3.87.19 \(\int \frac {-13 x-x^2+x^3+(390+19 x-63 x^2+4 x^3) \log (\frac {15}{-15+x})}{-2535 x^3-221 x^4+401 x^5+5 x^6-17 x^7+x^8} \, dx\)

Optimal. Leaf size=26 \[ \frac {\log \left (\frac {5}{-5+\frac {x}{3}}\right )}{x^2 \left (13+x-x^2\right )} \]

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Rubi [B]  time = 1.76, antiderivative size = 613, normalized size of antiderivative = 23.58, number of steps used = 76, number of rules used = 18, integrand size = 68, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.265, Rules used = {6741, 6742, 740, 800, 632, 31, 893, 638, 618, 206, 2418, 2395, 44, 36, 29, 2394, 2393, 2391} \begin {gather*} -\frac {209-365 x}{135733 \left (-x^2+x+13\right )}+\frac {12 x+365}{135733 \left (-x^2+x+13\right )}-\frac {29 x+12}{10441 \left (-x^2+x+13\right )}+\frac {\log \left (-\frac {15}{15-x}\right )}{13 x^2}+\frac {\left (77963+34959 \sqrt {53}\right ) \log \left (-2 x-\sqrt {53}+1\right )}{347612213}-\frac {\left (2809+9889 \sqrt {53}\right ) \log \left (-2 x-\sqrt {53}+1\right )}{218028962}-\frac {\left (68264+1457 \sqrt {53}\right ) \log \left (-2 x-\sqrt {53}+1\right )}{347612213}+\frac {27 \left (1-\sqrt {53}\right ) \log \left (-2 x-\sqrt {53}+1\right )}{8957 \left (29+\sqrt {53}\right )}-\frac {80 \log \left (-2 x-\sqrt {53}+1\right )}{8957 \left (29+\sqrt {53}\right )}+\frac {27 \left (1+\sqrt {53}\right ) \log \left (-2 x+\sqrt {53}+1\right )}{8957 \left (29-\sqrt {53}\right )}-\frac {80 \log \left (-2 x+\sqrt {53}+1\right )}{8957 \left (29-\sqrt {53}\right )}-\frac {\left (68264-1457 \sqrt {53}\right ) \log \left (-2 x+\sqrt {53}+1\right )}{347612213}-\frac {\left (2809-9889 \sqrt {53}\right ) \log \left (-2 x+\sqrt {53}+1\right )}{218028962}+\frac {\left (77963-34959 \sqrt {53}\right ) \log \left (-2 x+\sqrt {53}+1\right )}{347612213}+\frac {27 \left (1-\sqrt {53}\right ) \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x-\sqrt {53}+1\right )}-\frac {80 \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x-\sqrt {53}+1\right )}+\frac {27 \left (1+\sqrt {53}\right ) \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x+\sqrt {53}+1\right )}-\frac {80 \log \left (-\frac {15}{15-x}\right )}{8957 \left (-2 x+\sqrt {53}+1\right )}-\frac {\log \left (-\frac {15}{15-x}\right )}{169 x}-\frac {27 \left (1-\sqrt {53}\right ) \log (15-x)}{8957 \left (29+\sqrt {53}\right )}+\frac {80 \log (15-x)}{8957 \left (29+\sqrt {53}\right )}-\frac {27 \left (1+\sqrt {53}\right ) \log (15-x)}{8957 \left (29-\sqrt {53}\right )}+\frac {80 \log (15-x)}{8957 \left (29-\sqrt {53}\right )}-\frac {\log (15-x)}{33293}-\frac {58 \tanh ^{-1}\left (\frac {1-2 x}{\sqrt {53}}\right )}{10441 \sqrt {53}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-13*x - x^2 + x^3 + (390 + 19*x - 63*x^2 + 4*x^3)*Log[15/(-15 + x)])/(-2535*x^3 - 221*x^4 + 401*x^5 + 5*x
^6 - 17*x^7 + x^8),x]

[Out]

