\(\int \frac {(-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)) \log (x)+(16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+(-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)) \log (x)) \log (x^2)}{x^9 \log ^{16}(5) \log ^2(x^2)} \, dx\) [7634]

Optimal result

Integrand size = 85, antiderivative size = 22 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\left (x-\frac {4}{x^4 \log ^8(5)}\right )^2 \log (x)}{\log \left (x^2\right )} \]

[Out]

ln(x)/ln(x^2)*(x-4/x^4/ln(5)^8)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(47\) vs. \(2(22)=44\).

Time = 1.33 (sec) , antiderivative size = 47, normalized size of antiderivative = 2.14, number of steps used = 57, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules used = {12, 6873, 6874, 2395, 2343, 2347, 2209, 2344, 2335, 2413, 6617, 15, 6631} \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )} \]

[In]

Int[((-32 + 16*x^5*Log[5]^8 - 2*x^10*Log[5]^16)*Log[x] + (16 - 8*x^5*Log[5]^8 + x^10*Log[5]^16 + (-128 + 24*x^
5*Log[5]^8 + 2*x^10*Log[5]^16)*Log[x])*Log[x^2])/(x^9*Log[5]^16*Log[x^2]^2),x]

[Out]

(x^2*Log[x])/Log[x^2] + (16*Log[x])/(x^8*Log[5]^16*Log[x^2]) - (8*Log[x])/(x^3*Log[5]^8*Log[x^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 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[a^IntPart[m]*((a*x^n)^FracPart[m]/x^(n*FracPart[m])), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 2209

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - c*(f/d)))/d)*ExpInteg
ralEi[f*g*(c + d*x)*(Log[F]/d)], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2335

Int[Log[(c_.)*(x_)]^(-1), x_Symbol] :> Simp[LogIntegral[c*x]/c, x] /; FreeQ[c, x]

Rule 2343

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

Rule 2344

Int[(x_)^(m_.)/Log[(c_.)*(x_)^(n_)], x_Symbol] :> Dist[1/n, Subst[Int[1/Log[c*x], x], x, x^n], x] /; FreeQ[{c,
m, n}, x] && EqQ[m, n - 1]

Rule 2347

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rule 2413

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_.) + Log[(f_.)*(x_)^(r_.)]*(e_.))*((g_.)*(x_))^(m_.), x_Sy
mbol] :> With[{u = IntHide[(g*x)^m*(a + b*Log[c*x^n])^p, x]}, Dist[d + e*Log[f*x^r], u, x] - Dist[e*r, Int[Sim
plifyIntegrand[u/x, x], x], x]] /; FreeQ[{a, b, c, d, e, f, g, m, n, p, r}, x] &&  !(EqQ[p, 1] && EqQ[a, 0] &&
NeQ[d, 0])

Rule 6617

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

Rule 6631

Int[LogIntegral[(b_.)*(x_)]/(x_), x_Symbol] :> Simp[(-b)*x, x] + Simp[Log[b*x]*LogIntegral[b*x], x] /; FreeQ[b
, x]

