time = 1.48, antiderivative size = 89, normalized size of antiderivative = 3.07, number of steps used = 44, number of rules used = 17, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.198, Rules used = {12, 1594, 6688, 14, 2357, 2301, 2316, 2315, 2317, 2391, 2304, 2525, 800, 632, 31, 30, 2557} \begin {gather*} \frac {\log \left (-x^2-2 x+405\right ) \log (x)}{2 x \log (2)}-\frac {\log \left (\frac {1}{2} \left (-x^2-2 x+405\right )\right )}{2 x \log (2)}+\frac {\log \left (-x^2-2 x+405\right )}{2 x \log (2)}-\frac {1}{2 x}-\frac {\log (x)}{2 x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-405 + 2*x + x^2)*Log[(405 - 2*x - x^2)/2] + Log[x]*(2*x + 2*x^2 + (405 - 2*x - x^2)*Log[(405 - 2*x - x^
2)/2]))/((-810*x^2 + 4*x^3 + 2*x^4)*Log[2]),x]

[Out]

-1/2*1/x - Log[x]/(2*x) - Log[(405 - 2*x - x^2)/2]/(2*x*Log[2]) + Log[405 - 2*x - x^2]/(2*x*Log[2]) + (Log[x]*
Log[405 - 2*x - x^2])/(2*x*Log[2])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 31

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

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), 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 800

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 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 2301

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 2304

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 2315

Int[Log[(c_.)*(x_)]/((d_) + (e_.)*(x_)), x_Symbol] :> -Simp[PolyLog[2, 1 - c*x]/e, x] /; FreeQ[{c, d, e}, x] &
& EqQ[e + c*d, 0]

Rule 2316

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

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 0]

Rule 2357

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]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

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 2557

Int[Log[v_]*Log[w_]*(u_), x_Symbol] :> With[{z = IntHide[u, x]}, Dist[Log[v]*Log[w], z, x] + (-Int[SimplifyInt
egrand[(z*Log[w]*D[v, x])/v, x], x] - Int[SimplifyIntegrand[(z*Log[v]*D[w, x])/w, x], x]) /; InverseFunctionFr
eeQ[z, x]] /; InverseFunctionFreeQ[v, x] && InverseFunctionFreeQ[w, x]