-1/135733*(209 - 365*x)/(13 + x - x^2) + (365 + 12*x)/(135733*(13 + x - x^2)) - (12 + 29*x)/(10441*(13 + x - x
^2)) - (58*ArcTanh[(1 - 2*x)/Sqrt[53]])/(10441*Sqrt[53]) - (80*Log[1 - Sqrt[53] - 2*x])/(8957*(29 + Sqrt[53]))
+ (27*(1 - Sqrt[53])*Log[1 - Sqrt[53] - 2*x])/(8957*(29 + Sqrt[53])) - ((68264 + 1457*Sqrt[53])*Log[1 - Sqrt[
53] - 2*x])/347612213 - ((2809 + 9889*Sqrt[53])*Log[1 - Sqrt[53] - 2*x])/218028962 + ((77963 + 34959*Sqrt[53])
*Log[1 - Sqrt[53] - 2*x])/347612213 + ((77963 - 34959*Sqrt[53])*Log[1 + Sqrt[53] - 2*x])/347612213 - ((2809 -
9889*Sqrt[53])*Log[1 + Sqrt[53] - 2*x])/218028962 - ((68264 - 1457*Sqrt[53])*Log[1 + Sqrt[53] - 2*x])/34761221
3 - (80*Log[1 + Sqrt[53] - 2*x])/(8957*(29 - Sqrt[53])) + (27*(1 + Sqrt[53])*Log[1 + Sqrt[53] - 2*x])/(8957*(2
9 - Sqrt[53])) - (80*Log[-15/(15 - x)])/(8957*(1 - Sqrt[53] - 2*x)) + (27*(1 - Sqrt[53])*Log[-15/(15 - x)])/(8
957*(1 - Sqrt[53] - 2*x)) - (80*Log[-15/(15 - x)])/(8957*(1 + Sqrt[53] - 2*x)) + (27*(1 + Sqrt[53])*Log[-15/(1
5 - x)])/(8957*(1 + Sqrt[53] - 2*x)) + Log[-15/(15 - x)]/(13*x^2) - Log[-15/(15 - x)]/(169*x) - Log[15 - x]/33
293 + (80*Log[15 - x])/(8957*(29 - Sqrt[53])) - (27*(1 + Sqrt[53])*Log[15 - x])/(8957*(29 - Sqrt[53])) + (80*L
og[15 - x])/(8957*(29 + Sqrt[53])) - (27*(1 - Sqrt[53])*Log[15 - x])/(8957*(29 + Sqrt[53]))

Rule 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], x]

Rule 31

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

Rule 36

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

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

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 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 638

Int[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b*d - 2*a*e + (2*c*d -
b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] - Dist[((2*p + 3)*(2*c*d - b*e))/((p + 1)*(b^2
- 4*a*c)), Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^
2 - 4*a*c, 0] && LtQ[p, -1] && NeQ[p, -3/2]

Rule 740

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m + 1)*(
b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*Simp[b*c*d*e*(2*p - m
+ 2) + b^2*e^2*(m + p + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3) - c*e*(2*c*d - b*e)*(m + 2*p + 4)*x
, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b
*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && IntQuadraticQ[a, b, c, d, e, m, p, x]

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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 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 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 2395

