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

Optimal. Leaf size=22 \[ \frac {\left (x-\frac {4}{x^4 \log ^8(5)}\right )^2 \log (x)}{\log \left (x^2\right )} \]

________________________________________________________________________________________

Rubi [B]  time = 2.19, antiderivative size = 47, normalized size of antiderivative = 2.14, number of steps used = 57, number of rules used = 13, integrand size = 85, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.153, Rules used = {12, 6741, 6742, 2353, 2306, 2310, 2178, 2307, 2298, 2366, 6482, 15, 6496} \begin {gather*} \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 {gather*}

Antiderivative was successfully verified.

[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 2178

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

Rule 2298

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

Rule 2306

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 2307

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 2310

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)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2353

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 2366

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 6482

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 6496

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

Rule 6741

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

Rule 6742

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

Rubi steps

\begin {gather*} \begin {aligned} \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 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log (x)}{\log ^{16}(5)}-\frac {12 \left (x^2\right )^{3/2} \text {Ei}\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) \text {li}\left (x^2\right )+2 \int \left (-\frac {x}{2 \log \left (x^2\right )}+\frac {\text {li}\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 \text {Ei}\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 \text {Ei}\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 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log (x)}{\log ^{16}(5)}-\frac {12 \left (x^2\right )^{3/2} \text {Ei}\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) \text {li}\left (x^2\right )+2 \int \frac {x \log (x)}{\log \left (x^2\right )} \, dx-\frac {64 \int \frac {\text {Ei}\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 {\text {Ei}\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 {\text {li}\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} \operatorname {Subst}\left (\int \frac {\text {li}(x)}{x} \, dx,x,x^2\right )-2 \int \frac {\text {li}\left (x^2\right )}{2 x} \, dx-\frac {32 \operatorname {Subst}\left (\int \text {Ei}(-4 x) \, dx,x,\log \left (x^2\right )\right )}{\log ^{16}(5)}+\frac {128 \int \frac {\text {Ei}\left (-4 \log \left (x^2\right )\right )}{2 x} \, dx}{\log ^{16}(5)}+\frac {8 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{2 \sqrt {x^2}} \, dx}{\log ^8(5)}-\frac {32 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{2 \sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {(12 x) \int \frac {\text {Ei}\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 ) \operatorname {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 ) \operatorname {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 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log \left (x^2\right )}{\log ^{16}(5)}+\frac {1}{2} \log \left (x^2\right ) \text {li}\left (x^2\right )+\frac {64 \int \frac {\text {Ei}\left (-4 \log \left (x^2\right )\right )}{x} \, dx}{\log ^{16}(5)}+\frac {4 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}-\frac {16 \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{\sqrt {x^2}} \, dx}{\log ^8(5)}+\frac {(6 x) \operatorname {Subst}\left (\int \text {Ei}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}-\int \frac {\text {li}\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 \text {Ei}\left (-4 \log \left (x^2\right )\right ) \log \left (x^2\right )}{\log ^{16}(5)}+\frac {6 x \text {Ei}\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 ) \text {li}\left (x^2\right )-\frac {1}{2} \operatorname {Subst}\left (\int \frac {\text {li}(x)}{x} \, dx,x,x^2\right )+\frac {32 \operatorname {Subst}\left (\int \text {Ei}(-4 x) \, dx,x,\log \left (x^2\right )\right )}{\log ^{16}(5)}+\frac {(4 x) \int \frac {\text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right )}{x} \, dx}{\sqrt {x^2} \log ^8(5)}-\frac {(16 x) \int \frac {\text {Ei}\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 \text {Ei}\left (-\frac {3}{2} \log \left (x^2\right )\right ) \log \left (x^2\right )}{\sqrt {x^2} \log ^8(5)}+\frac {(2 x) \operatorname {Subst}\left (\int \text {Ei}\left (-\frac {3 x}{2}\right ) \, dx,x,\log \left (x^2\right )\right )}{\sqrt {x^2} \log ^8(5)}-\frac {(8 x) \operatorname {Subst}\left (\int \text {Ei}\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 {aligned} \end {gather*}

________________________________________________________________________________________

Mathematica [A]  time = 0.23, size = 28, normalized size = 1.27 \begin {gather*} \frac {\left (-4+x^5 \log ^8(5)\right )^2 \log (x)}{x^8 \log ^{16}(5) \log \left (x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[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])

________________________________________________________________________________________

fricas [A]  time = 1.96, size = 28, normalized size = 1.27 \begin {gather*} \frac {x^{10} \log \relax (5)^{16} - 8 \, x^{5} \log \relax (5)^{8} + 16}{2 \, x^{8} \log \relax (5)^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [A]  time = 0.19, size = 30, normalized size = 1.36 \begin {gather*} \frac {x^{2} \log \relax (5)^{16} - \frac {8 \, {\left (x^{5} \log \relax (5)^{8} - 2\right )}}{x^{8}}}{2 \, \log \relax (5)^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________

maple [C]  time = 0.14, size = 242, normalized size = 11.00

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

Verification of antiderivative is not currently implemented for this CAS.

[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]

1/2/ln(5)^16*(x^10*ln(5)^16-8*x^5*ln(5)^8+16)/x^8-1/2/ln(5)^16*Pi*csgn(I*x^2)*(ln(5)^16*x^10*csgn(I*x)^2-2*ln(
5)^16*x^10*csgn(I*x)*csgn(I*x^2)+ln(5)^16*x^10*csgn(I*x^2)^2-8*ln(5)^8*x^5*csgn(I*x)^2+16*ln(5)^8*x^5*csgn(I*x
)*csgn(I*x^2)-8*ln(5)^8*x^5*csgn(I*x^2)^2+16*csgn(I*x)^2-32*csgn(I*x^2)*csgn(I*x)+16*csgn(I*x^2)^2)/x^8/(4*I*l
n(x)+Pi*csgn(I*x^2)^3+Pi*csgn(I*x)^2*csgn(I*x^2)-2*Pi*csgn(I*x)*csgn(I*x^2)^2)

________________________________________________________________________________________

maxima [A]  time = 0.48, size = 28, normalized size = 1.27 \begin {gather*} \frac {x^{10} \log \relax (5)^{16} - 8 \, x^{5} \log \relax (5)^{8} + 16}{2 \, x^{8} \log \relax (5)^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 5.00, size = 28, normalized size = 1.27 \begin {gather*} \frac {\ln \relax (x)\,{\left (x^5\,{\ln \relax (5)}^8-4\right )}^2}{x^8\,\ln \left (x^2\right )\,{\ln \relax (5)}^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [A]  time = 0.37, size = 29, normalized size = 1.32 \begin {gather*} \frac {\frac {x^{2} \log {\relax (5 )}^{16}}{2} + \frac {- 4 x^{5} \log {\relax (5 )}^{8} + 8}{x^{8}}}{\log {\relax (5 )}^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

________________________________________________________________________________________