Rule 6688

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

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {\int \frac {\left (-405+2 x+x^2\right ) \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )+\log (x) \left (2 x+2 x^2+\left (405-2 x-x^2\right ) \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )\right )}{-810 x^2+4 x^3+2 x^4} \, dx}{\log (2)}\\ &=\frac {\int \frac {\left (-405+2 x+x^2\right ) \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )+\log (x) \left (2 x+2 x^2+\left (405-2 x-x^2\right ) \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )\right )}{x^2 \left (-810+4 x+2 x^2\right )} \, dx}{\log (2)}\\ &=\frac {\int \frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )+\log (x) \left (\frac {2 x (1+x)}{-405+2 x+x^2}+\log (2)-\log \left (405-2 x-x^2\right )\right )}{2 x^2} \, dx}{\log (2)}\\ &=\frac {\int \frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )+\log (x) \left (\frac {2 x (1+x)}{-405+2 x+x^2}+\log (2)-\log \left (405-2 x-x^2\right )\right )}{x^2} \, dx}{2 \log (2)}\\ &=\frac {\int \left (\frac {-2 x^2 \left (1+\frac {\log (2)}{2}\right ) \log (x)+405 \log (2) \log (x)-2 x (1+\log (2)) \log (x)+405 \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )-2 x \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )-x^2 \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{x^2 \left (405-2 x-x^2\right )}-\frac {\log (x) \log \left (405-2 x-x^2\right )}{x^2}\right ) \, dx}{2 \log (2)}\\ &=\frac {\int \frac {-2 x^2 \left (1+\frac {\log (2)}{2}\right ) \log (x)+405 \log (2) \log (x)-2 x (1+\log (2)) \log (x)+405 \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )-2 x \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )-x^2 \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{x^2 \left (405-2 x-x^2\right )} \, dx}{2 \log (2)}-\frac {\int \frac {\log (x) \log \left (405-2 x-x^2\right )}{x^2} \, dx}{2 \log (2)}\\ &=\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\int \frac {2 (1+x) \log (x)}{x \left (405-2 x-x^2\right )} \, dx}{2 \log (2)}+\frac {\int \frac {\frac {\left (-405 \log (2)+x^2 (2+\log (2))+x (2+\log (4))\right ) \log (x)}{-405+2 x+x^2}+\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{x^2} \, dx}{2 \log (2)}-\frac {\int \frac {\log \left (405-2 x-x^2\right )}{x^2} \, dx}{2 \log (2)}\\ &=\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}-\frac {\int \frac {2 (-1-x)}{x \left (405-2 x-x^2\right )} \, dx}{2 \log (2)}+\frac {\int \left (\frac {\left (405 \log (2)-x^2 (2+\log (2))-x (2+\log (4))\right ) \log (x)}{x^2 \left (405-2 x-x^2\right )}+\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{x^2}\right ) \, dx}{2 \log (2)}+\frac {\int \frac {(1+x) \log (x)}{x \left (405-2 x-x^2\right )} \, dx}{\log (2)}\\ &=\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\int \frac {\left (405 \log (2)-x^2 (2+\log (2))-x (2+\log (4))\right ) \log (x)}{x^2 \left (405-2 x-x^2\right )} \, dx}{2 \log (2)}+\frac {\int \frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{x^2} \, dx}{2 \log (2)}-\frac {\int \frac {-1-x}{x \left (405-2 x-x^2\right )} \, dx}{\log (2)}+\frac {\int \left (\frac {\log (x)}{405 x}+\frac {(-407-x) \log (x)}{405 \left (-405+2 x+x^2\right )}\right ) \, dx}{\log (2)}\\ &=-\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)}+\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\int \frac {\log (x)}{x} \, dx}{405 \log (2)}+\frac {\int \frac {(-407-x) \log (x)}{-405+2 x+x^2} \, dx}{405 \log (2)}+\frac {\int \frac {2 (-1-x)}{x \left (405-2 x-x^2\right )} \, dx}{2 \log (2)}+\frac {\int \left (-\frac {2 \log (x)}{405 x}+\frac {2 (407+x) \log (x)}{405 \left (-405+2 x+x^2\right )}+\frac {\log (2) \log (x)}{x^2}\right ) \, dx}{2 \log (2)}-\frac {\int \left (-\frac {1}{405 x}+\frac {407+x}{405 \left (-405+2 x+x^2\right )}\right ) \, dx}{\log (2)}\\ &=\frac {\log (x)}{405 \log (2)}+\frac {\log ^2(x)}{810 \log (2)}-\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)}+\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {1}{2} \int \frac {\log (x)}{x^2} \, dx-\frac {\int \frac {407+x}{-405+2 x+x^2} \, dx}{405 \log (2)}-\frac {\int \frac {\log (x)}{x} \, dx}{405 \log (2)}+\frac {\int \frac {(407+x) \log (x)}{-405+2 x+x^2} \, dx}{405 \log (2)}+\frac {\int \left (\frac {\left (-1-\sqrt {406}\right ) \log (x)}{2-2 \sqrt {406}+2 x}+\frac {\left (-1+\sqrt {406}\right ) \log (x)}{2+2 \sqrt {406}+2 x}\right ) \, dx}{405 \log (2)}+\frac {\int \frac {-1-x}{x \left (405-2 x-x^2\right )} \, dx}{\log (2)}\\ &=-\frac {1}{2 x}-\frac {\log (x)}{2 x}+\frac {\log (x)}{405 \log (2)}-\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)}+\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\int \left (\frac {\left (1+\sqrt {406}\right ) \log (x)}{2-2 \sqrt {406}+2 x}+\frac {\left (1-\sqrt {406}\right ) \log (x)}{2+2 \sqrt {406}+2 x}\right ) \, dx}{405 \log (2)}+\frac {\int \left (-\frac {1}{405 x}+\frac {407+x}{405 \left (-405+2 x+x^2\right )}\right ) \, dx}{\log (2)}-\frac {\left (1-\sqrt {406}\right ) \int \frac {1}{1+\sqrt {406}+x} \, dx}{810 \log (2)}-\frac {\left (1-\sqrt {406}\right ) \int \frac {\log (x)}{2+2 \sqrt {406}+2 x} \, dx}{405 \log (2)}-\frac {\left (1+\sqrt {406}\right ) \int \frac {1}{1-\sqrt {406}+x} \, dx}{810 \log (2)}-\frac {\left (1+\sqrt {406}\right ) \int \frac {\log (x)}{2-2 \sqrt {406}+2 x} \, dx}{405 \log (2)}\\ &=-\frac {1}{2 x}-\frac {\log (x)}{2 x}-\frac {\left (1+\sqrt {406}\right ) \log \left (1-\sqrt {406}+x\right )}{810 \log (2)}-\frac {\left (1-\sqrt {406}\right ) \log \left (1+\sqrt {406}+x\right )}{810 \log (2)}-\frac {\left (1+\sqrt {406}\right ) \log \left (-1+\sqrt {406}\right ) \log \left (2 \left (1-\sqrt {406}\right )+2 x\right )}{810 \log (2)}-\frac {\left (1-\sqrt {406}\right ) \log (x) \log \left (1+\frac {x}{1+\sqrt {406}}\right )}{810 \log (2)}-\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)}+\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\int \frac {407+x}{-405+2 x+x^2} \, dx}{405 \log (2)}+\frac {\left (1-\sqrt {406}\right ) \int \frac {\log \left (1+\frac {2 x}{2+2 \sqrt {406}}\right )}{x} \, dx}{810 \log (2)}+\frac {\left (1-\sqrt {406}\right ) \int \frac {\log (x)}{2+2 \sqrt {406}+2 x} \, dx}{405 \log (2)}+\frac {\left (1+\sqrt {406}\right ) \int \frac {\log (x)}{2-2 \sqrt {406}+2 x} \, dx}{405 \log (2)}-\frac {\left (1+\sqrt {406}\right ) \int \frac {\log \left (-\frac {2 x}{2-2 \sqrt {406}}\right )}{2-2 \sqrt {406}+2 x} \, dx}{405 \log (2)}\\ &=-\frac {1}{2 x}-\frac {\log (x)}{2 x}-\frac {\left (1+\sqrt {406}\right ) \log \left (1-\sqrt {406}+x\right )}{810 \log (2)}-\frac {\left (1-\sqrt {406}\right ) \log \left (1+\sqrt {406}+x\right )}{810 \log (2)}-\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)}+\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}-\frac {\left (1-\sqrt {406}\right ) \text {Li}_2\left (-\frac {x}{1+\sqrt {406}}\right )}{810 \log (2)}+\frac {\left (1+\sqrt {406}\right ) \text {Li}_2\left (1+\frac {x}{1-\sqrt {406}}\right )}{810 \log (2)}+\frac {\left (1-\sqrt {406}\right ) \int \frac {1}{1+\sqrt {406}+x} \, dx}{810 \log (2)}-\frac {\left (1-\sqrt {406}\right ) \int \frac {\log \left (1+\frac {2 x}{2+2 \sqrt {406}}\right )}{x} \, dx}{810 \log (2)}+\frac {\left (1+\sqrt {406}\right ) \int \frac {1}{1-\sqrt {406}+x} \, dx}{810 \log (2)}+\frac {\left (1+\sqrt {406}\right ) \int \frac {\log \left (-\frac {2 x}{2-2 \sqrt {406}}\right )}{2-2 \sqrt {406}+2 x} \, dx}{405 \log (2)}\\ &=-\frac {1}{2 x}-\frac {\log (x)}{2 x}-\frac {\log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)}+\frac {\log \left (405-2 x-x^2\right )}{2 x \log (2)}+\frac {\log (x) \log \left (405-2 x-x^2\right )}{2 x \log (2)}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.26, size = 28, normalized size = 0.97 \begin {gather*} \frac {\log (x) \log \left (\frac {1}{2} \left (405-2 x-x^2\right )\right )}{2 x \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-405 + 2*x + x^2)*Log[(405 - 2*x - x^2)/2] + Log[x]*(2*x + 2*x^2 + (405 - 2*x - x^2)*Log[(405 - 2*
x - x^2)/2]))/((-810*x^2 + 4*x^3 + 2*x^4)*Log[2]),x]