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

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 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} &=\int \frac {13 x+x^2-x^3-\left (390+19 x-63 x^2+4 x^3\right ) \log \left (\frac {15}{-15+x}\right )}{(15-x) x^3 \left (13+x-x^2\right )^2} \, dx\\ &=\int \left (\frac {1}{(-15+x) \left (-13-x+x^2\right )^2}-\frac {13}{(-15+x) x^2 \left (-13-x+x^2\right )^2}-\frac {1}{(-15+x) x \left (-13-x+x^2\right )^2}+\frac {\left (-26-3 x+4 x^2\right ) \log \left (\frac {15}{-15+x}\right )}{x^3 \left (-13-x+x^2\right )^2}\right ) \, dx\\ &=-\left (13 \int \frac {1}{(-15+x) x^2 \left (-13-x+x^2\right )^2} \, dx\right )+\int \frac {1}{(-15+x) \left (-13-x+x^2\right )^2} \, dx-\int \frac {1}{(-15+x) x \left (-13-x+x^2\right )^2} \, dx+\int \frac {\left (-26-3 x+4 x^2\right ) \log \left (\frac {15}{-15+x}\right )}{x^3 \left (-13-x+x^2\right )^2} \, dx\\ &=-\frac {12+29 x}{10441 \left (13+x-x^2\right )}-\frac {\int \frac {382-29 x}{(-15+x) \left (-13-x+x^2\right )} \, dx}{10441}-13 \int \left (\frac {1}{8732025 (-15+x)}-\frac {1}{2535 x^2}+\frac {17}{494325 x}+\frac {-183+x}{33293 \left (-13-x+x^2\right )^2}-\frac {2 (-18215+1471 x)}{85263373 \left (-13-x+x^2\right )}\right ) \, dx-\int \left (\frac {1}{582135 (-15+x)}-\frac {1}{2535 x}+\frac {1-14 x}{2561 \left (-13-x+x^2\right )^2}+\frac {-2745+2576 x}{6558721 \left (-13-x+x^2\right )}\right ) \, dx+\int \left (-\frac {2 \log \left (\frac {15}{-15+x}\right )}{13 x^3}+\frac {\log \left (\frac {15}{-15+x}\right )}{169 x^2}+\frac {(-40+27 x) \log \left (\frac {15}{-15+x}\right )}{169 \left (-13-x+x^2\right )^2}-\frac {\log \left (\frac {15}{-15+x}\right )}{169 \left (-13-x+x^2\right )}\right ) \, dx\\ &=-\frac {1}{195 x}-\frac {12+29 x}{10441 \left (13+x-x^2\right )}-\frac {28 \log (15-x)}{8732025}-\frac {2 \log (x)}{38025}-\frac {\int \frac {-2745+2576 x}{-13-x+x^2} \, dx}{6558721}+\frac {2 \int \frac {-18215+1471 x}{-13-x+x^2} \, dx}{6558721}-\frac {\int \left (-\frac {53}{197 (-15+x)}+\frac {-4971+53 x}{197 \left (-13-x+x^2\right )}\right ) \, dx}{10441}-\frac {\int \frac {1-14 x}{\left (-13-x+x^2\right )^2} \, dx}{2561}-\frac {\int \frac {-183+x}{\left (-13-x+x^2\right )^2} \, dx}{2561}+\frac {1}{169} \int \frac {\log \left (\frac {15}{-15+x}\right )}{x^2} \, dx+\frac {1}{169} \int \frac {(-40+27 x) \log \left (\frac {15}{-15+x}\right )}{\left (-13-x+x^2\right )^2} \, dx-\frac {1}{169} \int \frac {\log \left (\frac {15}{-15+x}\right )}{-13-x+x^2} \, dx-\frac {2}{13} \int \frac {\log \left (\frac {15}{-15+x}\right )}{x^3} \, dx\\ &=-\frac {1}{195 x}-\frac {209-365 x}{135733 \left (13+x-x^2\right )}+\frac {365+12 x}{135733 \left (13+x-x^2\right )}-\frac {12+29 x}{10441 \left (13+x-x^2\right )}+\frac {\log \left (-\frac {15}{15-x}\right )}{13 x^2}-\frac {\log \left (-\frac {15}{15-x}\right )}{169 x}+\frac {\log (15-x)}{44325}-\frac {2 \log (x)}{38025}-\frac {\int \frac {-4971+53 x}{-13-x+x^2} \, dx}{2056877}-\frac {12 \int \frac {1}{-13-x+x^2} \, dx}{135733}-\frac {365 \int \frac {1}{-13-x+x^2} \, dx}{135733}-\frac {1}{169} \int \frac {1}{(-15+x) x} \, dx-\frac {1}{169} \int \left (-\frac {2 \log \left (\frac {15}{-15+x}\right )}{\sqrt {53} \left (1+\sqrt {53}-2 x\right )}-\frac {2 \log \left (\frac {15}{-15+x}\right )}{\sqrt {53} \left (-1+\sqrt {53}+2 x\right )}\right ) \, dx+\frac {1}{169} \int \left (-\frac {40 \log \left (\frac {15}{-15+x}\right )}{\left (-13-x+x^2\right )^2}+\frac {27 x \log \left (\frac {15}{-15+x}\right )}{\left (-13-x+x^2\right )^2}\right ) \, dx+\frac {1}{13} \int \frac {1}{(-15+x) x^2} \, dx+\frac {\left (77963-34959 \sqrt {53}\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {53}}{2}+x} \, dx}{347612213}-\frac {\left (68264-1457 \sqrt {53}\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {53}}{2}+x} \, dx}{347612213}-\frac {\left (68264+1457 \sqrt {53}\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {53}}{2}+x} \, dx}{347612213}+\frac {\left (77963+34959 \sqrt {53}\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {53}}{2}+x} \, dx}{347612213}\\ &=-\frac {1}{195 x}-\frac {209-365 x}{135733 \left (13+x-x^2\right )}+\frac {365+12 x}{135733 \left (13+x-x^2\right )}-\frac {12+29 x}{10441 \left (13+x-x^2\right )}-\frac {\left (68264+1457 \sqrt {53}\right ) \log \left (1-\sqrt {53}-2 x\right )}{347612213}+\frac {\left (77963+34959 \sqrt {53}\right ) \log \left (1-\sqrt {53}-2 x\right )}{347612213}+\frac {\left (77963-34959 \sqrt {53}\right ) \log \left (1+\sqrt {53}-2 x\right )}{347612213}-\frac {\left (68264-1457 \sqrt {53}\right ) \log \left (1+\sqrt {53}-2 x\right )}{347612213}+\frac {\log \left (-\frac {15}{15-x}\right )}{13 x^2}-\frac {\log \left (-\frac {15}{15-x}\right )}{169 x}+\frac {\log (15-x)}{44325}-\frac {2 \log (x)}{38025}+\frac {24 \operatorname {Subst}\left (\int \frac {1}{53-x^2} \, dx,x,-1+2 x\right )}{135733}-\frac {\int \frac {1}{-15+x} \, dx}{2535}+\frac {\int \frac {1}{x} \, dx}{2535}+\frac {730 \operatorname {Subst}\left (\int \frac {1}{53-x^2} \, dx,x,-1+2 x\right )}{135733}+\frac {1}{13} \int \left (\frac {1}{225 (-15+x)}-\frac {1}{15 x^2}-\frac {1}{225 x}\right ) \, dx+\frac {27}{169} \int \frac {x \log \left (\frac {15}{-15+x}\right )}{\left (-13-x+x^2\right )^2} \, dx-\frac {40}{169} \int \frac {\log \left (\frac {15}{-15+x}\right )}{\left (-13-x+x^2\right )^2} \, dx+\frac {2 \int \frac {\log \left (\frac {15}{-15+x}\right )}{1+\sqrt {53}-2 x} \, dx}{169 \sqrt {53}}+\frac {2 \int \frac {\log \left (\frac {15}{-15+x}\right )}{-1+\sqrt {53}+2 x} \, dx}{169 \sqrt {53}}-\frac {\left (2809-9889 \sqrt {53}\right ) \int \frac {1}{-\frac {1}{2}-\frac {\sqrt {53}}{2}+x} \, dx}{218028962}-\frac {\left (2809+9889 \sqrt {53}\right ) \int \frac {1}{-\frac {1}{2}+\frac {\sqrt {53}}{2}+x} \, dx}{218028962}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.39, size = 23, normalized size = 0.88 \begin {gather*} -\frac {\log \left (\frac {15}{-15+x}\right )}{x^2 \left (-13-x+x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-13*x - x^2 + x^3 + (390 + 19*x - 63*x^2 + 4*x^3)*Log[15/(-15 + x)])/(-2535*x^3 - 221*x^4 + 401*x^5
+ 5*x^6 - 17*x^7 + x^8),x]