Rule 6873

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

Rule 6874

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

Rubi steps \begin{align*} \text {integral}& = \frac {\int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)} \\ & = \frac {\int \frac {\left (4-x^5 \log ^8(5)\right ) \left (-8 \log (x)+2 x^5 \log ^8(5) \log (x)+4 \log \left (x^2\right )-x^5 \log ^8(5) \log \left (x^2\right )-32 \log (x) \log \left (x^2\right )-2 x^5 \log ^8(5) \log (x) \log \left (x^2\right )\right )}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)} \\ & = \frac {\int \left (-\frac {2 \left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^9 \log ^2\left (x^2\right )}+\frac {\left (-4+x^5 \log ^8(5)\right ) \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^9 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)} \\ & = \frac {\int \frac {\left (-4+x^5 \log ^8(5)\right ) \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^9 \log \left (x^2\right )} \, dx}{\log ^{16}(5)}-\frac {2 \int \frac {\left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)} \\ & = \frac {\int \left (-\frac {4 \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^9 \log \left (x^2\right )}+\frac {\log ^8(5) \left (-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)\right )}{x^4 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}-\frac {2 \int \left (\frac {16 \log (x)}{x^9 \log ^2\left (x^2\right )}-\frac {8 \log ^8(5) \log (x)}{x^4 \log ^2\left (x^2\right )}+\frac {x \log ^{16}(5) \log (x)}{\log ^2\left (x^2\right )}\right ) \, dx}{\log ^{16}(5)} \\ & = -\left (2 \int \frac {x \log (x)}{\log ^2\left (x^2\right )} \, dx\right )-\frac {4 \int \frac {-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)}{x^9 \log \left (x^2\right )} \, dx}{\log ^{16}(5)}-\frac {32 \int \frac {\log (x)}{x^9 \log ^2\left (x^2\right )} \, dx}{\log ^{16}(5)}+\frac {\int \frac {-4+x^5 \log ^8(5)+32 \log (x)+2 x^5 \log ^8(5) \log (x)}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\frac {16 \int \frac {\log (x)}{x^4 \log ^2\left (x^2\right )} \, dx}{\log ^8(5)} \\ & = \frac {64 \operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right ) \log (x)}{\log ^{16}(5)}-\frac {12 \left (x^2\right )^{3/2} \operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log (x)}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\log (x) \operatorname {LogIntegral}\left (x^2\right )+2 \int \left (-\frac {x}{2 \log \left (x^2\right )}+\frac {\operatorname {LogIntegral}\left (x^2\right )}{2 x}\right ) \, dx-\frac {4 \int \left (-\frac {4}{x^9 \log \left (x^2\right )}+\frac {\log ^8(5)}{x^4 \log \left (x^2\right )}+\frac {32 \log (x)}{x^9 \log \left (x^2\right )}+\frac {2 \log ^8(5) \log (x)}{x^4 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}+\frac {32 \int \left (-\frac {2 \operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right )}{x}-\frac {1}{2 x^9 \log \left (x^2\right )}\right ) \, dx}{\log ^{16}(5)}+\frac {\int \left (-\frac {4}{x^4 \log \left (x^2\right )}+\frac {x \log ^8(5)}{\log \left (x^2\right )}+\frac {32 \log (x)}{x^4 \log \left (x^2\right )}+\frac {2 x \log ^8(5) \log (x)}{\log \left (x^2\right )}\right ) \, dx}{\log ^8(5)}-\frac {16 \int \left (-\frac {3 \operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{4 \sqrt {x^2}}-\frac {1}{2 x^4 \log \left (x^2\right )}\right ) \, dx}{\log ^8(5)} \\ & = \frac {64 \operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right ) \log (x)}{\log ^{16}(5)}-\frac {12 \left (x^2\right )^{3/2} \operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log (x)}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\log (x) \operatorname {LogIntegral}\left (x^2\right )+2 \int \frac {x \log (x)}{\log \left (x^2\right )} \, dx-\frac {64 \int \frac {\operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right )}{x} \, dx}{\log ^{16}(5)}-\frac {128 \int \frac {\log (x)}{x^9 \log \left (x^2\right )} \, dx}{\log ^{16}(5)}-2 \frac {4 \int \frac {1}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\frac {8 \int \frac {1}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}-\frac {8 \int \frac {\log (x)}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\frac {12 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {32 \int \frac {\log (x)}{x^4 \log \left (x^2\right )} \, dx}{\log ^8(5)}+\int \frac {\operatorname {LogIntegral}\left (x^2\right )}{x} \, dx \\ & = \frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}+\frac {1}{2} \text {Subst}\left (\int \frac {\operatorname {LogIntegral}(x)}{x} \, dx,x,x^2\right )-2 \int \frac {\operatorname {LogIntegral}\left (x^2\right )}{2 x} \, dx-\frac {32 \text {Subst}\left (\int \operatorname {ExpIntegralEi}(-4 x) \, dx,x,\log \left (x^2\right )\right )}{\log ^{16}(5)}+\frac {128 \int \frac {\operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right )}{2 x} \, dx}{\log ^{16}(5)}+\frac {8 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{2 \sqrt {x^2}} \, dx}{\log ^8(5)}-\frac {32 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{2 \sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {(12 x) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)}-2 \frac {\left (2 \left (x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{-3 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{x^3 \log ^8(5)}+\frac {\left (4 \left (x^2\right )^{3/2}\right ) \text {Subst}\left (\int \frac {e^{-3 x/2}}{x} \, dx,x,\log \left (x^2\right )\right )}{x^3 \log ^8(5)} \\ & = -\frac {x^2}{2}-\frac {8}{x^8 \log ^{16}(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\frac {32 \operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right ) \log \left (x^2\right )}{\log ^{16}(5)}+\frac {1}{2} \log \left (x^2\right ) \operatorname {LogIntegral}\left (x^2\right )+\frac {64 \int \frac {\operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right )}{x} \, dx}{\log ^{16}(5)}+\frac {4 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}-\frac {16 \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {(6 x) \text {Subst}\left (\int \operatorname {ExpIntegralEi}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}-\int \frac {\operatorname {LogIntegral}\left (x^2\right )}{x} \, dx \\ & = -\frac {x^2}{2}-\frac {8}{x^8 \log ^{16}(5)}+\frac {4}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}-\frac {32 \operatorname {ExpIntegralEi}\left (-4 \log \left (x^2\right )\right ) \log \left (x^2\right )}{\log ^{16}(5)}+\frac {6 x \operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log \left (x^2\right )}{\sqrt {x^2} \log ^8(5)}+\frac {1}{2} \log \left (x^2\right ) \operatorname {LogIntegral}\left (x^2\right )-\frac {1}{2} \text {Subst}\left (\int \frac {\operatorname {LogIntegral}(x)}{x} \, dx,x,x^2\right )+\frac {32 \text {Subst}\left (\int \operatorname {ExpIntegralEi}(-4 x) \, dx,x,\log \left (x^2\right )\right )}{\log ^{16}(5)}+\frac {(4 x) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)}-\frac {(16 x) \int \frac {\operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)} \\ & = \frac {4}{x^3 \log ^8(5)}+\frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )}+\frac {6 x \operatorname {ExpIntegralEi}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log \left (x^2\right )}{\sqrt {x^2} \log ^8(5)}+\frac {(2 x) \text {Subst}\left (\int \operatorname {ExpIntegralEi}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}-\frac {(8 x) \text {Subst}\left (\int \operatorname {ExpIntegralEi}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)} \\ & = \frac {x^2 \log (x)}{\log \left (x^2\right )}+\frac {16 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )}-\frac {8 \log (x)}{x^3 \log ^8(5) \log \left (x^2\right )} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )} \]

