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

Optimal. Leaf size=509 \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}} \]

[Out]

(x*(a + b*Log[c*x^n])^2)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b*Log[c*x^n])^2)/(4*(-d)^(3/2)*(Sqrt[
-d] + Sqrt[e]*x)) + (b*n*(a + b*Log[c*x^n])*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - ((a + b*Lo
g[c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (b*n*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x
)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + ((a + b*Log[c*x^n])^2*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[
e]) - (b^2*n^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b*n*(a + b*Log[c*x^n])*PolyLog[2
, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b^2*n^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*
Sqrt[e]) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (b^2*n^2*PolyLog
[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b^2*n^2*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2
)*Sqrt[e])

________________________________________________________________________________________

Rubi [A]  time = 0.611818, antiderivative size = 509, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 6, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273, Rules used = {2330, 2318, 2317, 2391, 2374, 6589} \[ \frac{b n \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (2,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{PolyLog}\left (3,-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(a + b*Log[c*x^n])^2)/(4*(-d)^(3/2)*(Sqrt[-d] - Sqrt[e]*x)) + (x*(a + b*Log[c*x^n])^2)/(4*(-d)^(3/2)*(Sqrt[
-d] + Sqrt[e]*x)) + (b*n*(a + b*Log[c*x^n])*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - ((a + b*Lo
g[c*x^n])^2*Log[1 - (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[e]) - (b*n*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x
)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) + ((a + b*Log[c*x^n])^2*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(4*(-d)^(3/2)*Sqrt[
e]) - (b^2*n^2*PolyLog[2, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b*n*(a + b*Log[c*x^n])*PolyLog[2
, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b^2*n^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*
Sqrt[e]) - (b*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2)*Sqrt[e]) - (b^2*n^2*PolyLog
[3, -((Sqrt[e]*x)/Sqrt[-d])])/(2*(-d)^(3/2)*Sqrt[e]) + (b^2*n^2*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(2*(-d)^(3/2
)*Sqrt[e])

Rule 2330

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

Rule 2318

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

Rule 2317

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/((d_) + (e_.)*(x_)), x_Symbol] :> Simp[(Log[1 + (e*x)/d]*(a +
b*Log[c*x^n])^p)/e, x] - Dist[(b*n*p)/e, Int[(Log[1 + (e*x)/d]*(a + b*Log[c*x^n])^(p - 1))/x, x], x] /; FreeQ[
{a, b, c, d, e, n}, x] && IGtQ[p, 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 2374

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

Rule 6589

Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_Symbol] :> Simp[PolyLog[n + 1, c*(a
+ b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d, e, n, p}, x] && EqQ[b*d, a*e]

Rubi steps

\begin{align*} \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (d+e x^2\right )^2} \, dx &=\int \left (-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{4 d \left (\sqrt{-d} \sqrt{e}-e x\right )^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{4 d \left (\sqrt{-d} \sqrt{e}+e x\right )^2}-\frac{e \left (a+b \log \left (c x^n\right )\right )^2}{2 d \left (-d e-e^2 x^2\right )}\right ) \, dx\\ &=-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt{-d} \sqrt{e}-e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\left (\sqrt{-d} \sqrt{e}+e x\right )^2} \, dx}{4 d}-\frac{e \int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{-d e-e^2 x^2} \, dx}{2 d}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}-\frac{e \int \left (-\frac{\sqrt{-d} \left (a+b \log \left (c x^n\right )\right )^2}{2 d e \left (\sqrt{-d}-\sqrt{e} x\right )}-\frac{\sqrt{-d} \left (a+b \log \left (c x^n\right )\right )^2}{2 d e \left (\sqrt{-d}+\sqrt{e} x\right )}\right ) \, dx}{2 d}-\frac{\left (b \sqrt{e} n\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{-d} \sqrt{e}-e x} \, dx}{2 (-d)^{3/2}}-\frac{\left (b \sqrt{e} n\right ) \int \frac{a+b \log \left (c x^n\right )}{\sqrt{-d} \sqrt{e}+e x} \, dx}{2 (-d)^{3/2}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d}-\sqrt{e} x} \, dx}{4 (-d)^{3/2}}+\frac{\int \frac{\left (a+b \log \left (c x^n\right )\right )^2}{\sqrt{-d}+\sqrt{e} x} \, dx}{4 (-d)^{3/2}}-\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (b^2 n^2\right ) \int \frac{\log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}-\frac{(b n) \int \frac{\left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (b^2 n^2\right ) \int \frac{\text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (b^2 n^2\right ) \int \frac{\text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{x} \, dx}{2 (-d)^{3/2} \sqrt{e}}\\ &=\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{x \left (a+b \log \left (c x^n\right )\right )^2}{4 (-d)^{3/2} \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{\left (a+b \log \left (c x^n\right )\right )^2 \log \left (1+\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{4 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b n \left (a+b \log \left (c x^n\right )\right ) \text{Li}_2\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}-\frac{b^2 n^2 \text{Li}_3\left (-\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}+\frac{b^2 n^2 \text{Li}_3\left (\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{2 (-d)^{3/2} \sqrt{e}}\\ \end{align*}

