3.98.52 \(\int \frac {(-2+2 e^{100} x^2) \log (\frac {1+e^{100} (27 x+x^2)}{e^{100} x})+(1+e^{100} (27 x+x^2)) \log ^2(\frac {1+e^{100} (27 x+x^2)}{e^{100} x})}{1+e^{100} (27 x+x^2)} \, dx\)

Optimal. Leaf size=15 \[ x \log ^2\left (27+\frac {1}{e^{100} x}+x\right ) \]

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Rubi [C]  time = 3.06, antiderivative size = 47, normalized size of antiderivative = 3.13, number of steps used = 74, number of rules used = 17, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.198, Rules used = {6688, 6728, 1657, 634, 618, 206, 628, 2523, 2528, 2524, 2418, 2392, 2391, 2390, 2301, 2394, 2393} \begin {gather*} 0 \text {Li}_2\left (-\frac {2 e^{50} x}{27 e^{50}-\sqrt {-4+729 e^{100}}}\right )+x \log ^2\left (x+\frac {1}{e^{100} x}+27\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((-2 + 2*E^100*x^2)*Log[(1 + E^100*(27*x + x^2))/(E^100*x)] + (1 + E^100*(27*x + x^2))*Log[(1 + E^100*(27*
x + x^2))/(E^100*x)]^2)/(1 + E^100*(27*x + x^2)),x]

[Out]

x*Log[27 + 1/(E^100*x) + x]^2 + 0*PolyLog[2, (-2*E^50*x)/(27*E^50 - Sqrt[-4 + 729*E^100])]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 1657

Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x + c*x^2)^p, x
], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

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 2390

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_))^(q_.), x_Symbol] :> Dist[1/
e, Subst[Int[((f*x)/d)^q*(a + b*Log[c*x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, q}, x]
&& EqQ[e*f - d*g, 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 2392

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

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2394

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

Rule 2418

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*(RFx_), x_Symbol] :> With[{u = ExpandIntegrand[
(a + b*Log[c*(d + e*x)^n])^p, RFx, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[{a, b, c, d, e, n}, x] && RationalFunct
ionQ[RFx, x] && IntegerQ[p]