[In]

Integrate[((-32 + 16*x^5*Log[5]^8 - 2*x^10*Log[5]^16)*Log[x] + (16 - 8*x^5*Log[5]^8 + x^10*Log[5]^16 + (-128 +
24*x^5*Log[5]^8 + 2*x^10*Log[5]^16)*Log[x])*Log[x^2])/(x^9*Log[5]^16*Log[x^2]^2),x]

[Out]

((-4 + x^5*Log[5]^8)^2*Log[x])/(x^8*Log[5]^16*Log[x^2])

Maple [A] (verified)

Time = 18.59 (sec) , antiderivative size = 41, normalized size of antiderivative = 1.86

method	result	size
parallelrisch	\(\frac {\ln \left (x \right ) \ln \left (5\right )^{16} x^{10}-8 x^{5} \ln \left (5\right )^{8} \ln \left (x \right )+16 \ln \left (x \right )}{\ln \left (5\right )^{16} x^{8} \ln \left (x^{2}\right )}\)	\(41\)
risch	\(\frac {x^{10} \ln \left (5\right )^{16}-8 x^{5} \ln \left (5\right )^{8}+16}{2 \ln \left (5\right )^{16} x^{8}}-\frac {\pi \,\operatorname {csgn}\left (i x^{2}\right ) \left (\ln \left (5\right )^{16} x^{10} \operatorname {csgn}\left (i x \right )^{2}-2 \ln \left (5\right )^{16} x^{10} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )+\ln \left (5\right )^{16} x^{10} \operatorname {csgn}\left (i x^{2}\right )^{2}-8 \ln \left (5\right )^{8} x^{5} \operatorname {csgn}\left (i x \right )^{2}+16 \ln \left (5\right )^{8} x^{5} \operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )-8 \ln \left (5\right )^{8} x^{5} \operatorname {csgn}\left (i x^{2}\right )^{2}+16 \operatorname {csgn}\left (i x \right )^{2}-32 \,\operatorname {csgn}\left (i x^{2}\right ) \operatorname {csgn}\left (i x \right )+16 \operatorname {csgn}\left (i x^{2}\right )^{2}\right )}{2 \ln \left (5\right )^{16} x^{8} \left (\pi \operatorname {csgn}\left (i x \right )^{2} \operatorname {csgn}\left (i x^{2}\right )-2 \pi \,\operatorname {csgn}\left (i x \right ) \operatorname {csgn}\left (i x^{2}\right )^{2}+\pi \operatorname {csgn}\left (i x^{2}\right )^{3}+4 i \ln \left (x \right )\right )}\)	\(242\)