[Out]

-(Log[15/(-15 + x)]/(x^2*(-13 - x + x^2)))

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fricas [A]  time = 0.54, size = 26, normalized size = 1.00 \begin {gather*} -\frac {\log \left (\frac {15}{x - 15}\right )}{x^{4} - x^{3} - 13 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^7+5*x^6+401*x^5-221*x^4-2535*x^3),x,
algorithm="fricas")

[Out]

-log(15/(x - 15))/(x^4 - x^3 - 13*x^2)

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giac [B]  time = 0.21, size = 80, normalized size = 3.08 \begin {gather*} -\frac {1}{7490925} \, {\left (\frac {197 \, {\left (\frac {165}{x - 15} - 2\right )}}{\frac {30}{x - 15} + \frac {225}{{\left (x - 15\right )}^{2}} + 1} - \frac {225 \, {\left (\frac {168}{x - 15} - 1\right )}}{\frac {29}{x - 15} + \frac {197}{{\left (x - 15\right )}^{2}} + 1}\right )} \log \left (\frac {15}{x - 15}\right ) - \frac {1}{44325} \, \log \left (\frac {15}{x - 15}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^7+5*x^6+401*x^5-221*x^4-2535*x^3),x,
algorithm="giac")

[Out]

-1/7490925*(197*(165/(x - 15) - 2)/(30/(x - 15) + 225/(x - 15)^2 + 1) - 225*(168/(x - 15) - 1)/(29/(x - 15) +
197/(x - 15)^2 + 1))*log(15/(x - 15)) - 1/44325*log(15/(x - 15))

