3.523 \(\int \frac{(a+b \log (c (d+\frac{e}{x^{2/3}})^n))^2}{x^2} \, dx\)

Optimal. Leaf size=361 \[ \frac{4 i b^2 d^{3/2} n^2 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}+\frac{32 b^2 d n^2}{3 e \sqrt [3]{x}}-\frac{8 b^2 n^2}{9 x} \]

[Out]

(-8*b^2*n^2)/(9*x) + (32*b^2*d*n^2)/(3*e*x^(1/3)) + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*
e^(3/2)) + ((4*I)*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^(3/2) - (8*b^2*d^(3/2)*n^2*ArcTan[(Sq
rt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) + (4*b*n*(a + b*Log[c*(d +
e/x^(2/3))^n]))/(3*x) - (4*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e*x^(1/3)) - (4*b*d^(3/2)*n*ArcTan[(Sqrt[
d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(3/2) - (a + b*Log[c*(d + e/x^(2/3))^n])^2/x + ((4*I)
*b^2*d^(3/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2)

________________________________________________________________________________________

Rubi [A]  time = 0.594036, antiderivative size = 361, normalized size of antiderivative = 1., number of steps used = 19, number of rules used = 14, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.583, Rules used = {2458, 2457, 2476, 2455, 263, 325, 205, 2470, 12, 260, 6688, 4924, 4868, 2447} \[ \frac{4 i b^2 d^{3/2} n^2 \text{PolyLog}\left (2,-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}+\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right ) \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}+\frac{32 b^2 d n^2}{3 e \sqrt [3]{x}}-\frac{8 b^2 n^2}{9 x} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^2,x]

[Out]

(-8*b^2*n^2)/(9*x) + (32*b^2*d*n^2)/(3*e*x^(1/3)) + (32*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]])/(3*
e^(3/2)) + ((4*I)*b^2*d^(3/2)*n^2*ArcTan[(Sqrt[d]*x^(1/3))/Sqrt[e]]^2)/e^(3/2) - (8*b^2*d^(3/2)*n^2*ArcTan[(Sq
rt[d]*x^(1/3))/Sqrt[e]]*Log[2 - (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2) + (4*b*n*(a + b*Log[c*(d +
e/x^(2/3))^n]))/(3*x) - (4*b*d*n*(a + b*Log[c*(d + e/x^(2/3))^n]))/(e*x^(1/3)) - (4*b*d^(3/2)*n*ArcTan[(Sqrt[
d]*x^(1/3))/Sqrt[e]]*(a + b*Log[c*(d + e/x^(2/3))^n]))/e^(3/2) - (a + b*Log[c*(d + e/x^(2/3))^n])^2/x + ((4*I)
*b^2*d^(3/2)*n^2*PolyLog[2, -1 + (2*Sqrt[e])/(Sqrt[e] - I*Sqrt[d]*x^(1/3))])/e^(3/2)

Rule 2458

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*(x_)^(m_.), x_Symbol] :> With[{k = Denomina
tor[n]}, Dist[k, Subst[Int[x^(k*(m + 1) - 1)*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x], x, x^(1/k)], x]] /; FreeQ
[{a, b, c, d, e, m, p, q}, x] && FractionQ[n]

Rule 2457

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

Rule 2476

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

Rule 2455

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

Rule 263

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rule 325

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a*
c*(m + 1)), x] - Dist[(b*(m + n*(p + 1) + 1))/(a*c^n*(m + 1)), Int[(c*x)^(m + n)*(a + b*x^n)^p, x], x] /; Free
Q[{a, b, c, p}, x] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, n, m, p, x]

Rule 205

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

Rule 2470

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))/((f_) + (g_.)*(x_)^2), x_Symbol] :> With[{u = In
tHide[1/(f + g*x^2), x]}, Simp[u*(a + b*Log[c*(d + e*x^n)^p]), x] - Dist[b*e*n*p, Int[(u*x^(n - 1))/(d + e*x^n
), x], x]] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && IntegerQ[n]