[In]

int((((2*x^10*ln(5)^16+24*x^5*ln(5)^8-128)*ln(x)+x^10*ln(5)^16-8*x^5*ln(5)^8+16)*ln(x^2)+(-2*x^10*ln(5)^16+16*
x^5*ln(5)^8-32)*ln(x))/x^9/ln(5)^16/ln(x^2)^2,x,method=_RETURNVERBOSE)

[Out]

1/ln(5)^16/x^8*(ln(x)*ln(5)^16*x^10-8*x^5*ln(5)^8*ln(x)+16*ln(x))/ln(x^2)

Fricas [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^{10} \log \left (5\right )^{16} - 8 \, x^{5} \log \left (5\right )^{8} + 16}{2 \, x^{8} \log \left (5\right )^{16}} \]

[In]

integrate((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8*x^5*log(5)^8+16)*log(x^2)+(-2*x^10*
log(5)^16+16*x^5*log(5)^8-32)*log(x))/x^9/log(5)^16/log(x^2)^2,x, algorithm="fricas")

[Out]

1/2*(x^10*log(5)^16 - 8*x^5*log(5)^8 + 16)/(x^8*log(5)^16)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.32 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\frac {x^{2} \log {\left (5 \right )}^{16}}{2} + \frac {- 4 x^{5} \log {\left (5 \right )}^{8} + 8}{x^{8}}}{\log {\left (5 \right )}^{16}} \]

[In]

integrate((((2*x**10*ln(5)**16+24*x**5*ln(5)**8-128)*ln(x)+x**10*ln(5)**16-8*x**5*ln(5)**8+16)*ln(x**2)+(-2*x*
*10*ln(5)**16+16*x**5*ln(5)**8-32)*ln(x))/x**9/ln(5)**16/ln(x**2)**2,x)

[Out]

(x**2*log(5)**16/2 + (-4*x**5*log(5)**8 + 8)/x**8)/log(5)**16

Maxima [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^{10} \log \left (5\right )^{16} - 8 \, x^{5} \log \left (5\right )^{8} + 16}{2 \, x^{8} \log \left (5\right )^{16}} \]

[In]

integrate((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8*x^5*log(5)^8+16)*log(x^2)+(-2*x^10*
log(5)^16+16*x^5*log(5)^8-32)*log(x))/x^9/log(5)^16/log(x^2)^2,x, algorithm="maxima")

[Out]

1/2*(x^10*log(5)^16 - 8*x^5*log(5)^8 + 16)/(x^8*log(5)^16)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.36 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {x^{2} \log \left (5\right )^{16} - \frac {8 \, {\left (x^{5} \log \left (5\right )^{8} - 2\right )}}{x^{8}}}{2 \, \log \left (5\right )^{16}} \]

[In]

integrate((((2*x^10*log(5)^16+24*x^5*log(5)^8-128)*log(x)+x^10*log(5)^16-8*x^5*log(5)^8+16)*log(x^2)+(-2*x^10*
log(5)^16+16*x^5*log(5)^8-32)*log(x))/x^9/log(5)^16/log(x^2)^2,x, algorithm="giac")

[Out]

1/2*(x^2*log(5)^16 - 8*(x^5*log(5)^8 - 2)/x^8)/log(5)^16

Mupad [B] (verification not implemented)

Time = 13.05 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.27 \[ \int \frac {\left (-32+16 x^5 \log ^8(5)-2 x^{10} \log ^{16}(5)\right ) \log (x)+\left (16-8 x^5 \log ^8(5)+x^{10} \log ^{16}(5)+\left (-128+24 x^5 \log ^8(5)+2 x^{10} \log ^{16}(5)\right ) \log (x)\right ) \log \left (x^2\right )}{x^9 \log ^{16}(5) \log ^2\left (x^2\right )} \, dx=\frac {\ln \left (x\right )\,{\left (x^5\,{\ln \left (5\right )}^8-4\right )}^2}{x^8\,\ln \left (x^2\right )\,{\ln \left (5\right )}^{16}} \]

[In]

int(-(log(x)*(2*x^10*log(5)^16 - 16*x^5*log(5)^8 + 32) - log(x^2)*(x^10*log(5)^16 - 8*x^5*log(5)^8 + log(x)*(2
4*x^5*log(5)^8 + 2*x^10*log(5)^16 - 128) + 16))/(x^9*log(x^2)^2*log(5)^16),x)

[Out]

(log(x)*(x^5*log(5)^8 - 4)^2)/(x^8*log(x^2)*log(5)^16)