Mathematica [A]  time = 0.775516, size = 432, normalized size = 0.85 \[ \frac{-\frac{2 b n \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{2 b n \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{2 b^2 n^2 \text{PolyLog}\left (2,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{3/2}}-\frac{2 b^2 n^2 \text{PolyLog}\left (2,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}+\frac{2 b^2 n^2 \text{PolyLog}\left (3,\frac{\sqrt{e} x}{\sqrt{-d}}\right )}{(-d)^{3/2}}-\frac{2 b^2 n^2 \text{PolyLog}\left (3,\frac{d \sqrt{e} x}{(-d)^{3/2}}\right )}{(-d)^{3/2}}-\frac{2 b n \log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}+\frac{2 b n \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )}{(-d)^{3/2}}-\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d \left (\sqrt{-d}-\sqrt{e} x\right )}+\frac{\left (a+b \log \left (c x^n\right )\right )^2}{d \left (\sqrt{-d}+\sqrt{e} x\right )}+\frac{\log \left (\frac{\sqrt{e} x}{\sqrt{-d}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{(-d)^{3/2}}+\frac{d \log \left (\frac{d \sqrt{e} x}{(-d)^{3/2}}+1\right ) \left (a+b \log \left (c x^n\right )\right )^2}{(-d)^{5/2}}}{4 \sqrt{e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-((a + b*Log[c*x^n])^2/(d*(Sqrt[-d] - Sqrt[e]*x))) + (a + b*Log[c*x^n])^2/(d*(Sqrt[-d] + Sqrt[e]*x)) - (2*b*n
*(a + b*Log[c*x^n])*Log[1 + (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3/2) + ((a + b*Log[c*x^n])^2*Log[1 + (Sqrt[e]*x)/Sqrt
[-d]])/(-d)^(3/2) + (2*b*n*(a + b*Log[c*x^n])*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(3/2) + (d*(a + b*Log[c*
x^n])^2*Log[1 + (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(5/2) + (2*b^2*n^2*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3/2
) - (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3/2) - (2*b^2*n^2*PolyLog[2, (d*Sqrt[e]*
x)/(-d)^(3/2)])/(-d)^(3/2) + (2*b*n*(a + b*Log[c*x^n])*PolyLog[2, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(3/2) + (2*b
^2*n^2*PolyLog[3, (Sqrt[e]*x)/Sqrt[-d]])/(-d)^(3/2) - (2*b^2*n^2*PolyLog[3, (d*Sqrt[e]*x)/(-d)^(3/2)])/(-d)^(3
/2))/(4*Sqrt[e])

________________________________________________________________________________________

Maple [F]  time = 4.638, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) ^{2}}{ \left ( e{x}^{2}+d \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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*x^n))^2/(e*x^2+d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b^{2} \log \left (c x^{n}\right )^{2} + 2 \, a b \log \left (c x^{n}\right ) + a^{2}}{e^{2} x^{4} + 2 \, d e x^{2} + d^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b^2*log(c*x^n)^2 + 2*a*b*log(c*x^n) + a^2)/(e^2*x^4 + 2*d*e*x^2 + d^2), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*log(c*x**n))**2/(d + e*x**2)**2, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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