Rule 2523

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 2524

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

Rule 2528

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 6688

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

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 {\log \left (27+\frac {1}{e^{100} x}+x\right ) \left (-2+2 e^{100} x^2+\left (1+e^{100} x (27+x)\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )\right )}{1+27 e^{100} x+e^{100} x^2} \, dx\\ &=\int \left (\frac {2 \left (-1+e^{100} x^2\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2}+\log ^2\left (27+\frac {1}{e^{100} x}+x\right )\right ) \, dx\\ &=2 \int \frac {\left (-1+e^{100} x^2\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2} \, dx+\int \log ^2\left (27+\frac {1}{e^{100} x}+x\right ) \, dx\\ &=x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )-2 \int \frac {\left (-1+e^{100} x^2\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2} \, dx+2 \int \left (\log \left (27+\frac {1}{e^{100} x}+x\right )-\frac {\left (2+27 e^{100} x\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx\\ &=x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )+2 \int \log \left (27+\frac {1}{e^{100} x}+x\right ) \, dx-2 \int \frac {\left (2+27 e^{100} x\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2} \, dx-2 \int \left (\log \left (27+\frac {1}{e^{100} x}+x\right )-\frac {\left (2+27 e^{100} x\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx\\ &=2 x \log \left (27+\frac {1}{e^{100} x}+x\right )+x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )-2 \int \frac {-1+e^{100} x^2}{1+27 e^{100} x+e^{100} x^2} \, dx-2 \int \log \left (27+\frac {1}{e^{100} x}+x\right ) \, dx+2 \int \frac {\left (2+27 e^{100} x\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{1+27 e^{100} x+e^{100} x^2} \, dx-2 \int \left (\frac {\left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x}+\frac {\left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x}\right ) \, dx\\ &=x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )+2 \int \frac {-1+e^{100} x^2}{1+27 e^{100} x+e^{100} x^2} \, dx-2 \int \left (1-\frac {2+27 e^{100} x}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx+2 \int \left (\frac {\left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x}+\frac {\left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}\right ) \log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x}\right ) \, dx-\left (2 e^{50} \left (27 e^{50}-\sqrt {-4+729 e^{100}}\right )\right ) \int \frac {\log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x} \, dx-\left (2 e^{50} \left (27 e^{50}+\sqrt {-4+729 e^{100}}\right )\right ) \int \frac {\log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x} \, dx\\ &=-2 x+x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )-\left (27-\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \log \left (27+\frac {1}{e^{100} x}+x\right ) \log \left (e^{50} \left (27 e^{50}-\sqrt {-4+729 e^{100}}\right )+2 e^{100} x\right )-\left (27+\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \log \left (27+\frac {1}{e^{100} x}+x\right ) \log \left (e^{50} \left (27 e^{50}+\sqrt {-4+729 e^{100}}\right )+2 e^{100} x\right )+2 \int \frac {2+27 e^{100} x}{1+27 e^{100} x+e^{100} x^2} \, dx+2 \int \left (1-\frac {2+27 e^{100} x}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx+\left (2 e^{50} \left (27 e^{50}-\sqrt {-4+729 e^{100}}\right )\right ) \int \frac {\log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x} \, dx+\left (2 e^{50} \left (27 e^{50}+\sqrt {-4+729 e^{100}}\right )\right ) \int \frac {\log \left (27+\frac {1}{e^{100} x}+x\right )}{27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x} \, dx+\left (27-\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \frac {\left (1-\frac {1}{e^{100} x^2}\right ) \log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{27+\frac {1}{e^{100} x}+x} \, dx+\left (27+\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \frac {\left (1-\frac {1}{e^{100} x^2}\right ) \log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{27+\frac {1}{e^{100} x}+x} \, dx\\ &=x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )-2 \int \frac {2+27 e^{100} x}{1+27 e^{100} x+e^{100} x^2} \, dx+27 \int \frac {27 e^{100}+2 e^{100} x}{1+27 e^{100} x+e^{100} x^2} \, dx+\left (4-729 e^{100}\right ) \int \frac {1}{1+27 e^{100} x+e^{100} x^2} \, dx-\left (27-\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \frac {\left (1-\frac {1}{e^{100} x^2}\right ) \log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{27+\frac {1}{e^{100} x}+x} \, dx+\left (27-\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \left (-\frac {\log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{x}+\frac {e^{100} (27+2 x) \log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx-\left (27+\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \frac {\left (1-\frac {1}{e^{100} x^2}\right ) \log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{27+\frac {1}{e^{100} x}+x} \, dx+\left (27+\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \left (-\frac {\log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{x}+\frac {e^{100} (27+2 x) \log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx\\ &=x \log ^2\left (27+\frac {1}{e^{100} x}+x\right )+27 \log \left (1+27 e^{100} x+e^{100} x^2\right )-27 \int \frac {27 e^{100}+2 e^{100} x}{1+27 e^{100} x+e^{100} x^2} \, dx-\left (4-729 e^{100}\right ) \int \frac {1}{1+27 e^{100} x+e^{100} x^2} \, dx-\left (2 \left (4-729 e^{100}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-e^{100} \left (4-729 e^{100}\right )-x^2} \, dx,x,27 e^{100}+2 e^{100} x\right )+\left (e^{50} \left (27 e^{50}-\sqrt {-4+729 e^{100}}\right )\right ) \int \frac {(27+2 x) \log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{1+27 e^{100} x+e^{100} x^2} \, dx+\left (e^{50} \left (27 e^{50}+\sqrt {-4+729 e^{100}}\right )\right ) \int \frac {(27+2 x) \log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{1+27 e^{100} x+e^{100} x^2} \, dx+\left (-27-\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \frac {\log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{x} \, dx-\left (27-\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \left (-\frac {\log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{x}+\frac {e^{100} (27+2 x) \log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx+\left (-27+\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \frac {\log \left (27 e^{100}-e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{x} \, dx-\left (27+\frac {\sqrt {-4+729 e^{100}}}{e^{50}}\right ) \int \left (-\frac {\log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{x}+\frac {e^{100} (27+2 x) \log \left (27 e^{100}+e^{50} \sqrt {-4+729 e^{100}}+2 e^{100} x\right )}{1+27 e^{100} x+e^{100} x^2}\right ) \, dx\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.55, size = 15, normalized size = 1.00 \begin {gather*} x \log ^2\left (27+\frac {1}{e^{100} x}+x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((-2 + 2*E^100*x^2)*Log[(1 + E^100*(27*x + x^2))/(E^100*x)] + (1 + E^100*(27*x + x^2))*Log[(1 + E^10
0*(27*x + x^2))/(E^100*x)]^2)/(1 + E^100*(27*x + x^2)),x]

[Out]

x*Log[27 + 1/(E^100*x) + x]^2

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fricas [A]  time = 0.69, size = 23, normalized size = 1.53 \begin {gather*} x \log \left (\frac {{\left ({\left (x^{2} + 27 \, x\right )} e^{100} + 1\right )} e^{\left (-100\right )}}{x}\right )^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+27*x)*exp(25)^4+1)*log(((x^2+27*x)*exp(25)^4+1)/x/exp(25)^4)^2+(2*x^2*exp(25)^4-2)*log(((x^2+
27*x)*exp(25)^4+1)/x/exp(25)^4))/((x^2+27*x)*exp(25)^4+1),x, algorithm="fricas")