Rule 12

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

Rule 260

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

Rule 6688

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

Rule 4924

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

Rule 4868

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

Rule 2447

Int[Log[u_]*(Pq_)^(m_.), x_Symbol] :> With[{C = FullSimplify[(Pq^m*(1 - u))/D[u, x]]}, Simp[C*PolyLog[2, 1 - u
], x] /; FreeQ[C, x]] /; IntegerQ[m] && PolyQ[Pq, x] && RationalFunctionQ[u, x] && LeQ[RationalFunctionExponen
ts[u, x][[2]], Expon[Pq, x]]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x^2} \, dx &=3 \operatorname{Subst}\left (\int \frac{\left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )^2}{x^4} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-(4 b e n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-(4 b e n) \operatorname{Subst}\left (\int \left (\frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e x^4}-\frac{d \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 x^2}+\frac{d^2 \left (a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )\right )}{e^2 \left (e+d x^2\right )}\right ) \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-(4 b n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^4} \, dx,x,\sqrt [3]{x}\right )+\frac{(4 b d n) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (4 b d^2 n\right ) \operatorname{Subst}\left (\int \frac{a+b \log \left (c \left (d+\frac{e}{x^2}\right )^n\right )}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}\\ &=\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^4} \, dx,x,\sqrt [3]{x}\right )-\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\sqrt{d} \sqrt{e} \left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )+\frac{1}{3} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{\left (d+\frac{e}{x^2}\right ) x^6} \, dx,x,\sqrt [3]{x}\right )\\ &=\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )-\frac{\left (8 b^2 d^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{\left (d+\frac{e}{x^2}\right ) x^3} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}+\frac{1}{3} \left (8 b^2 e n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^4 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )\\ &=-\frac{8 b^2 n^2}{9 x}+\frac{8 b^2 d n^2}{e \sqrt [3]{x}}+\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{1}{3} \left (8 b^2 d n^2\right ) \operatorname{Subst}\left (\int \frac{1}{x^2 \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{e}-\frac{\left (8 b^2 d^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (e+d x^2\right )} \, dx,x,\sqrt [3]{x}\right )}{\sqrt{e}}\\ &=-\frac{8 b^2 n^2}{9 x}+\frac{32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac{8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{e^{3/2}}+\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}+\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}-\frac{\left (8 i b^2 d^{3/2} n^2\right ) \operatorname{Subst}\left (\int \frac{\tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{e}}\right )}{x \left (i+\frac{\sqrt{d} x}{\sqrt{e}}\right )} \, dx,x,\sqrt [3]{x}\right )}{e^{3/2}}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{1}{e+d x^2} \, dx,x,\sqrt [3]{x}\right )}{3 e}\\ &=-\frac{8 b^2 n^2}{9 x}+\frac{32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}+\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac{\left (8 b^2 d^2 n^2\right ) \operatorname{Subst}\left (\int \frac{\log \left (2-\frac{2}{1-\frac{i \sqrt{d} x}{\sqrt{e}}}\right )}{1+\frac{d x^2}{e}} \, dx,x,\sqrt [3]{x}\right )}{e^2}\\ &=-\frac{8 b^2 n^2}{9 x}+\frac{32 b^2 d n^2}{3 e \sqrt [3]{x}}+\frac{32 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )}{3 e^{3/2}}+\frac{4 i b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right )^2}{e^{3/2}}-\frac{8 b^2 d^{3/2} n^2 \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \log \left (2-\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}+\frac{4 b n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{3 x}-\frac{4 b d n \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e \sqrt [3]{x}}-\frac{4 b d^{3/2} n \tan ^{-1}\left (\frac{\sqrt{d} \sqrt [3]{x}}{\sqrt{e}}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )}{e^{3/2}}-\frac{\left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{x}+\frac{4 i b^2 d^{3/2} n^2 \text{Li}_2\left (-1+\frac{2 \sqrt{e}}{\sqrt{e}-i \sqrt{d} \sqrt [3]{x}}\right )}{e^{3/2}}\\ \end{align*}