[Out]

(Log[x]*Log[(405 - 2*x - x^2)/2])/(2*x*Log[2])

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fricas [A]  time = 0.87, size = 22, normalized size = 0.76 \begin {gather*} \frac {\log \left (-\frac {1}{2} \, x^{2} - x + \frac {405}{2}\right ) \log \relax (x)}{2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x+405)*log(-1/2*x^2-x+405/2)+2*x^2+2*x)*log(x)+(x^2+2*x-405)*log(-1/2*x^2-x+405/2))/(2*x^4
+4*x^3-810*x^2)/log(2),x, algorithm="fricas")

[Out]

1/2*log(-1/2*x^2 - x + 405/2)*log(x)/(x*log(2))

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giac [A]  time = 0.19, size = 33, normalized size = 1.14 \begin {gather*} -\frac {\frac {\log \relax (2) \log \relax (x)}{x} - \frac {\log \left (-x^{2} - 2 \, x + 405\right ) \log \relax (x)}{x}}{2 \, \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x+405)*log(-1/2*x^2-x+405/2)+2*x^2+2*x)*log(x)+(x^2+2*x-405)*log(-1/2*x^2-x+405/2))/(2*x^4
+4*x^3-810*x^2)/log(2),x, algorithm="giac")

[Out]

-1/2*(log(2)*log(x)/x - log(-x^2 - 2*x + 405)*log(x)/x)/log(2)

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maple [A]  time = 0.05, size = 23, normalized size = 0.79

method	result	size
risch	\(\frac {\ln \relax (x ) \ln \left (-\frac {1}{2} x^{2}-x +\frac {405}{2}\right )}{2 \ln \relax (2) x}\)	\(23\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((-x^2-2*x+405)*ln(-1/2*x^2-x+405/2)+2*x^2+2*x)*ln(x)+(x^2+2*x-405)*ln(-1/2*x^2-x+405/2))/(2*x^4+4*x^3-81
0*x^2)/ln(2),x,method=_RETURNVERBOSE)

[Out]

1/2/ln(2)*ln(x)*ln(-1/2*x^2-x+405/2)/x

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maxima [A]  time = 0.59, size = 30, normalized size = 1.03 \begin {gather*} -\frac {\log \relax (2) \log \relax (x) - \log \left (-x^{2} - 2 \, x + 405\right ) \log \relax (x)}{2 \, x \log \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x^2-2*x+405)*log(-1/2*x^2-x+405/2)+2*x^2+2*x)*log(x)+(x^2+2*x-405)*log(-1/2*x^2-x+405/2))/(2*x^4
+4*x^3-810*x^2)/log(2),x, algorithm="maxima")

[Out]

-1/2*(log(2)*log(x) - log(-x^2 - 2*x + 405)*log(x))/(x*log(2))

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mupad [B]  time = 3.42, size = 22, normalized size = 0.76 \begin {gather*} \frac {\ln \left (-\frac {x^2}{2}-x+\frac {405}{2}\right )\,\ln \relax (x)}{2\,x\,\ln \relax (2)} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log(405/2 - x^2/2 - x)*(2*x + x^2 - 405) + log(x)*(2*x - log(405/2 - x^2/2 - x)*(2*x + x^2 - 405) + 2*x^2
))/(log(2)*(4*x^3 - 810*x^2 + 2*x^4)),x)

[Out]

(log(405/2 - x^2/2 - x)*log(x))/(2*x*log(2))

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sympy [A]  time = 0.74, size = 20, normalized size = 0.69 \begin {gather*} \frac {\log {\relax (x )} \log {\left (- \frac {x^{2}}{2} - x + \frac {405}{2} \right )}}{2 x \log {\relax (2 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((-x**2-2*x+405)*ln(-1/2*x**2-x+405/2)+2*x**2+2*x)*ln(x)+(x**2+2*x-405)*ln(-1/2*x**2-x+405/2))/(2*x
**4+4*x**3-810*x**2)/ln(2),x)

[Out]

log(x)*log(-x**2/2 - x + 405/2)/(2*x*log(2))

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