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maple [A]  time = 0.16, size = 24, normalized size = 0.92

method	result	size
norman	\(-\frac {\ln \left (\frac {15}{x -15}\right )}{x^{2} \left (x^{2}-x -13\right )}\)	\(24\)
risch	\(-\frac {\ln \left (\frac {15}{x -15}\right )}{x^{2} \left (x^{2}-x -13\right )}\)	\(24\)
derivativedivides	\(\frac {11 \ln \left (\frac {15}{x -15}\right )}{2535 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )}-\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{1764529}+\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{1764529}-\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {15}{x -15}+2\right )}{195 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )^{2}}+\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {7446600 \sqrt {53}\, \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{\left (x -15\right )^{2}}-\frac {7446600 \sqrt {53}\, \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{\left (x -15\right )^{2}}+\frac {1096200 \sqrt {53}\, \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{x -15}-\frac {1096200 \sqrt {53}\, \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{x -15}+37800 \sqrt {53}\, \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )-37800 \sqrt {53}\, \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )+\frac {2349225}{\left (x -15\right )^{2}}+\frac {2349225}{x -15}\right )}{\frac {78212747925}{\left (x -15\right )^{2}}+\frac {11513551725}{x -15}[phone]}\)	\(374\)
default	\(\frac {11 \ln \left (\frac {15}{x -15}\right )}{2535 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )}-\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{1764529}+\frac {168 \sqrt {53}\, \ln \left (\frac {15}{x -15}\right ) \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{1764529}-\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {15}{x -15}+2\right )}{195 \left (x -15\right ) \left (1+\frac {15}{x -15}\right )^{2}}+\frac {\ln \left (\frac {15}{x -15}\right ) \left (\frac {7446600 \sqrt {53}\, \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{\left (x -15\right )^{2}}-\frac {7446600 \sqrt {53}\, \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{\left (x -15\right )^{2}}+\frac {1096200 \sqrt {53}\, \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )}{x -15}-\frac {1096200 \sqrt {53}\, \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )}{x -15}+37800 \sqrt {53}\, \ln \left (\frac {-435+15 \sqrt {53}-\frac {5910}{x -15}}{-435+15 \sqrt {53}}\right )-37800 \sqrt {53}\, \ln \left (\frac {435+15 \sqrt {53}+\frac {5910}{x -15}}{435+15 \sqrt {53}}\right )+\frac {2349225}{\left (x -15\right )^{2}}+\frac {2349225}{x -15}\right )}{\frac {78212747925}{\left (x -15\right )^{2}}+\frac {11513551725}{x -15}[phone]}\)	\(374\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3-63*x^2+19*x+390)*ln(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^7+5*x^6+401*x^5-221*x^4-2535*x^3),x,method=
_RETURNVERBOSE)

[Out]

-ln(15/(x-15))/x^2/(x^2-x-13)

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maxima [A]  time = 0.51, size = 29, normalized size = 1.12 \begin {gather*} -\frac {\log \relax (5) + \log \relax (3) - \log \left (x - 15\right )}{x^{4} - x^{3} - 13 \, x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3-63*x^2+19*x+390)*log(15/(x-15))+x^3-x^2-13*x)/(x^8-17*x^7+5*x^6+401*x^5-221*x^4-2535*x^3),x,
algorithm="maxima")

[Out]

-(log(5) + log(3) - log(x - 15))/(x^4 - x^3 - 13*x^2)

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mupad [B]  time = 5.45, size = 26, normalized size = 1.00 \begin {gather*} \frac {\ln \left (15\right )+\ln \left (\frac {1}{x-15}\right )}{-x^4+x^3+13\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((13*x - log(15/(x - 15))*(19*x - 63*x^2 + 4*x^3 + 390) + x^2 - x^3)/(2535*x^3 + 221*x^4 - 401*x^5 - 5*x^6
+ 17*x^7 - x^8),x)

[Out]

(log(15) + log(1/(x - 15)))/(13*x^2 + x^3 - x^4)

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sympy [A]  time = 0.17, size = 19, normalized size = 0.73 \begin {gather*} - \frac {\log {\left (\frac {15}{x - 15} \right )}}{x^{4} - x^{3} - 13 x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3-63*x**2+19*x+390)*ln(15/(x-15))+x**3-x**2-13*x)/(x**8-17*x**7+5*x**6+401*x**5-221*x**4-2535
*x**3),x)

[Out]

-log(15/(x - 15))/(x**4 - x**3 - 13*x**2)

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