[Out]

x*log(((x^2 + 27*x)*e^100 + 1)*e^(-100)/x)^2

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giac [B]  time = 1.39, size = 47, normalized size = 3.13 \begin {gather*} x \log \left (\frac {x^{2} e^{100} + 27 \, x e^{100} + 1}{x}\right )^{2} - 200 \, x \log \left (\frac {x^{2} e^{100} + 27 \, x e^{100} + 1}{x}\right ) + 10000 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+27*x)*exp(25)^4+1)*log(((x^2+27*x)*exp(25)^4+1)/x/exp(25)^4)^2+(2*x^2*exp(25)^4-2)*log(((x^2+
27*x)*exp(25)^4+1)/x/exp(25)^4))/((x^2+27*x)*exp(25)^4+1),x, algorithm="giac")

[Out]

x*log((x^2*e^100 + 27*x*e^100 + 1)/x)^2 - 200*x*log((x^2*e^100 + 27*x*e^100 + 1)/x) + 10000*x

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maple [A]  time = 0.17, size = 24, normalized size = 1.60

method	result	size
risch	\(x \ln \left (\frac {\left (\left (x^{2}+27 x \right ) {\mathrm e}^{100}+1\right ) {\mathrm e}^{-100}}{x}\right )^{2}\)	\(24\)
norman	\(x \ln \left (\frac {\left (\left (x^{2}+27 x \right ) {\mathrm e}^{100}+1\right ) {\mathrm e}^{-100}}{x}\right )^{2}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((((x^2+27*x)*exp(25)^4+1)*ln(((x^2+27*x)*exp(25)^4+1)/x/exp(25)^4)^2+(2*x^2*exp(25)^4-2)*ln(((x^2+27*x)*ex
p(25)^4+1)/x/exp(25)^4))/((x^2+27*x)*exp(25)^4+1),x,method=_RETURNVERBOSE)

[Out]

x*ln(((x^2+27*x)*exp(100)+1)/x*exp(-100))^2

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maxima [B]  time = 1.13, size = 57, normalized size = 3.80 \begin {gather*} x \log \left (x^{2} e^{100} + 27 \, x e^{100} + 1\right )^{2} + x \log \relax (x)^{2} - 2 \, {\left (x \log \relax (x) + 100 \, x\right )} \log \left (x^{2} e^{100} + 27 \, x e^{100} + 1\right ) + 200 \, x \log \relax (x) + 10000 \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x^2+27*x)*exp(25)^4+1)*log(((x^2+27*x)*exp(25)^4+1)/x/exp(25)^4)^2+(2*x^2*exp(25)^4-2)*log(((x^2+
27*x)*exp(25)^4+1)/x/exp(25)^4))/((x^2+27*x)*exp(25)^4+1),x, algorithm="maxima")

[Out]

x*log(x^2*e^100 + 27*x*e^100 + 1)^2 + x*log(x)^2 - 2*(x*log(x) + 100*x)*log(x^2*e^100 + 27*x*e^100 + 1) + 200*
x*log(x) + 10000*x

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mupad [B]  time = 6.83, size = 14, normalized size = 0.93 \begin {gather*} x\,{\ln \left (x+\frac {{\mathrm {e}}^{-100}}{x}+27\right )}^2 \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((log((exp(-100)*(exp(100)*(27*x + x^2) + 1))/x)^2*(exp(100)*(27*x + x^2) + 1) + log((exp(-100)*(exp(100)*(
27*x + x^2) + 1))/x)*(2*x^2*exp(100) - 2))/(exp(100)*(27*x + x^2) + 1),x)

[Out]

x*log(x + exp(-100)/x + 27)^2

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sympy [A]  time = 0.24, size = 20, normalized size = 1.33 \begin {gather*} x \log {\left (\frac {\left (x^{2} + 27 x\right ) e^{100} + 1}{x e^{100}} \right )}^{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((((x**2+27*x)*exp(25)**4+1)*ln(((x**2+27*x)*exp(25)**4+1)/x/exp(25)**4)**2+(2*x**2*exp(25)**4-2)*ln(
((x**2+27*x)*exp(25)**4+1)/x/exp(25)**4))/((x**2+27*x)*exp(25)**4+1),x)

[Out]

x*log(((x**2 + 27*x)*exp(100) + 1)*exp(-100)/x)**2

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