Mathematica [A]  time = 1.2665, size = 598, normalized size = 1.66 \[ \frac{\frac{b n \left (9 b (-d)^{3/2} n x \left (2 \text{PolyLog}\left (2,\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \text{PolyLog}\left (2,\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )+\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (\log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right )+2 \log \left (\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )-4 \log \left (-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )+9 b d \sqrt{-d} n x \left (-4 \text{PolyLog}\left (2,1-\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )+2 \text{PolyLog}\left (2,\frac{1}{2}-\frac{\sqrt{-d} \sqrt [3]{x}}{2 \sqrt{e}}\right )+\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (\log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right )+2 \log \left (\frac{1}{2} \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}+1\right )\right )-4 \log \left (\frac{\sqrt{-d} \sqrt [3]{x}}{\sqrt{e}}\right )\right )\right )+12 e^{3/2} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+18 (-d)^{3/2} x \log \left (\sqrt{e}-\sqrt{-d} \sqrt [3]{x}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )+18 d \sqrt{-d} x \log \left (\sqrt{-d} \sqrt [3]{x}+\sqrt{e}\right ) \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-36 d \sqrt{e} x^{2/3} \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )-8 b n \left (3 d^{3/2} x \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )+\sqrt{e} \left (e-3 d x^{2/3}\right )\right )-72 b d^{3/2} n x \tan ^{-1}\left (\frac{\sqrt{e}}{\sqrt{d} \sqrt [3]{x}}\right )+72 b d \sqrt{e} n x^{2/3}\right )}{e^{3/2}}-9 \left (a+b \log \left (c \left (d+\frac{e}{x^{2/3}}\right )^n\right )\right )^2}{9 x} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Log[c*(d + e/x^(2/3))^n])^2/x^2,x]

[Out]

(-9*(a + b*Log[c*(d + e/x^(2/3))^n])^2 + (b*n*(72*b*d*Sqrt[e]*n*x^(2/3) - 72*b*d^(3/2)*n*x*ArcTan[Sqrt[e]/(Sqr
t[d]*x^(1/3))] - 8*b*n*(Sqrt[e]*(e - 3*d*x^(2/3)) + 3*d^(3/2)*x*ArcTan[Sqrt[e]/(Sqrt[d]*x^(1/3))]) + 12*e^(3/2
)*(a + b*Log[c*(d + e/x^(2/3))^n]) - 36*d*Sqrt[e]*x^(2/3)*(a + b*Log[c*(d + e/x^(2/3))^n]) + 18*(-d)^(3/2)*x*(
a + b*Log[c*(d + e/x^(2/3))^n])*Log[Sqrt[e] - Sqrt[-d]*x^(1/3)] + 18*Sqrt[-d]*d*x*(a + b*Log[c*(d + e/x^(2/3))
^n])*Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 9*b*Sqrt[-d]*d*n*x*(Log[Sqrt[e] - Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] - Sqrt
[-d]*x^(1/3)] + 2*Log[(1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*Log[(Sqrt[-d]*x^(1/3))/Sqrt[e]]) - 4*PolyLog[2,
1 - (Sqrt[-d]*x^(1/3))/Sqrt[e]] + 2*PolyLog[2, 1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])]) + 9*b*(-d)^(3/2)*n*x*(Lo
g[Sqrt[e] + Sqrt[-d]*x^(1/3)]*(Log[Sqrt[e] + Sqrt[-d]*x^(1/3)] + 2*Log[1/2 - (Sqrt[-d]*x^(1/3))/(2*Sqrt[e])] -
4*Log[-((Sqrt[-d]*x^(1/3))/Sqrt[e])]) + 2*PolyLog[2, (1 + (Sqrt[-d]*x^(1/3))/Sqrt[e])/2] - 4*PolyLog[2, 1 + (
Sqrt[-d]*x^(1/3))/Sqrt[e]])))/e^(3/2))/(9*x)

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Maple [F]  time = 0.36, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{{x}^{2}} \left ( a+b\ln \left ( c \left ( d+{e{x}^{-{\frac{2}{3}}}} \right ) ^{n} \right ) \right ) ^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*ln(c*(d+e/x^(2/3))^n))^2/x^2,x)

[Out]

int((a+b*ln(c*(d+e/x^(2/3))^n))^2/x^2,x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^2/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right )^{2} + 2 \, a b \log \left (c \left (\frac{d x + e x^{\frac{1}{3}}}{x}\right )^{n}\right ) + a^{2}}{x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^2/x^2,x, algorithm="fricas")

[Out]

integral((b^2*log(c*((d*x + e*x^(1/3))/x)^n)^2 + 2*a*b*log(c*((d*x + e*x^(1/3))/x)^n) + a^2)/x^2, x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*ln(c*(d+e/x**(2/3))**n))**2/x**2,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (b \log \left (c{\left (d + \frac{e}{x^{\frac{2}{3}}}\right )}^{n}\right ) + a\right )}^{2}}{x^{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*log(c*(d+e/x^(2/3))^n))^2/x^2,x, algorithm="giac")

[Out]

integrate((b*log(c*(d + e/x^(2/3))^n) + a)^2/x^2, x)